Non-Euclidean Laguerre geometry and incircular nets
Alexander I. Bobenko, Carl O. R. Lutz, Helmut Pottmann, Jan Techter
NNon-Euclidean Laguerre geometry and incircular nets
Alexander I. Bobenko , Carl O. R. Lutz , Helmut Pottmann , Jan Techter Institut f¨ur Mathematik, TU Berlin,Str. des 17. Juni 136, 10623 Berlin, Germany Visual Computing Center, KAUST,Thuwal 23955-6900, Saudi ArabiaSeptember 3, 2020
Abstract.
Classical (Euclidean) Laguerre geometry studies oriented hyperplanes, oriented hy-perspheres, and their oriented contact in Euclidean space. We describe how this can be general-ized to arbitrary Cayley-Klein spaces, in particular hyperbolic and elliptic space, and study thecorresponding groups of Laguerre transformations. We give an introduction to Lie geometry anddescribe how these Laguerre geometries can be obtained as subgeometries. As an application oftwo-dimensional Lie and Laguerre geometry we study the properties of incircular nets.
Acknowledgement.
This research was supported by the DFG Collaborative Research CenterTRR 109 “Discretization in Geometry and Dynamics”. H. Pottmann’s participation in thisprogram has been supported through grant I2978 of the Austrian Science Fund. We would liketo thank Oliver Gross and Nina Smeenk for their assistance in creating the figures.1 a r X i v : . [ m a t h . M G ] S e p ontents S n . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 477.2 Laguerre geometry from Lie geometry . . . . . . . . . . . . . . . . . . . . . . . . 507.3 Subgeometries of Lie geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 Euclidean cases 83
A.1 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.2 Euclidean geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83A.3 Euclidean geometry from M¨obius geometry . . . . . . . . . . . . . . . . . . . . . 85A.4 Euclidean Laguerre geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.5 Lie geometry in Euclidean space . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
B Generalized signed inversive distance 90
B.1 Invariant on a quadric induced by a point . . . . . . . . . . . . . . . . . . . . . . 90B.2 Signed inversive distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.3 Geometric interpretation for sphere complexes . . . . . . . . . . . . . . . . . . . . 933
Introduction
The discovery of non-Euclidean geometry by Lobachevsky, Bolyai and Gauss was a revolutionwhich might be compared with the discovery of the spherical form of the Earth. It turned outthat there exist other geometric worlds with points, straight lines and circles, and they havenatural geometric properties generalizing the ones of classical Euclidean geometry. The latteris recovered in the limit when the curvature of the space goes to zero. Almost immediatelyafter the invention of hyperbolic geometry Lobachevsky and Gauss posed the question aboutthe real geometry of our world and even tried to measure it experimentally. This played acrucial role in the further development of geometry and physics. Indeed, in the hyperbolicspace conventional Euclidean translations and rotations are replaced by the group of hyperbolicisometric transformations. In the three-dimensional case this group coincides with the Lorentzgroup of our space-time, which is central in Einstein’s special theory of relativity.Felix Klein in his
Erlangen program of 1872 [Kle1893] revolutionized the point of view ongeometry by declaring the transformation group as the conceptually central notion. The tradi-tional view is that geometry studies the space around us. Due to Klein, geometry is the studyof invariants under a group of transformations. This was the organizing principle which broughtan order into various facts accumulated in geometry, or rather, into different geometries thathad been discovered.Various transformation groups naturally lead to various geometries including projective,affine, spherical, hyperbolic, M¨obius, Lie, Pl¨ucker, and Laguerre geometries. Many beautifulresults were obtained during the classical period of the theory. A good presentation can befound in the books by Wilhelm Blaschke [Bla1923, Bla1929], which is probably the most com-prehensive source of knowledge of the corresponding geometries. Unfortunately till now thesebooks exist only in German.Modern revival of the interest in classical geometries and their recent development is inmuch extent due to the possibility of their investigation by computational methods. Computersenable experimental and numerical investigations of geometries as well as their visualization.Classical geometries became visible! Also physics contributed with more and more involvedtransformation groups and problems.Last but not least are applications in computer graphics, geometry processing, architecturalgeometry and even computer simulation of dynamics and other physical processes. M¨obiusgeometry is probably the most popular geometry in this context. For numerous applications ofclassical geometries we refer in particular to [BS2008, PW2001].This small book is on a rather “exotic” geometry called non-Euclidean Laguerre geometry .Euclidean Laguerre geometry, M¨obius geometry and Lie geometry belong to its close environmentand also appear in this book. Before we come to precise mathematical explanations let us givea rough idea of these geometries in the plane. The basic geometric objects in these geometriesare points, straight lines and circles. Whereas M¨obius geometry is dealing with points andcircles and has no notion of a straight line, Laguerre geometry is the geometry of circles andstraight lines and has no notion of a point. Incidences in M¨obius geometry, like points lieon circles, in Laguerre geometry correspond to the tangency condition between circles andstraight lines (more precisely, oriented circles and lines which are in oriented contact). In thenon-Euclidean case, straight lines are replaced by geodesics (see Figure 1). Generalizations ofLaguerre geometry to non-Euclidean space have already been studied by Beck [Bec1910], Graf[Gra1934, Gra1937, Gra1939] and Fladt [Fla1956, Fla1957], mainly in dimension 2.Classically, (Euclidean)
Laguerre geometry is the geometry of oriented hyperplanes, orientedhyperspheres, and their oriented contact in Euclidean space [Lag1885]. It is named after Laguerre[HPR1898], and was actively studied in dimensions 2 and 3 in the early twentieth century, see,e.g., [Bla1910, Bla1929]. In [Ben1973] and [Yag1968] the relation between Laguerre geometryand projective planes over commutative rings, e.g. dual numbers, is investigated.More recently, Laguerre geometry has been employed in specific applications, most notably4 igure 1.
Euclidean and elliptic checkerboard incircular nets as instances of Euclidean and non-Euclidean (spherical) Laguerre geometries. Straight lines and circles are tangent and can be orientedso that their orientations coincide at the points of tangency (oriented contact). The “straight lines”,or geodesics, on the sphere are great circles. in connection with offsets. These are curves or surfaces which lie at constant normal distance toeach other and have various applications in Computer-Aided Design and Manufacturing (see e.g.[Far2008]). Viewing a curve or surface as a set of oriented tangents or tangent planes, respec-tively, the offsetting operation is a special Laguerre transformation and thus Laguerre geometryis a natural geometry for the study of offsets. Examples of its use include the determination ofall families of offsets that are rational algebraic and therefore possess exact representations inNURBS-based 3D modeling systems [Far2008, PP1998a, PP1998b]. Discrete versions of offsetsurfaces play an important role in discrete differential geometry in connection with the definitionof discrete curvatures [BPW2010] and in architectural geometry [PL*2007].The knowledge of Laguerre geometry as a counterpart to the more familiar M¨obius geometryis a useful tool in research. It allows one to study sphere geometric concepts within both ofthese two geometries, which may open up new applications. An example for that is furnishedby circular meshes , a M¨obius geometric concept, and conical meshes , their Laguerre geometriccounterparts [BS2007, BS2008, PW2008]. Both of them are discrete versions of curvature lineparameterizations of surfaces, but have different properties in view of applications. It turnedout that conical meshes are preferable for the realization of architectural freeform structures.The main reason is an offset property which facilitates the design and fabrication of supportingbeam layouts [PL*2007]. Even more remarkable is the fact that the supporting structures withthe cleanest node geometry are based on so-called edge offset meshes and are also of a Laguerregeometric nature [PGB2010]. Quadrilateral structures of this type impose a shape restriction.They are discrete versions of Laguerre isothermic surfaces [Bla1929, BS2006], a special caseof which are Laguerre minimal surfaces [Bla1929, PGM2009, PGB2010, SPG2012]. The “dual”viewpoints of M¨obius and Laguerre geometry also led to different discretizations and applicationsof surface parameterizations which run symmetrically to the principal directions [PW*2020].The most comprehensive text on Laguerre geometry is the classical book by Blaschke [Bla1929],where however only the Euclidean case is treated. There exists no systematic presentation ofnon-Euclidean Laguerre geometry in the literature. The goal of the present book is twofold.On one hand, it is supposed to be a comprehensive presentation of non-Euclidean Laguerregeometry, and thus has the character of a textbook. On the other hand, Section 8 presents newresults. We demonstrate the power of Laguerre geometry on the example of checkerboard incir- ular nets introduced in [AB2018], give a unified treatment of these nets in all space forms, anddescribe them explicitly. Checkerboard incircular nets are Laguerre geometric generalizations ofincircular nets introduced by B¨ohm [B¨oh1970], which are defined as congruences of straight linesin the plane with the combinatorics of the square grid such that each elementary quadrilateraladmits an incircle. They are closely related to (discrete) confocal conics [BSST2016, BSST2018].The construction and geometry of incircular nets and their Laguerre geometric generalizationto checkerboard incircular nets have been discussed in great detail. Explicit parametrizationsfor the Euclidean cases were derived in [BST2018], while different higher dimensional analoguesof incircular nets were studied in [ABST2019] and [AB2018]. In this book we further generalizeplanar checkerboard incircular nets to Lie geometry, and show that these may be classified interms of checkerboard incircular nets in hyperbolic/elliptic/Euclidean Laguerre geometry. Weprove incidence theorems of Miquel type and show that all lines of a checkerboard incircular netare tangent to a hypercycle. This generalizes the results from [BST2018] and leads to a unifiedtreatment of checkerboard incircular nets in all space forms. Visualizations and geometric datafor checkerboard incircular nets can also be found at [DGDGallery].In Section 2 we begin our treatment of non-Euclidean Laguerre geometry by introducingelementary models for Laguerre geometry in the elliptic and hyperbolic plane. The intentionhere is to enable the reader to quickly get a glimpse of this geometry without reference to thefollowing more general discussions.In Section 6 we show how Laguerre geometry can be obtained in a unified way for an arbitrary Cayley-Klein space of any dimension. In the spirit of Klein’s Erlangen program this is done in apurely projective setup for which we introduce the foundations on quadrics (Sections 3), Cayley-Klein spaces (Section 4), and central projections (Section 5) [Kle1928, Bla1954, Gie1982]. For aCayley-Klein space
K ⊂ R P n the space of hyperplanes is lifted to a quadric B ⊂ R P n +1 , which wecall the Laguerre quadric. Vice versa, the projection from the Laguerre quadric yields a doublecover of the space of K -hyperplanes which can be interpreted as carrying their orientation. Inthe projection hyperplanar sections of B correspond to spheres of the Cayley-Klein space K . Thecorresponding group of quadric preserving transformations, which maps hyperplanar sections of B to hyperplanar sections of B , naturally induces the group of transformations of oriented K -hyperplanes, which preserves the oriented contact to Cayley-Klein spheres. We explicitly carryout this general construction in the cases of hyperbolic and elliptic geometry, yielding hyperbolicLaguerre geometry and elliptic Laguerre geometry, respectively. The (classical) Euclidean caseconstitutes a degenerate case of this construction, which we treat in Appendix A. In AppendixB we treat an invariant of two points on a quadric, which is closely related to the Cayley-Kleindistance, and of which the classical inversive distance introduced Coxeter [Cox1971] turns outto be a special case.In Section 7 we show how the different Laguerre geometries appear as subgeometries ofLie geometry. Lie (sphere) geometry is the geometry of oriented hyperspheres of the n -sphere S n ⊂ R n +1 , and their oriented contact [Bla1929, Cec1992, BS2008]. Laguerre geometry can beobtained by distinguishing the set of “oriented hyperplanes” as a sphere complex among the setof oriented hyperspheres, the so called plane complex. Classically, the plane complex is taken tobe parabolic, which leads to the notion of Euclidean Laguerre geometry, where elements of theplane complex are interpreted as oriented hyperplanes of Euclidean space. Choosing an ellipticor hyperbolic sphere complex, on the other hand, allows for the interpretation of the elementsof the plane complex as oriented hyperplanes in hyperbolic or elliptic space, respectively. Thegroup of Lie transformations that preserve the set of “oriented hyperplanes” covers the group ofLaguerre transformations. 6 Two-dimensional non-Euclidean Laguerre geometry
In this section we present Laguerre geometry in the elliptic and hyperbolic plane in an elementaryway, without reference to the following more general discussions. The intention here is to enablethe reader to quickly get a glimpse of this geometry without diving into the details.In Laguerre geometry in these planes, we consider oriented lines and oriented circles as thebasic objects and orientated contact (tangency) as the basic relation between them. A pointin the elliptic or hyperbolic plane is considered as an oriented circle, being in contact with alloriented lines passing through it.A
Laguerre transformation is bijective in the sets of oriented lines and oriented circles,respectively, and preserves oriented contact. It is important to note that in general Laguerretransformations do not preserve points. Points are special oriented circles and are thereforemapped to oriented circles.We will now present elementary quadric models of Laguerre geometry in the elliptic andhyperbolic plane, in which oriented lines are represented by points of a quadric in projective 3-space and oriented circles appear as the planar sections of that quadric. In this quadric model,Laguerre transformations are seen as projective transformations which map the quadric ontoitself.We first discuss the simpler case of elliptic Laguerre geometry and then proceed towards hyperbolic Laguerre geometry . We use the sphere model of the elliptic plane E . Points of E are seen as pairs of antipodal pointsof the unit sphere S ⊂ R . Oriented lines of E appear as oriented great circles in S and orientedcircles in E correspond to oriented circles (different from great circles) in S . S ‘ G ( c ) c x Figure 2.
The unit sphere S as a model of elliptic Laguerre geometry. An oriented line ‘ isrepresented by the spherical center x of a great circle. An oriented circle c is represented by aplanar section G ( c ), which is composed of the spherical centers of the great circles in orientedcontact with the circle. Oriented lines
An oriented great circle ‘ ⊂ S defines two half-spheres, one of which lies onthe positive side, which shall be the left side when running on ‘ in terms of the orientation (seeFigure 2). We now represent an oriented great circle ‘ by its spherical center x . This is theintersection point of the circle’s rotational axis with S which lies on the positive side of ‘ .7 riented circles Let us now consider all oriented lines ‘ of E which are tangent to an orientedcircle. In the sphere model, this yields all oriented great circles of S which are tangent to anoriented circle c ⊂ S . By rotational symmetry, their centers x form a circle G ( c ). The set ofcenters G ( c ) does not degenerate to a point as c is not a great circle, while it is a great circle if c is a point. Laguerre transformations
It now becomes clear how to realize Laguerre transformations ofthe elliptic plane in the sphere model: These are projective transformations that map the sphere S to itself. Planar sections G ( c ) ⊂ S (oriented circles) are mapped to planar sections, butin general great circles are not mapped to great circles. This expresses the fact that Laguerretransformations map oriented circles to oriented circles, but do in general not preserve points.In M¨obius geometry these projective automorphisms of S also appear as M¨obius transforma-tions, but the meaning of points is different. Summarizing, we can say that the elliptic Laguerregroup is isomorphic to the (Euclidean) M¨obius group. Both appear as projective automorphismsof the sphere, but in elliptic Laguerre geometry, points of the sphere have the meaning of centersof great circles, representing oriented straight lines of E . We employ the projective model of the hyperbolic plane H with an absolute circle S . Recallthat points of H are points inside the circle S , with the points of S playing the role of points atinfinity (ideal points). Straight lines ‘ ⊂ H are seen as straight line segments bounded by twoideal points m − and m + . The line obtains an orientation by traversing it from m − to m + (seeFigure 3).To obtain a quadric model for Laguerre geometry in H , we view the absolute circle S x + y = 1as the smallest circle on the rotational hyperboloid B hyp x + y − z = 1 , which lies in its symmetry plane z = 0. This hyperboloid carries two families of straight lines(rulings). Two rulings are obtained by intersecting B hyp with the tangent plane x = 1, yielding z = ± y . By rotation about the z -axis, the line x = 1 , z = y generates the family of rulings R + ,and likewise x = 1 , z = − y generates the rulings R − . Through each point of B hyp there passesexactly one line of R + and one line of R − . Oriented lines
We now lift an oriented straight line ‘ of H to a point on the quadric B hyp as follows (see Figure 3): We intersect the ruling of R + through m − with the ruling of R − through m + , yielding a point x ∈ B hyp . The plane spanned by R − and R + is the tangent planeof B hyp at x and this tangent plane intersects the base plane z = 0 in the line ‘ . Changing theorientation of ‘ to ‘ yields a point x which is the reflection of x in the base plane z = 0. Theconnecting line x ∧ x is the polar line of ‘ with respect to the quadric B hyp , and the orthogonalprojection of x , respectively x , onto the base plane z = 0 is the pole of ‘ with respect to theabsolute circle S .Parallel oriented straight lines share an ideal point and thus correspond to points which lieon the same ruling of B hyp . Oriented circles
For the oriented circles of H the situation is slightly more complicatedsince different types of circles come into play (see Figures 5 and 6). The first three types (seeFigure 5 (a,b,c)) arise from generalized hyperbolic circles. Those appear as special conics c inthe projective model. For visualization, it is probably easiest to employ the sphere model of8 hyp xx ‘ H S m + m − Figure 3.
A rotational hyperboloid B hyp as a model of hyperbolic Laguerre geometry. It containsthe absolute circle S of the projective model of the hyperbolic plane H at z = 0. An oriented line ‘ is represented by the intersection x of two rulings of B hyp through the two ideal points m − and m + .The tangent plane of B hyp at the point x intersects the hyperbolic plane in the line ‘ . A differentchoice of rulings yields the point x which represents the same line with opposite orientation. hyperbolic geometry and view the conic c as the orthogonal projection of a circle on the sphere x + y + z = 1 onto the plane z = 0. Such a conic c together with the absolute circle S spansa pencil which contains a doubly counted line L . The conic c and the absolute circle S touchin two points on the line L . These may be two different real points as in Figure 5 (c), in whichcase c is a curve of constant distance to the hyperbolic line given by L . But they can also betwo complex conjugate points as in Figure 5 (a), in which case c is a hyperbolic circle with itscenter being the pole of the line L with respect to the absolute circle S . Lastly, the two pointsmay coincide as in Figure 5 (b), which corresponds to the case of a horocycle, represented bythe conic c which has third order contact with S . Hyperbolic Laguerre geometry, however, alsoconsiders circles as in Figure 5 (d), whose tangent lines correspond to real straight lines in H ,but the envelope lies in the deSitter space (outsite S ). B hyp Γ G ( c ) SH cc ⊥ Figure 4.
Lifting an oriented hyperbolic circle c to the Laguerre quadric B hyp . The polar conic c ⊥ of c with respect to S is a deSitter circle. The quadratic cylinder Γ over c ⊥ intersects the hyperboloid B hyp in two conics. The planar section G ( c ) represents the circle c in its given orientation. B hyp G ( c ) H S c (b) L (c)(d) Figure 5.
Oriented circles in hyperbolic Laguerre geometry. On the Laguerre quadric B hyp ori-ented circles are represented by planar sections, or, by polarity, the pole of the corresponding planewith respect to B hyp . The first three types arise from generalized hyperbolic circles: (a) ordinaryhyperbolic circle, (b) horocycle, (c) curve of constant distance to a hyperbolic line. The fourth typeis a deSitter circle with hyperbolic tangent lines (d). Their envelope lies outside H (in the deSitterplane). B hyp G ( c ) SH (b)(c) Figure 6.
Points in hyperbolic Laguerre geometry. In Laguerre geometry, points are oriented circlesas well. Their images in the quadric model B hyp are sections with planes that are orthogonal to thebase plane of H . Every tangent line appears with both orientations. (a) An ordinary point in thehyperbolic plane H . (b) An ideal point on the absolute circle S . It defines two pencils of ”parallel”oriented lines. In the quadric model these pencils correspond to the rulings of B hyp . (c) A deSitterpoint. Only part of the lines through this point define hyperbolic lines. To transform these oriented hyperbolic circles c to the quadric model we apply polaritywith respect to B hyp . The tangents of c are mapped to the lines of a quadratic cylinder Γ(parallel to the z -axis, see Figure 4). Its intersection with B hyp is composed of two conics (inplanes symmetric to z = 0). This follows from the tangency of Γ and B hyp in the ideal points c ∩ S = L ∩ S . One of the two planar sections, G ( c ), represents the oriented circle c in its givenorientation; the other, symmetric section corresponds to the reverse orientation. The orthogonalprojection c ⊥ of such a planar section is polar to c with respect to S and thus a conic whichtouches S in the same points as c does. However, this conic c ⊥ is not a hyperbolic circle, but adeSitter circle (outside S ).Let us now turn to points which also define Laguerre circles and are presented in Figure 6. Inthe quadric model these circles correspond to sections with planes parallel to the z -axis (passing11hrough the polar line of the point c with respect to S ). Note that beside hyperbolic points(Figure 6 (a)) there is a further type of point-like circles, shown in Figure 6 (c), where thecommon point of tangents lies in the deSitter space, outside S . Figure 6 (b) illustrates the caseof an ideal point x ∈ S viewed as set of lines. This set can be oriented in two ways and suchdefines two pencils of parallel oriented lines. They are not considered as oriented Laguerre circlesand correspond to the intersection of B hyp with its tangent plane at x , which decomposes intotwo rulings. Each of the two pencils corresponds to a ruling of B hyp .Having discussed all these cases we can state that oriented Laguerre circles of the hyperbolicplane H correspond precisely to the planar sections of B hyp different from rulings. Laguerre transformations
Finally, having the quadric model at our disposal, we turn toLaguerre transformations. Laguerre transformations of the hyperbolic plane appear as projectivetransformations that map the hyperboloid B hyp to itself. Those are exactly the maps that actbijectively on the set of points and planar sections of the projectively extended quadric B hyp .Again we see that Laguerre transformations do not preserve the special circles whose en-velopes are points. Those belong to planar sections of B hyp in z -parallel planes and this specialproperty of a plane is in general not preserved under a projective automorphism of B hyp .A projective automorphism of B hyp maps rulings to rulings. Thus, hyperbolic Laguerretransformations preserve parallelity of oriented straight lines. The same is true in EuclideanLaguerre geometry but does not apply in the elliptic plane where there is no parallelism ofstraight lines. 12 Quadrics in projective space
We begin our general discussions with the introduction of quadrics in projective space, see, e.g.,[Kle1928, Bla1954, Gie1982].
Consider the n -dimensional real projective space R P n := P( R n +1 ) := (cid:16) R n +1 \ { } (cid:17) (cid:30) ∼ as it is generated via projectivization from its homogeneous coordinate space R n +1 using theequivalence relation x ∼ y ⇔ x = λy for some λ ∈ R . We denote points in R P n and its homogeneous coordinates by x = [ x ] = [ x , . . . , x n +1 ] . Affine coordinates are given by normalizing one homogeneous coordinate to be equal to one andthen dropping this coordinate, e.g., (cid:18) x x n +1 , . . . , x n x n +1 (cid:19) . Points with x n +1 = 0, for which this normalization is not possible, are said to lie on the hyperplane at infinity .The projectivization operator P acts on any subset of the homogeneous coordinate space. Inparticular, a projective subspace U ⊂ R P n is given by the projectivization of a linear subspace U ⊂ R n +1 , U = P( U ) , dim U = dim U − . To denote projective subspaces spanned by a given set of points a , . . . a k with linear independenthomogeneous coordinate vectors we use the exterior product a ∧ · · · ∧ a k := [ a ∧ · · · ∧ a k ] = P(span { a , . . . , a k } ) . The group of projective transformations is induced by the group of linear transformations of R n +1 and denoted by PGL( n + 1). A projective transformation maps projective subspaces toprojective subspaces, while preserving their dimension and incidences. The fundamental theoremof real projective geometry states that this property characterizes projective transformations. Theorem 3.1.
Let n ≥ , and W ⊂ R P n be a non-empty open subset. Let f : W → R P n be an injective map that maps intersections of k -dimensional projective subspaces with W tointersections of k -dimensional projective subspaces with f ( W ) for some ≤ k ≤ n − . Then f is the restriction of a unique projective transformation of R P n . For a projective subgroup G ⊂ PGL( n + 1) we denote the stabilizer of a finite number ofpoints v , . . . v m ∈ R P n by G v ,..., v m := { g ∈ G | g ( v i ) = v i , for i = 1 , . . . , m } . (1)13 .2 Quadrics Let h· , ·i be a non-zero symmetric bilinear form on R n +1 . A vector x ∈ R n +1 is called (cid:73) spacelike if h x, x i > (cid:73) timelike if h x, x i < (cid:73) lightlike , or isotropic , if h x, x i = 0.There always exists an orthogonal basis with respect to h· , ·i , i.e. a basis ( e i ) i =1 ,...,n +1 satisfying h e i , e j i = 0 if i = j . The triple ( r, s, t ), consisting of the numbers of spacelike ( r ), timelike ( s ),and lightlike ( t ) vectors in ( e i ) i =1 ,...,n +1 is called the signature of h· , ·i . It is invariant underlinear transformations. If t = 0, the bilinear form h· , ·i is called non-degenerate , in which casewe might omit its value in the signature. We alternatively write the signature in the form(+ · · · + | {z } r − · · · − | {z } s · · · | {z } t ) . The space R n +1 together with a bilinear form of signature ( r, s, t ) is denoted by R r,s,t . Thezero set of the quadratic form corresponding to h· , ·i L r,s,t := n x ∈ R n +1 (cid:12)(cid:12)(cid:12) h x, x i = 0 o is called the light cone , or isotropic cone . Its projectivization Q := P( L r,s,t ) = { x ∈ R P n | h x, x i = 0 } ⊂ R P n defines a quadric in R P n (quadrics in R P are called conics ).A point x ∈ Q contained in the kernel of the corresponding bilinear form h· , ·i , i.e. h x, y i = 0 for all y ∈ R n +1 is called a vertex of Q . A quadric is called non-degenerate if it contains no vertices, or equivalentlyif t = 0. If Q is degenerate, i.e. t >
0, its set of vertices is a projective subspace of dimension t − h· , ·i defines the same quadric Q . Vice versa, if Q is non-emptyand does not solely consist of vertices it uniquely determines its corresponding symmetric bilinearform up to a non-zero scalar multiple. Upon considering the complexification of real quadrics Q C := { x ∈ C P n | h x, x i = 0 } ⊂ C P n this correspondence holds in all cases, and it is convenient to generally identify the term quadricsand symmetric bilinear forms up to non-zero scalar multiples.The signature of a quadric is well-defined up to interchanging r and s . The signature of aprojective subspace U = P( U ) is defined by the signature of the bilinear form restricted to U .After a choice of the sign for the bilinear form of Q the signs for the signature of U are fixed.A quadric Q naturally defines two regions in the projective space R P n , Q + := { x ∈ R P n | h x, x i > } , Q − := { x ∈ R P n | h x, x i < } , (2)which we call the two sides of the quadric. Which side is “+” and which side is “-” is onlydetermined after choosing the sign for the bilinear form of Q .A projective subspace entirely contained in the quadric Q is called an isotropic subspace .A quadric with signature ( r, s, t ) contains isotropic subspaces of dimension min { r, s } + t − R P n with different signatures.14 xample 3.1. (i) A quadric with signature ( n + 1 ,
0) is empty in R P n . By either identifying the quadricwith its bilinear form up to non-zero scalar multiples or by complexification Q C ⊂ C P n ,we consider this to be an admissible non-degenerate quadric, which only happens to havean empty real part. Note that one side of the quadric Q + = R P n is the whole space, whilethe other side Q − = ∅ is empty.(ii) A quadric with signature ( n,
1) is an “oval quadric”. It is projectively equivalent to the( n − S n − .(iii) A quadric with signature ( n − ,
2) is a higher dimensional analogue of a doubly ruledquadric in R P . It contains lines as isotropic subspaces through every point, but noplanes.(iv) A quadric with signature ( r, s,
1) is a cone. It consists of all lines connecting its vertex toa non-degenerate quadric of signature ( r, s ), given by its intersection with a hyperplanenot containing the vertex. Note that if r = 0 or s = 0 (the real part of) the cone onlyconsists of the vertex. The remaining part of the cone can be considered as imaginary (cf.Example (i)).(v) A quadric with signature (1 , , n ) is a “doubly counted hyperplane”.For non-neutral signature, i.e. r = s , and rs = 0, the subgroup of projective transformationspreserving the quadric Q is exactly the projective orthogonal group PO( r, s, t ), i.e. the projec-tivization of all linear transformations that preserve the bilinear form h· , ·i . For simplicity, wecall PO( r, s, t ) the “group of transformations that preserve the quadric Q ” for all signatures. Remark . In the case r = s the statement remains true if we exclude projective transforma-tions that interchange the two sides (2) of the quadric. In the case rs = 0 the statement remainstrue upon complexification.The fundamental theorem of real projective geometry (see Theorem 3.1) may be specializedto quadrics. Theorem 3.2.
Let n ≥ , Q ⊂ R P n be a non-degenerate non-empty quadric in R P n , and W ⊂ Q be a non-empty open subset of the quadric. Let f : W → Q be an injective mapthat maps intersections of k -dimensional projective subspaces with W to intersections of k -dimensional projective subspaces with f ( W ) for some ≤ k ≤ n − . Then f is the restrictionof a unique projective transformation of R P n that preserves the quadric Q . For a non-degenerate quadric every such transformation can be decomposed into a finitenumber of reflections in hyperplanes by the theorem of Cartan and Dieudonn´e.
Theorem 3.3.
Let
Q ⊂ R P n be a non-degenerate quadric of signature ( r, s ) . Then each elementof the corresponding projective orthogonal group PO( r, s ) is the composition of at most n + 1 reflections in hyperplanes, i.e. transformations of the form σ q : R P n → R P n , [ x ] (cid:20) x − h x, q ih q, q i q (cid:21) for some q ∈ R P n \ Q . A quadric induces the notion of polarity between projective subspaces (see Figure 7). For aprojective subspace U = P( U ) ⊂ R P n , where U ⊂ R n +1 is a linear subspace, the polar subspace of U is defined as U ⊥ := { x ∈ R P n | h x, y i = 0 for all y ∈ U } . x ⊥ (+0)(+ − )(++) Q xx ⊥ (+ − )(++) (+0) Q Figure 7.
Polarity with respect to a conic Q with signature (+ + − ) in R P ( left ) and a quadricof signature (+ + + − ) in R P ( right ). The point x and its polar hyperplane x ⊥ are shown as wellas the cone of contact from the point x . Lines through x that are “inside” (signature (+ − )), “on”(signature (+0)), and “outside” (signature (++)) the cone intersect the quadric in 2, 1, or 0 pointsrespectively. If Q is non-degenerate, the dimensions of two polar subspaces satisfy the following relation:dim U + dim U ⊥ = n − . A refinement of this statement, which includes the signatures of the two polar subspaces, iscaptured in the following lemma.
Lemma 3.1.
Let
Q ⊂ R P n be a non-degenerate quadric of signature ( r, s ) . Then the signature (˜ r, ˜ s, ˜ t ) of a subspace U ⊂ R P n and the signature (˜ r ⊥ , ˜ s ⊥ , ˜ t ⊥ ) of its polar subspace U ⊥ withrespect to Q satisfy r = ˜ r + ˜ r ⊥ + ˜ t, s = ˜ s + ˜ s ⊥ + ˜ t, ˜ t = ˜ t ⊥ . In particular, ˜ t ≤ min { r, s } . For a point x ∈ Q on a quadric, which is not a vertex, the tangent hyperplane of Q at x is given by its polar hyperplane x ⊥ . If Q has signature ( r, s, t ) then the tangent plane hassignature ( r − , s − , t + 1). Furthermore, for a non-degenerate quadric a projective subspaceis tangent to Q if and only if its signature is degenerate.A projective line not contained in a quadric can intersect the quadric in either zero, one, ortwo points (see Figure 7). Lemma 3.2.
Let
Q ⊂ R P n be a quadric, x , y ∈ R P n , x = y be two points, and define ∆ := h x, y i − h x, x i h y, y i . (cid:73) If ∆ > , then the line x ∧ y has signature (+ − ) and intersects Q in two points x ± = h h y, y i x + (cid:16) − h x, y i ± √ ∆ (cid:17) y i . (cid:73) If ∆ < , then the line x ∧ y has signature (++) or ( −− ) and intersects Q in no real points,but in two complex conjugate points x ± = h h y, y i x + (cid:16) − h x, y i ± i √− ∆ (cid:17) y i . If ∆ = 0 , then the line x ∧ y has signature (+0) or ( − and it is tangent to Q in the point ˜ x = [ h y, y i x − h x, y i y ] , or it has signature (00) and is contained in Q (isotropic line). The last point of the preceding lemma gives rise to the following definition of the cone ofcontact (see Figure 7).
Definition 3.1.
Let
Q ⊂ R P n be a quadric with corresponding bilinear form h· , ·i , and x ∈ R P n \ Q . Define the quadratic form∆ x ( y ) := h x, y i − h x, x i h y, y i . Then the corresponding quadric C Q ( x ) := { y ∈ R P n | ∆ x ( y ) = 0 } is called the cone of contact , or tangent cone , to Q from the point x .The points of tangency of the cone of contact lie in the polar hyperplane of its vertex. Lemma 3.3.
Let
Q ⊂ R P n be a quadric. For a point x ∈ R P n \ Q the cone of contact to Q from x is given by C Q ( x ) = [ y ∈ x ⊥ ∩Q x ∧ y . Remark . For a non-degenerate quadric Q the intersection x ⊥ ∩ Q always results in a non-degenerate quadric in x ⊥ . If the restriction of the corresponding bilinear form has signature( n,
0) or (0 , n ) the intersection can be considered as imaginary. The real part of the cone onlyconsists of the vertex in this case (cf. Example 3.1 (iv)).
Let Q , Q ⊂ R P n be two distinct quadrics with corresponding bilinear forms h· , ·i , h· , ·i re-spectively. Every linear combination of these two bilinear forms yields a quadric. The familyof quadrics obtained by all linear combinations of the two bilinear forms is called a pencil ofquadrics (see Figure 8): Q ∧ Q := (cid:16) Q [ λ ,λ ] (cid:17) [ λ ,λ ] ∈ R P , Q [ λ ,λ ] := { x ∈ R P n | λ h x, x i + λ h x, x i = 0 } . This is a line in the projective space of quadrics of R P n .A pencil of quadrics is called non-degenerate if it does not consist exclusively of degeneratequadrics. It contains at most n + 1 degenerate quadrics.A point contained in the intersection of two quadrics from a pencil of quadrics is called a base point . It is then contained in every quadric of the pencil. The variety of base points has(at least) codimension 2. Example 3.2.
The pencil of quadrics
Q ∧ C Q ( x ) spanned by a non-degenerate quadric Q andthe cone of contact C Q ( x ) from a point x ∈ R P n \ Q contains as degenerate quadrics onlythe cone C Q ( x ) and the polar hyperplane x ⊥ . It is comprised of exactly the quadrics that aretangent to Q in Q ∩ x ⊥ . 17 Cayley-Klein spaces
In Klein’s
Erlangen program
Euclidean and non-Euclidean geometries are considered as subge-ometries of projective geometry. Projective models for, e.g., hyperbolic, deSitter, and ellipticspace can be obtained by using a quadric to induce the corresponding metric [Kle1928]. Inthis section we introduce the corresponding general notion of
Cayley-Klein spaces and theirgroups of isometries , see, e.g., [Kle1928, Bla1954, Gie1982]. We put a particular emphasis onthe description of hyperplanes, hyperspheres, and their mutual relations.
A quadric within a projective space induces an invariant for pairs of points.
Definition 4.1.
Let
Q ⊂ R P n be a quadric with corresponding bilinear form h· , ·i . Then wedenote by K Q ( x , y ) := h x, y i h x, x i h y, y i the Cayley-Klein distance of any two points x , y ∈ R P n \ Q that are not on the quadric.In the presence of a Cayley-Klein distance the quadric Q is called the absolute quadric . Remark . The name Cayley-Klein distance, or Cayley-Klein metric, is usually assigned toan actual metric derived from the above quantity as, for example, the hyperbolic metric (cf.Section 4.4). Nevertheless, we prefer to assign it to this basic quantity associated with anarbitrary quadric.The Cayley-Klein distance is projectively well-defined, in the sense that it depends neither onthe choice of the bilinear form corresponding to the quadric Q nor on the choice of homogeneouscoordinate vectors for the points x and y . Furthermore, it is invariant under the group ofprojective transformations that preserve the quadric Q , which we call the corresponding groupof isometries .The Cayley-Klein distance can be positive or negative depending on the relative location ofthe two points with respect to the quadric, cf. (2). Proposition 4.1.
For two points x , y ∈ R P n \ Q with h x, y i 6 = 0 : (cid:73) K Q ( x , y ) > if x and y are on the same side of Q , (cid:73) K Q ( x , y ) < if x and y are on opposite sides of Q . A Cayley-Klein space is usually considered to be one side of the quadric, i.e. Q + or Q − ,together with a (pseudo-)metric derived from the Cayley-Klein distance, or equivalently, togetherwith the transformation group of isometries. Having a notion of “distance” allows for the definition of corresponding spheres (see Figure 8).
Definition 4.2.
Let
Q ⊂ R P n be a quadric, x ∈ R P n \ Q , and µ ∈ R ∪ {∞} . Then we call theset S µ ( x ) := { y ∈ R P n | K Q ( x , y ) = µ } the Cayley-Klein hypersphere with center x and Cayley-Klein radius µ . Remark . (i) Due to the fact that the Cayley-Klein sphere equation can be written as h x, y i − µ h x, x i h y, y i = 0 , (3)we may include into the set S µ ( x ) points y ∈ Q on the quadric, and also allow for µ = ∞ (cf. Proposition 4.3). 18 igure 8. Concentric Cayley-Klein circles in the hyperbolic/deSitter plane.
Left:
Concentriccircles with hyperbolic center.
Middle:
Concentric circles with deSitter center.
Right:
Concentrichorocycles with center on the absolute conic. (ii) Given the center x of a Cayley-Klein sphere one can further rewrite the Cayley-Kleinsphere equation (3) as h x, y i − ˜ µ h y, y i = 0 , where ˜ µ := µ h x, x i . While ˜ µ is not projectively invariant anymore, the solution set of thisequation still invariantly describes a Cayley-Klein sphere. We may now allow for centerson the quadric x ∈ Q which gives rise to Cayley-Klein horospheres (see Figure 8, right).(iii) We will occasionally denote all these cases including Cayley-Klein horospheres by the term
Cayley-Klein spheres . Proposition 4.2.
For a Cayley-Klein sphere with center x ∈ R P n \ Q and Cayley-Klein radius µ one has: (cid:73) If µ < the center and the points of a Cayley-Klein sphere are on opposite sides of thequadric. (cid:73) If µ > the center and the points of a Cayley-Klein sphere are on the same side of thequadric. (cid:73) If µ = 0 the Cayley-Klein sphere is given by the (doubly counted) polar hyperplane x ⊥ . (cid:73) If µ = 1 the Cayley-Klein sphere is the cone of contact C Q ( x ) touching Q , which is alsocalled the null-sphere with center x . (cid:73) If µ = ∞ the Cayley-Klein sphere is the absolute quadric Q .Proof. Follows from Proposition 4.1 and Lemma 3.3.Fixing the center and varying the radius of a Cayley-Klein sphere results in a family ofconcentric spheres (see Figure 8).
Definition 4.3.
Given a quadric
Q ⊂ R P n and a point x ∈ R P n \ Q we call the family( S µ ( x )) µ ∈ R ∪{∞} concentric Cayley-Klein spheres with center x . Proposition 4.3.
Let
Q ⊂ R P n be a quadric. Then the family of concentric Cayley-Kleinspheres with center x ∈ R P n \ Q is the pencil of quadrics Q ∧ C Q ( x ) spanned by the absolutequadric Q and the cone of contact C Q ( x ) , or equivalently, by Q and the (doubly counted) polarhyperplane x ⊥ (cf. Example 3.2). y Q S µ ( x ) S µ ( x ) y z Q S ˜ µ ( x ) Figure 9.
Left:
Polarity with respect to a Cayley-Klein sphere S µ ( x ) and the absolute quadric Q . Right:
A Cayley-Klein sphere S µ ( x ) and its (concentric) polar Cayley-Klein sphere S ˜ µ ( x ). Proof.
Writing the Cayley-Klein sphere equation as (3) we find that it is a linear equation in µ describing a pencil of quadrics. As observed in Proposition 4.2 it contains, in particular, thequadric Q , the cone C Q ( x ), and the hyperplane x ⊥ .This leads to a further characterization of Cayley-Klein spheres among all quadrics. Corollary 4.1.
Let
Q ⊂ R P n be a non-degenerate quadric. Then another quadric is a Cayley-Klein sphere if and only if it is tangent to Q in the (possibly imaginary) intersection with ahyperplane.Proof. Follows from Proposition 4.3 and Example 3.2.
Remark . A pencil of concentric Cayley-Klein horospheres with center x ∈ Q is spannedby the absolute quadric Q and the (doubly counted) tangent hyperplane x ⊥ , which yields thirdorder contact between each horosphere and the absolute quadric. To describe spheres in terms of their tangent planes we turn our attention towards polarity inCayley-Klein spheres (see Figure 9).
Lemma 4.1.
The bilinear form corresponding to a Cayley-Klein sphere S µ ( x ) with center x ∈ R P n \ Q and Cayley-Klein radius µ is given by b ( y, ˜ y ) = h x, y i h x, ˜ y i − µ h x, x i h y, ˜ y i , y, ˜ y ∈ R n +1 . Thus, for a point y ∈ R P n the pole z with respect to the absolute quadric Q of the polarhyperplane of y with respect to S µ ( x ) is given by z = h x, y i x − µ h x, x i y. (4) Proof.
The quadratic form of the Cayley-Klein sphere S µ ( x ) is given by (3):∆( y ) := h x, y i − µ h x, x i h y, y i . The corresponding bilinear form can be obtained by b ( y, ˜ y ) = (∆( y + ˜ y ) − ∆( y ) − ∆(˜ y )).20or every point on a Cayley-Klein sphere the tangent hyperplane in that point is given bypolarity in the Cayley-Klein sphere. Now the tangent hyperplanes of a Cayley-Klein sphere, inturn, may equivalently be described by their poles with respect to the absolute quadric Q (seeFigure 9). Proposition 4.4.
Let x ∈ R P n \ Q and µ ∈ R \ { , } . Then the poles (with respect to theabsolute quadric Q ) of the tangent hyperplanes of the Cayley-Klein sphere S µ ( x ) are the pointsof a concentric Cayley-Klein sphere S ˜ µ ( x ) with µ + ˜ µ = 1 , and vice versa.Proof. Let y ∈ S µ ( x ) be a point on the Cayley-Klein sphere. Then the tangent plane to S µ ( x )at the point y is the polar plane of y with respect to S µ ( x ). According to Lemma 4.1 the pole z of that tangent plane is given by (4). Computing the Cayley-Klein distance of this point tothe center x we obtain K Q ( x , y ) = h x, z i h x, x i h z, z i = h x, y i (1 − µ ) h x, y i (1 − µ ) + µ h x, x i h y, y i = 1 − µ, where we used h x, y i = µ h x, x i h y, y i . Definition 4.4.
For a Cayley-Klein sphere S µ ( x ) we call the Cayley-Klein sphere S − µ ( x ),consisting of all poles (with respect to the absolute quadric Q ) of tangent planes of S µ ( x ), its polar Cayley-Klein sphere . Remark . The two degenerate Cayley-Klein spheres x ⊥ and C Q ( x ) corresponding to thevalues µ = 0 and µ = 1 respectively, may be treated as being mutually polar. Then polaritydefines a projective involution on a pencil of concentric Cayley-Klein spheres with fixed pointsat µ = and µ = ∞ . Let h· , ·i be the standard non-degenerate bilinear form of signature ( n, h x, y i := x y + . . . + x n y n − x n +1 y n +1 for x, y ∈ R n +1 , and denote by S ⊂ R P n the corresponding quadric. We identify the “inside” of S , cf. (2), with the n -dimensional hyperbolic space H := S − . For two points x , y ∈ H one has K S ( x , y ) ≥
1, and the quantity d given by K S ( x , y ) = cosh d ( x , y )defines a metric on H of constant negative sectional curvature. The corresponding group ofisometries is given by PO( n,
1) and called the group of hyperbolic motions . The absolute quadric S consists of the points at (metric) infinity. We call the union H := H ∪ S the compactified hyperbolic space . 21n this projective model of hyperbolic geometry geodesics are given by intersections of pro-jective lines in R P n with H , while, more generally, hyperbolic subspaces (totally geodesic sub-manifolds) are given by intersections of projective subspaces in R P n with H . Thus, by polarity,every point m ∈ dS in the “outside” of hyperbolic space,dS := S + , which is called deSitter space , corresponds to a hyperbolic hyperplane m ⊥ ∩ H .Consider two hyperbolic hyperplanes with poles k , m ∈ dS. (cid:73) If K S ( k , m ) <
1, the two hyperplanes intersect in H , and their hyperbolic intersectionangle α , or equivalently its conjugate angle π − α is given by K S ( k , m ) = cos α ( k ⊥ , m ⊥ ) . (cid:73) If K S ( k , m ) >
1, the two hyperplanes do not intersect in H , and their hyperbolic distanceis given by K S ( k , m ) = cosh d ( k ⊥ , m ⊥ ) . The corresponding projective hyperplanes intersect in ( k ∧ m ) ⊥ ⊂ dS. (cid:73) If K S ( k , m ) = 0, the two hyperplanes are parallel, i.e., they intersect on S .Finally, the hyperbolic distance of a point x ∈ H and a hyperbolic hyperplane with pole m ∈ dSis given by K S ( x , m ) = − sinh d ( x , m ⊥ ) . It is occasionally useful to employ a certain normalization of the homogeneous coordinatevectors: H n := n x = ( x , . . . , x n +1 ) ∈ R n, (cid:12)(cid:12)(cid:12) h x, x i = − , x n +1 ≥ o , f d S n := n m = ( m , . . . , m n +1 ) ∈ R n, (cid:12)(cid:12)(cid:12) h m, m i = 1 o . Then P( H n ) = H is an embedding and P( f d S n ) = dS is a double cover. For x, y ∈ H n and k, m ∈ f d S n above distance formulas become h x, y i = − cosh d ( x , y ) , |h k, m i| = cos α ( k ⊥ , m ⊥ ) , if |h k, m i| ≤ |h k, m i| = cosh d ( k ⊥ , m ⊥ ) , if |h k, m i| ≥ |h x, m i| = sinh d ( x , m ⊥ ) . Remark . The double cover P( f d S n ) = dS of deSitter space can be used to encode the ori-entation of the corresponding polar hyperplanes, e.g., by endowing the hyperbolic hyperplanecorresponding to m ∈ f d S n with a normal vector in the direction of the hyperbolic halfspace onwhich the bilinear form with points x ∈ H n is positive: h x, m i >
0. Using the double coverto encode orientation one may omit the absolute value in h x, m i = cos d to obtain an orientedhyperbolic distance d between a point and an hyperbolic hyperplane. Similarly, one may omitthe absolute value in h k, m i = cos α which allows to distinguish the intersection angle α and itsconjugate angle π − α .We now turn our attention to the Cayley-Klein spheres of hyperbolic/deSitter geometry.First, consider a pencil of concentric Cayley-Klein spheres S µ ( x ) with center inside hyperbolicspace x ∈ H , x ∈ H n . Depending on the value of µ ∈ R ∪ {∞} we obtain the following types ofhyperbolic/deSitter spheres (see Figure 8, left): (cid:73) µ <
0: A deSitter sphere with hyperbolic center.22 igure 10.
Concentric Cayley-Klein circles in the elliptic plane. (cid:73) < µ < S µ ( x ) is empty. (cid:73) < µ < ∞ : A hyperbolic sphere with center x ∈ H and hyperbolic radius r = arcosh √ µ > S µ ( x ) = n y ∈ H (cid:12)(cid:12)(cid:12) K S ( x , y ) = cosh r o = P ( { y ∈ H n | h x, y i = − cosh r } ) . Second, consider a pencil of concentric Cayley-Klein spheres S µ ( m ) with center outside hyper-bolic space m ∈ dS, m ∈ f d S n (see Figure 8, middle): (cid:73) µ <
0: A hypersurface of constant hyperbolic distance r = arsinh √ µ > m ⊥ ∩ H : S µ ( m ) = n y ∈ H (cid:12)(cid:12)(cid:12) K S ( m , y ) = − sinh r o = P (cid:16)n y ∈ H N (cid:12)(cid:12)(cid:12) |h m, y i| = sinh r o(cid:17) . (cid:73) < µ <
1: A deSitter sphere tangent to S . All its tangent hyperplanes are hyperbolichyperplanes. (cid:73) < µ < ∞ : A deSitter sphere tangent to S with no hyperbolic tangent hyperplanes.Third, a pencil of concentric Cayley-Klein horospheres with center on the absolute quadric x ∈ S , x ∈ L n, consists of hyperbolic horospheres and deSitter horospheres (see Figure 8, right). For x, y ∈ R n +1 we denote by x · y := x y + . . . x n y n + x n +1 y n +1 the standard (positive definite) scalar product on R n +1 , i.e. the standard non-degenerate bilinearform of signature ( n + 1 , O ⊂ R P n is empty (or purely imaginary,cf. Example 3.1 (i)), as well as the set O − = ∅ , while E := O + = R P n is the whole projective space, which we identify with the n -dimensional elliptic space . For twopoints x , y ∈ E one always has 0 ≤ K O ( x , y ) ≤ d given by K O ( x , y ) = cos d ( x , y )23efines a metric on E of constant positive sectional curvature. The corresponding group ofisometries is given by PO( n + 1) and called the group of elliptic motions .In this projective model of elliptic geometry geodesics are given by projective lines, while,more generally, elliptic subspaces are given by projective subspaces. By polarity, there is aone-to-one correspondence of points x ∈ E in elliptic space and elliptic hyperplanes x ⊥ .Two hyperplanes in elliptic space always intersect. If x , y ∈ E are the poles of two elliptichyperplanes, then their intersection angle α , or equivalently its conjugate angle π − α is givenby K O ( x , y ) = cos α ( x ⊥ , y ⊥ ) . The distance of a point x ∈ R P n and an elliptic hyperplane with pole y ∈ R P n is given by K O ( x , y ) = sin α ( x , y ⊥ ) . One may normalize the homogeneous coordinate vectors of points in elliptic space to lie ona sphere: S n := n x ∈ R n +1 (cid:12)(cid:12)(cid:12) x · x = 1 o . Then P( S n ) = E is a double cover, where antipodal points of the sphere are identified. In thisnormalization elliptic planes correspond to great spheres of S n , and it turns out that ellipticgeometry is a double cover of spherical geometry . For x, y ∈ S n above distance formulas become | x · y | = cos d ( x , y ) , | x · y | = cos α ( x ⊥ , y ⊥ ) , | x · y | = sin d ( x , y ⊥ ) , Remark . The pole x ∈ E of an elliptic hyperplane x ⊥ has two lifts to the sphere, x, − x ∈ S n ,which may be used to encode the orientation of the hyperplane (cf. Remark 4.5). This allowsfor omitting the absolute values in above distance formulas, while taking distances to be signedand distinguishing between intersection angles and their conjugate angles.A Cayley-Klein sphere in elliptic space S µ ( x ) with center x ∈ E , x ∈ S n , is not empty if andonly if 0 ≤ µ ≤ elliptic sphere with center x ∈ E and elliptic radius 0 ≤ r = arccos √ µ ≤ π : S µ ( x ) = n y ∈ E (cid:12)(cid:12)(cid:12) K O ( x , y ) = cos r o = P ( { y ∈ S n | | x · y | = cos r } ) . ⊥ q x π q ( x ) σ q ( x ) Q q x π q ( x ) σ q ( x ) q ⊥ Q Figure 11.
The involution and projection of an oval quadric
Q ⊂ R P induced by a point q noton the quadric. Left:
The point q lies “outside” the quadric. Right:
The point q lies “inside” thequadric. In this section we study the general construction of central projection of a quadric from a pointonto its polar hyperplane, see, e.g., [Kle1928, Bla1954, Gie1982]. This leads to a double coverof a Cayley-Klein space in the hyperplane such that the spheres in that Cayley-Klein spacecorrespond to hyperplanar sections of the quadric. Vice versa, a Cayley-Klein space can belifted to a quadric in a projective space of one dimension higher, such that Cayley-Klein sphereslift to hyperplanar sections of the quadric. In this way, hyperbolic and elliptic geometry canbe lifted to
M¨obius geometry , and M¨obius geometry may be seen as the geometry of pointsand spheres of the hyperbolic or elliptic space, respectively. We demonstrate how the groupof
M¨obius transformations can be decomposed into the respective isometries and scalings alongconcentric spheres.
Let h· , ·i be a bilinear form on R n +2 of signature ( r, s, t ), and denote by Q ⊂ R P n +1 the corre-sponding quadric. We introduce the central projection of Q from a point q not on the quadriconto a hyperplane of R P n +1 which is canonically chosen to be the polar hyperplane of q . Definition 5.1.
A point q ∈ R P n +1 \ Q not on the quadric induces two maps σ q , π q : R P n +1 → R P n +1 σ q : [ x ] [ σ q ( x )] = (cid:20) x − h x, q ih q, q i q (cid:21) , π q : [ x ] [ π q ( x )] = (cid:20) x − h x, q ih q, q i q (cid:21) , which we call the associated involution and projection respectively. Remark . The involution σ q is also called reflection in the hyperplane q ⊥ (cf. Theorem 3.3).We summarize the main properties of this involution and projection in the following propo-sition. Proposition 5.1. (i) The map σ q is a projective involution that fixes q , i.e., σ q ∈ PO( r, s, t ) q , σ q ◦ σ q = id , t further fixes every point on the polar hyperplane q ⊥ .For every line through q that intersects the quadric Q the involution σ q interchanges thetwo intersection points, while for a line through q that touches the quadric Q it fixes thetouching point (cf. Lemma 3.2).(ii) The map π q is a projection onto q ⊥ ’ R P n . Its restriction onto the quadric π q (cid:12)(cid:12) Q : Q → π q ( Q ) is a double cover with branch locus Q ∩ q ⊥ .(iii) The involution and projection together satisfy π q ◦ σ q = π q . Vice versa, if two distinct points x , y ∈ R P n +1 project to the same point π q ( x ) = π q ( y ) ,then x = σ q ( y ) . This gives rise to a one-to-one correspondence of the projection and thequotient π q ( Q ) ’ Q (cid:30) σ q . Remark . The involution σ q and projection π q act in the same way as described in Propo-sition 5.1 on every quadric from the pencil Q ∧ C Q ( q ) spanned by Q and the cone of contact C Q ( q ) with vertex q (cf. Example 3.2).The intersection e Q := Q ∩ q ⊥ is a quadric of signature (cid:73) ( r − , s, t ) if h q, q i >
0, or (cid:73) ( r, s − , t ) if h q, q i < Q ⊂ R P n +1 from a point q ∈ R P n +1 \ Q onto its polar hyperplane q ⊥ is a double cover of the “inside” or the “outside”, cf. (2), of e Q = q ⊥ ∩ Q depending on thesignature of q . Proposition 5.2.
Let q ∈ R P n +1 \ Q . Then (cid:73) π q ( Q ) = e Q − ∪ e Q , if h q, q i > , (cid:73) π q ( Q ) = e Q + ∪ e Q , if h q, q i < ,Proof. Decompose the homogeneous coordinate vector of a point x ∈ Q into its projection onto q and q ⊥ x = αq + π q ( x ) , with some α ∈ R . Then 0 = h x, x i = α h q, q i + h π q ( x ) , π q ( x ) i and thus h π q ( x ) , π q ( x ) i = − α h q, q i ( < , if h q, q i ≥ > , if h q, q i ≤ . The following proposition shows how the Cayley-Klein distance induced by e Q for points inthe projection π q ( Q ) can be lifted to the points on Q .26 roposition 5.3. Let q ∈ R P n +1 \ Q and x , y ∈ Q . Then the Cayley-Klein distance withrespect to e Q of their projections π q ( x ) , π q ( y ) is given by K e Q ( π q ( x ) , π q ( y )) = (cid:18) − h x, y i h q, q ih x, q i h y, q i (cid:19) . (5) Proof.
We decompose the homogeneous coordinate vectors of x, y into their projections onto q and q ⊥ x = αq + π q ( x ) , y = βq + π q ( y )with some α, β ∈ R . Then,1 − h x, y i h q, q ih x, q i h y, q i = 1 − (cid:16) αβ h q, q i + h x, y i q (cid:17) h q, q i αβ h q, q i = − h π q ( x ) , π q ( y ) i αβ h q, q i . Now with 0 = h x, x i = α h q, q i + h π q ( x ) , π q ( x ) i , and the analogous equation for y we obtain h π q ( x ) , π q ( y ) i α β h q, q i = h π q ( x ) , π q ( y ) i h π q ( x ) , π q ( x ) i h π q ( y ) , π q ( y ) i . Remark . Omitting the square for the quantity on the right hand side of equation (5) leadsto a signed version of the lifted Cayley-Klein distance (see Appendix B).While the Cayley-Klein distance can, in general, be both positive or negative, the right handside of equation (5) is always positive. This corresponds to the fact that the projection of Q only always covers one side of e Q . Though having no real preimages the points on the other sideof e Q may be viewed as projections of certain imaginary points of Q (see Proposition B.4).The transformation group induced by PO( r, s, t ) q , cf. (1), onto q ⊥ is exactly the group ofprojective transformations PO(˜ r, ˜ s, ˜ t ) that preserve the quadric e Q . It is doubly covered byPO( r, s, t ) q and can be identified with the quotientPO(˜ r, ˜ s, ˜ t ) ’ PO( r, s, t ) q (cid:30) σ q . Note that PO( r, s, t ) q is the largest subgroup of PO( r, s, t ) admitting this quotient, i.e. thesubgroup of transformations that commute with σ q . From now on, let Q be a non-degenerate quadric of signature ( r, s ). Then each section of thequadric Q with a hyperplane can be identified with the pole of that hyperplane. Definition 5.2.
We call a non-empty intersection of the quadric Q with a hyperplane a Q -sphere , and identify it with the pole of the hyperplane. Thus, we call S := n x ∈ R P n +1 (cid:12)(cid:12)(cid:12) x ⊥ ∩ Q 6 = ∅ o the space of Q -spheres . Remark . (i) The intersection of Q with a tangent hyperplane only consists of one point, or a cone (seeExample 3.1 (iv)). To exclude these degenerate cases one might want to take S \ Q insteadas the “space of spheres”. 27 x x ⊥ ∩ Q π q ( x ) π S q ( x ) q ⊥ Q e Q qx x ⊥ ∩ Q π q ( x ) π S q ( x ) q ⊥ Q Figure 12.
The central projection of a hyperplanar section x ⊥ ∩ Q of a quadric Q ⊂ R P from apoint q . Its image is a Cayley-Klein sphere π S q ( x ) ⊂ π q ( Q ) with respect to the absolute quadric e Q .Its center is given by π q ( x ). The cone of contact can be used to distinguish the type of Cayley-Kleinsphere that is obtained in the projection. (ii) Depending on the signature of of the quadric Q only the following three cases can occur(w.l.o.g., r ≥ s ): (cid:73) S = ∅ if Q has signature ( n + 2 , (cid:73) S = Q + ∪ Q if Q has signature ( n + 1 , (cid:73) S = R P n +1 else.It turns out that every Q -sphere projects down to a Cayley-Klein sphere in π q ( Q ), wherethe type of sphere can be distinguished by the two sides of the cone of contact C Q ( q ). Denoteby ∆ q ( x ) = h x, q i − h x, x i h q, q i = − h q, q i h π q ( x ) , π q ( x ) i (6)the quadratic form of the cone of contact C Q ( q ) (see Definition 3.1). Proposition 5.4.
Consider the map π S q : x π q ( x ⊥ ∩ Q ) , for x ∈ S . Then for every x ∈ S the image π S q ( x ) is a Cayley-Klein sphere with points in π q ( Q ) (see Figure 12). (cid:73) For ∆ q ( x ) = 0 the image is a Cayley-Klein sphere with center π q ( x ) and Cayley-Kleinradius µ = h x, q i ∆ q ( x ) , i.e. π q ( x ⊥ ∩ Q ) = S µ ( π q ( x )) . • If ∆ q ( x ) > , then π q ( x ) ∈ π q ( Q ) \ e Q . If ∆ q ( x ) < , then π q ( x ) ∈ q ⊥ \ π q ( Q ) . (cid:73) For ∆ q ( x ) = 0 the image is a Cayley-Klein horosphere with center π q ( x ) ∈ e Q . (cid:73) For x ∈ S ∩ q ⊥ the image is a hyperplane in π q ( Q ) with pole x . (cid:73) For x ∈ Q the image is the cone of contact C e Q ( π q ( x )) ⊂ π q ( Q ) . (cid:73) For x = q the image is the absolute quadric e Q .Proof. We show the claim for points not on the cone of contact. Thus, let x ∈ S \ C Q ( q ), i.e.,∆ q ( x ) = 0. Let y ∈ x ⊥ ∩ Q be a point on the corresponding Q -sphere. Then we find for theprojections of their homogeneous coordinate vectors h π q ( x ) , π q ( y ) i = − h x, q i h y, q ih q, q i , h π q ( x ) , π q ( x ) i = − ∆ q ( x ) h q, q i , h π q ( y ) , π q ( y ) i = − h y, q i h q, q i , and thus K e Q ( π q ( x ) , π q ( y )) = h π q ( x ) , π q ( y ) i h π q ( x ) , π q ( x ) i h π q ( y ) , π q ( y ) i = h x, q i ∆ q ( x ) = µ. Therefore, π S q ( x ) is a Cayley-Klein sphere with center π q ( x ) and radius µ .We know that π q ( y ) ∈ π q ( Q ). Hence, according to Proposition 4.2, the sign of µ , which isequal to the sign of ∆ q ( x ), determines which side of e Q the center π q ( x ) lies on. Further we findthat, µ = 0 ⇔ x ∈ q ⊥ , and µ = 1 ⇔ x ∈ Q , which, again according to Proposition 4.2, corresponds to a hyperplane and the cone of contactrespectively.The map π S q covers the whole space of Cayley-Klein spheres with points in π q ( Q ). Proposition 5.5.
The map π S q constitutes a double cover of the set of Cayley-Klein spheres in π q ( Q ) with respect to e Q . Its ramification points are given by ( q ⊥ ∪ { q } ) ∩ S , and its coveringinvolution is σ q .Proof. We show that every Cayley-Klein sphere with points in π q ( Q ) possesses exactly twopreimages, which are interchanged by σ q , unless it is a hyperplane. The same is true for Cayley-Klein horospheres.Consider a Cayley-Klein sphere S µ ( e x ) with center e x ∈ q ⊥ \ e Q , Cayley-Klein radius µ ∈ R and points in π q ( Q ). Then, according to Proposition 5.4, a preimage x ∈ S , π S q ( x ) = S µ ( e x )must satisfy π q ( x ) = e x , i.e. x = e x + λq for some λ ∈ R , and h x, q i ∆ q ( x ) = µ, which is equivalent to λ = − µ h e x, e x ih q, q i . According to Lemma 5.1 we have − µ h e x, e x i h q,q i ≥ S µ ( e x ) ⊂ π q ( Q ), and thus x ± := e x ± s − µ h e x, e x ih q, q i q x y ˜ y S S ˜ y ˜ y Figure 13.
The Cayley-Klein distance with respect to Q corresponds to the Cayley-Klein intersec-tion angle in the central projection to π q ( Q ) (see Proposition 5.6). defines one or two (real) points x ± provided that x ± ∈ S .The two points are interchanged by the involution, σ q ( x ± ) = x ∓ , and we have x + = x − ⇔ µ = 0 , in which case x ± = e x ∈ q ⊥ .To see that x ⊥± ∩ Q 6 = ∅ , first assume µ = 0. We show that any point e y ∈ S µ ( e x ) onthe Cayley-Klein sphere, has (real) preimages y ± ∈ Q , i.e. π q ( y ± ) = e y , that lie in the polarhyperplane of x ± respectively. Indeed, the points y ± := ± h q, q i s − µ h e x, e x ih q, q i e y − h e x, e y i q satisfy h y ± , y ± i = − h q, q i (cid:16) µ h e x, e x i h e y, e y i − h e x, e y i (cid:17) = 0 , and h x ± , y ± i = ± h q, q i h e x, e y i s − µ h e x, e x ih q, q i ∓ h q, q i h e x, e y i s − µ h e x, e x ih q, q i = 0 . If µ = 0, then e x = x + = x − , and the whole line e y ∧ q lies in the polar hyperplane of e x .Since y ∈ π q ( Q ) the line e y ∧ q has two real intersection points with Q , which serve as preimagesfor e y . Lemma 5.1.
A Cayley-Klein sphere with center e x ∈ q ⊥ \ e Q and Cayley-Klein radius µ ∈ R haspoints in π q ( Q ) if and only if − µ h e x, e x ih q, q i ≥ . Proof.
Follows from Proposition 4.2 and Proposition 5.2.Thus, we have found that the lift of the Cayley-Klein space π q ( Q ) to the quadric Q leads toa linearization of the corresponding Cayley-Klein spheres, in the sense that they become planarsections of Q , which we represent by their polar points.30or two intersecting Cayley-Klein spheres we call the Cayley-Klein distance of the poles of thetwo tangent hyperplanes (with respect to the absolute quadric) their Cayley-Klein intersectionangle . It is independent of the chosen intersection point. The Cayley-Klein distance of twopoints in S describes exactly this Cayley-Klein intersection angle in the projection to π q ( Q )(see Figure 13). Proposition 5.6.
Let x , x ∈ S such that the corresponding Q -spheres intersect. Let y ∈ Q ∩ x ⊥ ∩ x ⊥ be a point in that intersection, and ˜ y := π q ( y ) its projection. Let S , S be the two projectedCayley-Klein spheres corresponding to x , x respectively S := π S q ( x ) , S := π S q ( x ) . Let ˜ y , ˜ y be the two poles of the tangent hyperplanes of S , S at ˜ y respectively. Then K Q ( x , x ) = K e Q ( ˜ y , ˜ y ) . Proof.
First, we express the Cayley-Klein distance K Q ( x , x ) in terms of the projected centers ˜ x := π q ( x ), ˜ x := π q ( x ) and the projected intersection point ˜ y . To this end, we write x = ˜ x + α q, x = ˜ x + α q, y = ˜ y + λq, for some α , α , λ ∈ R . From h y, y i = h x , y i = h x , y i = 0 we obtain λ = − h ˜ y, ˜ y ih q, q i , α λ = − h ˜ y, ˜ x ih q, q i , α λ = − h ˜ y, ˜ x ih q, q i , and therefore α α = − h ˜ y, ˜ x i h ˜ y, ˜ x ih ˜ y, ˜ y i h q, q i , ( α ) = − h ˜ y, ˜ x i h ˜ y, ˜ y i h q, q i , ( α ) = − h ˜ y, ˜ x i h ˜ y, ˜ y i h q, q i . Using this we find h x , x i = h ˜ x , ˜ x i− h ˜ y, ˜ x i h ˜ y, ˜ x ih ˜ y, ˜ y i , h x , x i = h ˜ x , ˜ x i− h ˜ y, ˜ x i h ˜ y, ˜ y i , h x , x i = h ˜ x , ˜ x i− h ˜ y, ˜ x i h ˜ y, ˜ y i , and thus K Q ( x , x ) = h x , x i h x , x i h x , x i = ( h ˜ x , ˜ x i h ˜ y, ˜ y i − h ˜ y, ˜ x i h ˜ y, ˜ x i ) (cid:16) h ˜ x , ˜ x i h ˜ y, ˜ y i − h ˜ y, ˜ x i (cid:17) (cid:16) h ˜ x , ˜ x i h ˜ y, ˜ y i − h ˜ y, ˜ x i (cid:17) . (7)Secondly, we express the right hand side K e Q ( ˜ y , ˜ y ) in terms of the same quantities. From(4) we know that the poles ˜ y , ˜ y of the tangent planes (with respect to e Q ) are given by˜ y = h ˜ x , ˜ y i ˜ x − µ h ˜ x , ˜ x i ˜ y, ˜ y = h ˜ x , ˜ y i ˜ x − µ h ˜ x , ˜ x i ˜ y, where µ = h ˜ x , ˜ y i h ˜ x , ˜ x i h ˜ y, ˜ y i , µ = h ˜ x , ˜ y i h ˜ x , ˜ x i h ˜ y, ˜ y i are the Cayley-Klein radii of S and S . From this we obtain h ˜ y , ˜ y i = h ˜ x , ˜ y i h ˜ x , ˜ y i (cid:18) h ˜ x , ˜ x i − h ˜ x , ˜ y i h ˜ x , ˜ y ih ˜ y, ˜ y i (cid:19) , h ˜ y , ˜ y i = h ˜ x , ˜ y i h ˜ x , ˜ x i − h ˜ x , ˜ y i h ˜ y, ˜ y i ! , h ˜ y , ˜ y i = h ˜ x , ˜ y i h ˜ x , ˜ x i − h ˜ x , ˜ y i h ˜ y, ˜ y i ! K e Q ( ˜ y , ˜ y ) = h ˜ y , ˜ y i h ˜ y , ˜ y i h ˜ y , ˜ y i leads to the same as in (7). Remark . (i) Starting with two intersecting Cayley-Klein spheres in π q ( Q ) the lifted Q -spheres must bechosen such that they intersect as well. Only then will the Cayley-Klein distance of thepoles of the lifted spheres recover the Cayley-Klein intersection angle.(ii) Every quadric comes with a naturally induced (pseudo-)conformal structure, see e.g.[Por1995]. The Cayley-Klein distance between the two points x , x ∈ S also coincideswith the angle measured in this conformal structure.As a corollary of Theorem 3.2 we can now characterize the (local) transformations of aCayley-Klein space π q ( Q ) that map hyperspheres to hyperspheres as the projective orthogonaltransformations in the lift to the quadric Q . Theorem 5.1.
Let n ≥ , Q ⊂ R P n +1 be a non-degenerate quadric, and q ∈ R P n +1 \ Q . Con-sider the Cayley-Klein space π q ( Q ) endowed with the Cayley-Klein metric induced by e Q = Q∩ q ⊥ .Let W ⊂ π q ( Q ) be a non-empty open subset, and f : W → π q ( Q ) be an injective map that mapsintersections of Cayley-Klein hyperspheres with W to intersections of Cayley-Klein hypersphereswith f ( W ) . Then f is the restriction of a projective transformation R P n +1 → R P n +1 that pre-serves the quadric Q .Proof. After lifting the open sets W and f ( W ) to Q the statement follows from Theorem 3.2. Remark . (i) If the transformation f is defined on the whole space π q ( Q ) its lift must fix the point q .Thus, in this case f must be an isometry of π q ( Q ).(ii) If n ≥ f of mapping hyperspheres to hyperspheres may be weakened to f being a conformal transformation, i.e. preserving Cayley-Klein angles between arbitraryhypersurfaces (generalized Liouville’s theorem, see [Por1995, Ben1992]).(iii) The group of projective transformations PO( r, s ) that preserve the quadric Q maps Q -spheres to Q -spheres. In the projection to π q ( Q ) it may be interpreted as the groupof transformations that map “oriented points” of π q ( Q ) to “oriented points” of π q ( Q ),while preserving Cayley-Klein spheres. It contains the subgroup PO( r, s ) q of isometries of π q ( Q ). The involution σ q plays the role of “orientation reversion”. For “hyperbolic M¨obiusgeometry” see, e.g. [Som1914], and for “oriented points” of the hyperbolic plane [Yag1968]. The transformation group PO( r, s ) contains the isometries of π q ( Q ), given by PO( r, s ) q . It turnsout that the only transformations additionally needed to generate the whole group PO( r, s ) are“scalings” along concentric spheres.In the lift to S pencils of concentric Cayley-Klein spheres in π q ( Q ) correspond to lines in S through q (cf. Proposition 5.4). Proposition 5.7.
The preimage under the map π S q of a family of concentric Cayley-Kleinspheres in π q ( Q ) with center e x ∈ q ⊥ is given by the line ‘ := ( e x ∧ q ) ∩ S .For every x ∈ ‘ the hyperplane x ⊥ that defines the Q -sphere by intersection with Q containsthe polar subspace ( e x ∧ q ) ⊥ . x T x , x Figure 14.
Scaling along a pencil of concentric Cayley-Klein spheres in the lift and in the projection.
Definition 5.3.
We call a line in S a pencil of Q -spheres , and a line in S containing the point q a pencil of concentric Q -spheres (with respect to q ) .For every pencil of Q -spheres there is a distinguished one-parameter family of projectiveorthogonal transformations that preserve the pencil and each hyperplane through the corre-sponding line (see Figure 14). Proposition 5.8.
Let x , x ∈ S with K Q ( x , x ) > . Then there is a unique transformation T x , x ∈ PO( r, s ) that maps x to x and preserves every hyperplane through the line x ∧ x .It satisfies ( T x , x ) − = T x , x . Definition 5.4.
For two points x , x ∈ S on a pencil of concentric Q -spheres with respect to q , i.e. q ∈ x ∧ x , we call the transformation T x , x ∈ PO( r, s ) a scaling along the pencil ofconcentric spheres x ∧ x . Remark . There are three types of scalings along concentric spheres depending on the signa-ture of the line x ∧ x .In the projection π S q to π q ( Q ) the line x ∧ x through q corresponds to a pencil of concentricCayley-Klein spheres. The transformation T x , x maps spheres of this pencil to spheres of thispencil.Every transformation from PO( r, s ) may be decomposed into a (lift of an) isometry of π q ( Q )and a scaling along a pencil of concentric spheres. Proposition 5.9.
Let f ∈ PO( r, s ) . Then f can be written as f = T q , x ◦ Φ = Ψ ◦ T y , q with x := f ( q ) , y := f − ( q ) and some Φ , Ψ ∈ PO( r, s ) q .Proof. We observe that T x , q ◦ f, f ◦ T q , y ∈ PO( r, s ) q . Remark . To generate all transformations of PO( r, s ) one may further restrict to (at most)three arbitrarily chosen one-parameter families of scalings (one of each type, cf. Remark 5.7).Then a transformation f ∈ PO( r, s ) can be written as f = Φ ◦ T ◦ Ψwhere Φ , Ψ ∈ PO( r, s ) q and T is exactly one of the three chosen scalings.33 igure 15. Hyperbolic geometry and its lift to M¨obius geometry.
Left:
Concentric hyperboliccircles.
Middle:
Constant distance curves to a common line.
Right:
Concentric horocycles withcenter on the absolute conic.
Let h· , ·i be the standard non-degenerate bilinear form of signature ( n + 1 , h x, y i := x y + . . . + x n +1 y n +1 − x n +2 y n +2 for x, y ∈ R n +2 , and denote by S ⊂ R P n +1 the corresponding quadric, which we call the M¨obiusquadric .The M¨obius quadric is projectively equivalent to the standard round sphere
S ’ S n ⊂ R n +1 .In this correspondence intersections of S with hyperplanes of R P n +1 , i.e. the S -spheres, areidentified with hyperspheres of S n (cf. Proposition 7.2). The corresponding transformationgroup Mob := PO( n + 1 , M¨obius transformations leaves the quadric S invariant and maps hyperplanes to hyperplanes.Thus M¨obius geometry may be seen as the geometry of points on S n in which hyperspheres aremapped to hyperspheres.The set of poles of hyperplanes that have non-empty intersections with the M¨obius quadric S , i.e., the space of S -spheres, is given by S = S + ∪ S . where S + is the “outside” of S . Remark . The Cayley-Klein metric on S that is induced by the M¨obius quadric S is calledthe inversive distance , see [Cox1971]. For two intersecting hyperspheres of S n it is equal tothe cosine of their intersection angle. For a signed version of this quantity see Section B.2.Comparing with Section 4.4 this same Cayley-Klein metric also induces ( n + 1)-dimensionalhyperbolic geometry on the “inside” S − of the M¨obius quadric, and ( n + 1)-dimensional deSittergeometry on the “outside” S + of the M¨obius quadric.Central projection of the ( n + 1)-dimensional M¨obius quadric from a point leads to a doublecover of n -dimensional hyperbolic/elliptic space. Given the M¨obius quadric
S ⊂ R P n choose a point q ∈ R P n +1 , h q, q i >
0, w.l.o.g. q := [ e n +1 ] = [0 , . . . , , . σ q : [ x , . . . , x n , x n +1 , x n +2 ] [ x , . . . , x n , − x n +1 , x n +2 ] ,π q : [ x , . . . , x n , x n +1 , x n +2 ] [ x , . . . , x n , , x n +2 ] . The quadric in the polar hyperplane of q e S = S ∩ q ⊥ has signature ( n, H = e S − can be identified with n -dimensional hyperbolic space(cf. Section 4.4), and the M¨obius quadric projects down to the compactified hyperbolic space H = π q ( S ) . According to Proposition 5.4, an S -sphere, which we identify with a point in S = S + ∪ S projects to the different types of generalized hyperbolic spheres in H (see Figure 15 and Table 1). Proposition 5.10.
Under the map π S q : x π q ( x ⊥ ∩ S ) a point x ∈ S = S + ∪ S (cid:73) with x ∈ S is mapped to a point π q ( x ) ∈ H , (cid:73) with x ∈ q ⊥ , i.e. x n +1 = 0 , is mapped to a hyperbolic hyperplane in H with pole x , (cid:73) with h π q ( x ) , π q ( x ) i < is mapped to a hyperbolic sphere in H with center π q ( x ) . In thenormalization h π q ( x ) , π q ( x ) i = − its hyperbolic radius is given by r ≥ , where cosh r = x n +1 , (cid:73) with h π q ( x ) , π q ( x ) i > is mapped to a hyperbolic surface of constant distance in H to ahyperbolic hyperplane with pole π q ( x ) , In the normalization h π q ( x ) , π q ( x ) i = 1 its hyperbolicdistance is given by r ≥ , where sinh r = x n +1 . (cid:73) with h π q ( x ) , π q ( x ) i = 0 is mapped to a hyperbolic horosphere .Proof. Compare Section 4.4 for the different types of possible Cayley-Klein spheres in hyperbolicspace. Following Proposition 5.4 they can be distinguished by the sign of the quadratic form∆ q ( x ), or, comparing with equation (6), by the sign of h π q ( x ) , π q ( x ) i . Furthermore, the centerof the Cayley-Klein sphere corresponding to x is given by π q ( x ), while its Cayley-Klein radiusis given by µ = h x, q i ∆ q ( x ) = − h x, q i h q, q i h π q ( x ) , π q ( x ) i = − x n +1 h π q ( x ) , π q ( x ) i . Remark . (i) The map π S q is a double cover of the set of generalized hyperbolic spheres, branching onthe subset of hyperbolic planes (see Proposition 5.5).(ii) The Cayley-Klein distance induced on S by S measures the Cayley-Klein angle betweenthe corresponding generalized hyperbolic spheres (see Proposition 5.6) if their lifts intersect(see Remark 5.5 (i)), and more generally their inversive distance (see Remark 5.9).(iii) In the projection to H M¨obius transformations map generalized hyperbolic spheres to gen-eralized hyperbolic spheres (see Remark 5.6 (iii)). Vice versa, every (local) transformationof the hyperbolic space that maps generalized hyperbolic spheres to generalized hyperbolicspheres is the restriction of a M¨obius transformation (see Theorem 5.1)35 igure 16.
Concentric circles in elliptic geometry and its lift to M¨obius geometry. (iv) Every M¨obius transformation can be decomposed into two hyperbolic isometries and ascaling along either a fixed pencil of concentric hyperbolic spheres, distance surfaces, orhorospheres (see Remark 5.8). hyperbolic geometry M¨obius geometry point y ∈ H , y = (ˆ y, y n +1 ) ∈ H n [ˆ y, ± , y n +1 ] ∈ S hyperplane with pole y ∈ dS, y = (ˆ y, y n +1 ) ∈ d S n [ˆ y, , y n +1 ] ∈ S + ∩ q ⊥ sphere with center y ∈ H , y = (ˆ y, y n +1 ) ∈ H n and radius r > y, ± cosh r, y n +1 ] ∈ S + ∩ C S ( q ) + surface of constant distance r > y ∈ dS, y = (ˆ y, y n +1 ) ∈ d S n [ˆ y, ± sinh r, y n +1 ] ∈ S + ∩ C S ( q ) − horosphere with center y ∈ e S , y = (ˆ y, y n +1 ) ∈ L n, [ˆ y, ± e r , y n +1 ] ∈ S + ∩ C S ( q ) Table 1.
The lifts of generalized hyperbolic spheres to M¨obius geometry.
Given the M¨obius quadric
S ⊂ R P n choose a point q ∈ R P n +1 , h q, q i <
0, w.l.o.g. q := [ e n +1 ] = [0 , . . . , , . The corresponding involution and projection take the form σ q : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , x n +1 , − x n +2 ] π q : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , x n +1 , q e S = S ∩ q ⊥
36s imaginary, and has signature ( n + 1 , E = e S + , which can be identified with n -dimensional elliptic space (cf. Section 4.5).According to Proposition 5.4 an S -sphere projects to an elliptic sphere in E (see Figure 16and Table 2). Proposition 5.11.
Under the map π S q : x π q ( x ⊥ ∩ S ) a point x ∈ S = S + ∪ S (cid:73) with x ∈ S is mapped to a point π q ( x ) ∈ E , (cid:73) with x ∈ q ⊥ , i.e. x n +2 = 0 , is mapped to an elliptic plane in E with pole x , (cid:73) with h π q ( x ) , π q ( x ) i > is mapped to an elliptic sphere in E with center π q ( x ) , In thenormalization h π q ( x ) , π q ( x ) i = 1 its elliptic radius is given by r ≥ , where cos r = x n +2 .It also has constant elliptic distance R ≥ , where sin R = x n +1 , to the polar hyperplane of π q ( x ) .Remark . (i) The map π S q is a double cover of the set of elliptic spheres, branching on the subset ofelliptic planes (see Proposition 5.5).(ii) The Cayley-Klein distance induced on S by S measures the Cayley-Klein angle betweenthe corresponding elliptic spheres (see Proposition 5.6) if their lifts intersect (see Re-mark 5.5 (i)), and more generally their inversive distance (see Remark 5.9).(iii) In the projection to E , M¨obius transformations map elliptic spheres to elliptic spheres(see Remark 5.6 (iii)). Vice versa, every (local) transformation of elliptic space thatmaps elliptic spheres to elliptic spheres is the restriction of a M¨obius transformation (seeTheorem 5.1).(iv) Every M¨obius transformation can be decomposed into two elliptic isometries and a scalingalong one fixed pencil of concentric elliptic spheres (see Remark 5.8). Remark . Upon the identification of the M¨obius quadric with the sphere
S ’ S n the groupof M¨obius transformations fixing the point q is the group of spherical transformations, yieldingspherical geometry, which is a double cover of elliptic geometry. elliptic geometry M¨obius geometry point y ∈ E , y ∈ S n [ y, ± ∈ S hyperplane with pole y ∈ E , y ∈ S n [ y, ∈ q ⊥ sphere with center y ∈ E , y ∈ S n and radius r > y, ± cos r ] ∈ S + Table 2.
The lifts of elliptic spheres to M¨obius geometry. p e Q π p ( x ) π ∗ p ( x ) e Q π S p ( x ) π S ∗ p ( x ) p x x ⊥ ∩ Q π p ( x ) Figure 17.
Left:
Polar projection of points on the quadric Q . Right:
Polar projection of Q -spheres. The primary objects in
M¨obius geometry are points on S , which yield a double cover of thepoints in hyperbolic/elliptic space, and spheres, which yield a double cover of the spheres inhyperbolic/elliptic space. The primary incidence between these objects is a point lying on asphere . Laguerre geometry is dual to M¨obius geometry in the sense that the primary objects arehyperplanes, and spheres, both being a double cover of the corresponding objects in hyper-bolic/elliptic space, while the primary incidence between these objects is a plane being tangentto a sphere .In this section we introduce the concept of polar projection of a quadric. Similar to thecentral projection of a quadric it yields a double cover of certain hyperplanes of a Cayley-Kleinspace. While the double cover of points in a space form in the case of M¨obius geometry maybe interpreted as “oriented points” (cf. Remark 5.6 (iii)), in the case of Laguerre geometry thisleads to the perhaps more intuitive notion of “oriented hyperplanes”.A decomposition of the corresponding groups of
Laguerre transformations into isometriesand scalings can be obtained in an analogous way to the decomposition of the M¨obius group.We discuss the cases of hyperbolic and elliptic Laguerre geometry in detail, including coor-dinate representations for the different geometric objects appearing in each case. A treatmentof the Euclidean case can be found in Appendix A.
Let
Q ⊂ R P n +1 be a quadric. We have seen that the projection π p of the quadric Q leads toa double cover of the points of π p ( Q ) ⊂ p ⊥ (cf. Proposition 5.1), i.e. the points “inside” or“outside” the quadric e Q = Q ∩ p ⊥ . Correspondingly, the Q -spheres yield a double cover of the Cayley-Klein spheres in π p ( Q ) (cf.Proposition 5.5). We now investigate the corresponding properties for polar hyperplanes andpolar Cayley-Klein spheres. Definition 6.1.
Let
Q ⊂ R P n +1 be a quadric and p ∈ R P n +1 \ Q . Then we call the map π ∗ p : x x ⊥ ∩ p ⊥ = ( x ∧ p ) ⊥ , x ∈ R P n +1 to the intersection of its polar hyperplane x ⊥ with p ⊥ , the polarprojection (associated with the point p ) .The projection π p and the polar projection π ∗ p map the same point to a point in p ⊥ and itspolar hyperplane respectively. Proposition 6.1.
For a point x ∈ R P n +1 its projection π p ( x ) ∈ p ⊥ is the pole of its polarprojection π ∗ p ( x ) ⊂ p ⊥ , where polarity in p ⊥ ’ R P n is taken with respect to e Q . If we restrict the polar projection π ∗ p to the quadric Q we obtain a map to the hyperplanesof p ⊥ , which are poles of image points of the projection π p (see Figure 17). This map leads toa double cover of the polar hyperplanes (cf. Proposition 5.1). Proposition 6.2.
The restriction of the polar projection onto the quadric π ∗ p (cid:12)(cid:12) Q is a double coverof the set of all hyperplanes that are polar to the points in π p ( Q ) with branch locus Q ∩ p ⊥ .Remark . The double cover can be interpreted as carrying the additional information of theorientation of these hyperplanes, where the involution σ p plays the role of orientation reversion .By polarity every point x ∈ S corresponds to a Q -sphere x ⊥ ∩ Q (see Definition 5.2). In theprojection to π p ( Q ) it becomes a Cayley-Klein sphere (see Proposition 5.4), which is obtainedfrom the point x by the map π S p : x π p ( x ⊥ ∩ Q )The polar projection π ∗ p of each point of a Q -sphere yields a tangent plane of the polar Cayley-Klein sphere of π S p ( x ) (see Definition 4.4). The points of the polar Cayley-Klein sphere aretherefore obtained by the map π S ∗ p : x C Q ( x ) ∩ p ⊥ , where C Q ( x ) is the cone of contact (see Definition 3.1) to Q with vertex x (see Figure 17). Proposition 6.3.
For x ∈ S the two Cayley-Klein spheres π S p ( x ) and π S ∗ p ( x ) are mutuallypolar Cayley-Klein spheres in p ⊥ with respect to e Q . This leads to a polar version of Proposition 5.5.
Proposition 6.4.
The map π S ∗ p constitutes a double cover of the set of Cayley-Klein sphereswhich are polar to Cayley-Klein spheres in π p ( Q ) with respect to e Q . Its ramification points aregiven by ( p ⊥ ∪ { p } ) ∩ S , and its covering involution is σ p .Remark . Following Remark 6.1 we may endow the Cayley-Klein spheres that are polar toCayley-Klein spheres in π p ( Q ) with an orientation by lifting them to planar sections of Q , i.e. Q -spheres. We call the planar section, or equivalently their oriented projections, Laguerre spheres(of π p ( Q ) ) . The involution σ p acts on Laguerre spheres as orientation reversion.The Cayley-Klein distance of two points in S describes the Cayley-Klein tangent distancebetween the two corresponding Cayley-Klein spheres in the projection to π p ( Q ). This is thepolar version of Proposition 5.6. Proposition 6.5.
Let x , x ∈ S such that the corresponding Q -spheres intersect. Let y ∈ Q ∩ x ⊥ ∩ x ⊥ be a point in that intersection, and ˜ y := π ∗ p ( y ) its polar projection. Let S , S be the two polarprojected Cayley-Klein spheres corresponding to x , x respectively S := π S ∗ p ( x ) , S := π S ∗ p ( x ) . Let ˜ y , ˜ y be the two tangent points of ˜ y to S , S respectively. Then K Q ( x , x ) = K e Q ( ˜ y , ˜ y ) . roof. Consider Proposition 5.6. By polarity, the intersection point of the spheres becomes acommon tangent hyperplane, and the intersection angle becomes the distance of the two tangentpoints.
Remark . Following Remark 6.1 and Remark 6.2, a common point in the lift of two (oriented)Laguerre spheres corresponds to a common oriented tangent hyperplane. Thus the Cayley-Kleindistance on S is the Cayley-Klein tangent distance between two (oriented) Laguerre spheres (cf.Remark 5.5 (i)). When projecting down from M¨obius geometry to hyperbolic geometry (cf. Section 5.5) we obtaina double cover of the points in hyperbolic space. Hyperbolic planes, on the other hand, arerepresented by points in deSitter space, or “outside” hyperbolic space, by polarity. Thus, toobtain hyperbolic Laguerre geometry, instead of the M¨obius quadric, we choose a quadric thatprojects to deSitter space.
Definition 6.2. (i) We call the quadric B hyp ⊂ R P n +1 corresponding to the standard bilinear form of signature ( n,
2) in R n +2 , i.e., h x, y i := x y + . . . + x n y n − x n +1 y n +1 − x n +2 y n +2 for x, y ∈ R n +2 , the hyperbolic Laguerre quadric .(ii) The corresponding transformation group Lag hyp := PO( n, hyperbolic Laguerre transformations .To recover hyperbolic space in the projection, choose a point p ∈ R P n +1 with h p, p i < p := [ e n +2 ] = [0 , . . . , , . The corresponding involution and projection take the form σ p : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , , x n +1 , − x n +2 ] ,π p : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , , x n +1 , . The quadric in the polar hyperplane of p e S := B hyp ∩ p ⊥ has signature ( n, H = e S − can be identified with n -dimensional hyperbolicspace, while its “outside” dS = e S + can be identified with n -dimensional deSitter space (cf. Sec-tion 4.4).Under the projection π p the hyperbolic Laguerre quadric projects down to the compactifieddeSitter space π p ( B hyp ) = dS = dS ∪ e S . Thus the polar projection π ∗ p of a point on B hyp yields a hyperbolic hyperplane, where the doublecover can be interpreted as encoding the orientation of that hyperplane (see Figure 3). Remark . The quadric B hyp is the projective version of the hyperboloid f d S n introduced inSection 4.4 as a double cover of deSitter space.40e call the hyperplanar sections of B hyp , i.e. the B hyp -spheres, hyperbolic Laguerre spheres .By polarity, we identify the space of hyperbolic Laguerre spheres with the whole space S = R P n +1 . (8) Remark . As we have done in Section 2, one might want to exclude oriented hyperplanes fromthe space of hyperbolic Laguerre spheres and thus take S = R P n +1 \ B hyp (cf. Remark 5.4 (i)).Under the polar projection π S ∗ p points in S are mapped to the spheres that are polar todeSitter spheres (see Table 3 and Figure 5). Theorem 6.1.
Under the map π S ∗ p : x C B hyp ( x ) ∩ p ⊥ a point x ∈ S = R P n +1 (cid:73) with x ∈ B hyp is mapped to a hyperbolic hyperplane with pole π p ( x ) ∈ dS , (cid:73) with h π p ( x ) , π p ( x ) i < is mapped to a hyperbolic sphere in H with center π p ( x ) . In thenormalization h π p ( x ) , π p ( x ) i = − its hyperbolic radius is given by r ≥ , where sinh r = x n +2 , (cid:73) with h π p ( x ) , π p ( x ) i = 0 is mapped to a hyperbolic horosphere . (cid:73) with h π p ( x ) , π p ( x ) i > and h x, x i < is mapped to a hyperbolic surface of con-stant distance in H to a hyperbolic hyperplane with pole π p ( x ) , In the normalization h π p ( x ) , π p ( x ) i = 1 its hyperbolic distance is given by r ≥ , where cosh r = x n +2 . (cid:73) with h π p ( x ) , π p ( x ) i > and h x, x i > is mapped to a deSitter sphere in dS with center π p ( x ) . In the normalization h π p ( x ) , π p ( x ) i = 1 its deSitter radius is given by r ≥ , where cos r = x n +2 .Remark . (i) The points x representing hyperbolic spheres/distance hypersurfaces/horospheres can bedistinguished from the points representing deSitter spheres by the first satisfying h x, x i < h x, x i >
0, i.e.lying “outside” the hyperbolic Laguerre quadric.(ii) The map π S ∗ p is a double cover of the spheres described in Theorem 6.1, branching on thesubset of hyperbolic points, and deSitter null-spheres (see Proposition 6.4). We interpretthe lift to carry the orientation of the hyperbolic Laguerre spheres. Upon the normalizationgiven in Theorem 6.1 the orientation is encoded in the sign of the x n +2 -component. Theinvolution σ p acts on the set of hyperbolic Laguerre spheres as orientation reversion.(iii) The Cayley-Klein distance induced on S by the hyperbolic Laguerre quadric B hyp mea-sures the Cayley-Klein tangent distance between the corresponding hyperbolic Laguerrespheres (see Proposition 6.5) if they possess a common oriented tangent hyperplane (seeRemark 6.3).(iv) Using the projection π p instead of the polar projection π ∗ p hyperbolic Laguerre geometrymay be interpreted as the “M¨obius geometry” of deSitter space (cf. Section 7.3).41 igure 18. Concentric circles in the hyperbolic plane.
Left:
Concentric hyperbolic circles.
Middle:
Curves of constant distance to a common line.
Right:
Concentric horocycles with center on theabsolute conic. hyperbolic geometry Laguerre geometry hyperplane with pole y ∈ dS, y ∈ d S n [ y, ± ∈ B hyp sphere with center y ∈ H , y ∈ H n and radius r > y, ± sinh r ] ∈ B − hyp ∩ C B hyp ( p ) − horosphere with center y ∈ e S [ y, ± e r ] ∈ B − hyp ∩ C B hyp ( p ) surface of constant distance r > y ∈ dS, y ∈ d S n [ y, ± cosh r ] ∈ B − hyp ∩ C B hyp ( p ) + deSitter sphere with center y ∈ dS, y ∈ d S n and deSitter radius r > y, ± cos r ] ∈ B +hyp Table 3.
Laguerre spheres in hyperbolic Laguerre geometry.
Every (local) transformation mapping (non-oriented) hyperbolic hyperplanes to hyperbolic hy-perplanes (not necessarily points to points) while preserving (tangency to) hyperbolic spherescan be lifted and extended to a hyperbolic Laguerre transformation (see Theorem 5.1).The hyperbolic Laguerre group
Lag hyp = PO( n, B hyp and maps planar sections of B hyp to planarsections of B hyp . Under the polar projection this means it maps oriented hyperplanes to orientedhyperplanes, or hyperbolic Laguerre spheres to hyperbolic Laguerre spheres, while preservingthe tangent distance and in particular the oriented contact (see Remark 6.3).The hyperbolic Laguerre group contains (doubly covers) the group of hyperbolic isometriesas PO( n, p . To generate the whole Laguerre group we only need to add three specific one-parameter families of scalings along concentric Laguerre spheres (see Remark 5.8 and Figure 18).42 Consider the family of transformations T (s) t := I n · · · t sin t · · · − sin t cos t for t ∈ [ − π/ , π/ . It maps the absolute p = [0 , . . . , ,
1] to T (s) t ( p ) = [0 , . . . , , sin t, cos t ] , which is a hyperbolic sphere with center [0 , . . . , , t = 0into a point for t = ± π/
2, while changing orientation when it passes through the center orthrough the absolute, i.e. when cos t · sin t changes sign. (cid:73) Consider the family of transformations T (c) t := I n − · · · t t · · · · · · t t for t ∈ R It maps the absolute p = [0 , . . . , ,
1] to T (c) t ( p ) = [0 , . . . , , sinh t, , cosh t ] , which is an oriented hypersurface of constant distance to the hyperbolic hyperplane [0 , . . . , , , , t = 0 into the hyperplane for t = ∞ , while changing orientationwhen it passes through the absolute, i.e. when t changes sign. (cid:73) Consider the family of transformations T (h) t := I n − · · · t t t · · · − t − t − t · · · t t for t ∈ R . It maps the absolute p = [0 , . . . , ,
1] to T (h) t ( s ) = [0 , . . . , , t, − t, , which is a horosphere with center [0 , . . . , , , − ,
0] on the absolute. It turns from theabsolute for t = 0 into the center for t = ∞ , while changing orientation when t changes itssign. Remark . While Laguerre transformations preserve oriented contact they do not preserve thenotion of sphere, horosphere and constant distance surface. For example the transformation T (s) π/ transforms the origin into the absolute and thus turns all spheres which contain the origininto horospheres. 43ow the hyperbolic Laguerre group can be generated by hyperbolic motions and the threeintroduced one-parameter families of scalings (see Remark 5.8). Theorem 6.2.
Every hyperbolic Laguerre transformation f ∈ PO( n, can be written as f = Φ T t Ψ , where Φ , Ψ ∈ PO( n, p are hyperbolic motions and T t ∈ { T (s) t , T (c) t , T (h) t } a scaling for some t ∈ R . When projecting down from M¨obius geometry to elliptic geometry (cf. Section 5.6) we obtaina double cover of the points in elliptic space. Since every elliptic hyperplane has a pole in theelliptic space, this equivalently leads to a double cover of the elliptic hyperplanes.
Definition 6.3. (i) We call the quadric B ell ⊂ R P n +1 corresponding to the standard bilinear form of signature ( n + 1 ,
1) in R n +2 , i.e., h x, y i := x y + . . . + x n +1 y n +1 − x n +2 y n +2 for x, y ∈ R n +2 the elliptic Laguerre quadric .(ii) The corresponding transformation group Lag ell := PO( n + 1 , ’ Mob is called the group of elliptic Laguerre transformations . Remark . Thus, n -dimensional elliptic Laguerre geometry is isomorphic to n -dimensionalM¨obius geometry.To recover elliptic space in the projection, choose a point p ∈ R P n +1 with h p, p i <
1, w.l.o.g., p := [ e n +2 ] = [0 , . . . , , . The corresponding involution and projection take the form σ p : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , , x n +1 , − x n +2 ] ,π p : [ x , . . . , x n +1 , x n +2 ] [ x , . . . , , x n , . The quadric in the polar hyperplane of p O = B ell ∩ p ⊥ has signature ( n + 1 , E = O + can be identified with n -dimensionalelliptic space (see Section 4.5)Under the projection π p the elliptic Laguerre quadric projects down to the elliptic space π p ( B ell ) = E Thus, the polar projection π ∗ p of every point on B ell yields an elliptic hyperplane, where thedouble cover can be interpreted as carrying the orientation of that hyperplane. Remark . The quadric B ell is the projective version of the sphere S n introduced in Section 4.5as a double cover of elliptic space. 44e call the hyperplanar sections of B ell , i.e. the B ell -spheres, elliptic Laguerre spheres . Bypolarity, we identify the space of hyperbolic Laguerre spheres with the the outside of B ell S = B +ell ∪ B ell . (9) Remark . As we have done in Section 2, one might want to exclude oriented hyperplanesfrom the space of elliptic Laguerre spheres and thus take S = B +ell (cf. Remark 5.4 (i)).Under the polar projection π S ∗ p points in S are mapped to spheres that are polar to ellipticspheres, i.e. they are mapped to all elliptic spheres (Table 4 and Figure 2). Theorem 6.3.
Under the map π S ∗ p : x C B hyp ( x ) ∩ p ⊥ a point x ∈ S = R P n +1 is mapped to an elliptic sphere in E with center π p ( x ) . In thenormalization h π p ( x ) , π p ( x ) i = 1 its elliptic radius is given by r ≥ , where x n +2 = sin r . Inparticular, x ∈ B ell is mapped to an elliptic hyperplane with pole π p ( x ) ∈ E .Remark . (i) The map π S ∗ p is a double cover of elliptic spheres, branching on the subset of elliptic points(see Proposition 6.4). We interpret the lift to carry the orientation of the elliptic Laguerrespheres. Upon the normalization given in Theorem 6.3 the orientation is encoded in thesign of the x n +2 -component. The involution σ p acts on the set of elliptic Laguerre spheresas orientation reversion.(ii) The Cayley-Klein distance induced on S by the elliptic Laguerre quadric B ell measuresthe Cayley-Klein tangent distance between the corresponding elliptic Laguerre spheres (seeProposition 6.5) if they possess a common oriented tangent hyperplane (see Remark 6.3).(iii) Using the projection π q instead of the polar projection π ∗ q elliptic Laguerre geometrycoincides with M¨obius geometry (cf. Section 5.4 and Section 7.3). elliptic geometry Laguerre geometry hyperplane with pole y ∈ E , y ∈ S n [ y, ± ∈ B ell sphere with center y ∈ E , y ∈ S n and radius r > y, ± sin r ] ∈ B +ell Table 4.
The lifts of elliptic spheres to elliptic Laguerre geometry.
Every (local) transformation mapping (non-oriented) elliptic hyperplanes to elliptic hyperplanes(not necessarily points to points) while preserving (tangency to) elliptic spheres can be liftedand extended to an elliptic Laguerre transformation (see Theorem 5.1).The elliptic Laguerre group
Lag ell = PO( n + 1 , B ell and maps planar sections of B ell to planar sections of B ell . Under the polar projection this means it maps oriented hyperplanes to oriented hyperplanes,or elliptic Laguerre spheres to elliptic Laguerre spheres, while preserving the tangent distanceand in particular the oriented contact (see Remark 6.3).45 igure 19. Concentric circles in the elliptic plane.
The elliptic Laguerre group contains (doubly covers) the group of elliptic isometries asPO( n + 1 , p . To generate the whole Laguerre group we only need to add one specific one-parameter family of scalings along concentric Laguerre spheres (see Remark 5.8 and Figure 19). (cid:73) Consider the family of transformations S t := I n · · · t sinh t · · · t cosh t for t ∈ R . It maps the absolute p = [0 , . . . , ,
1] to T (s) t ( p ) = [0 , . . . , , sinh t, cosh t ] , which is an elliptic sphere with center [0 , . . . , , t = 0 intoa point for t = ∞ , while changing orientation when it passes through the center or throughthe absolute, i.e. when t changes sign.Now the elliptic Laguerre group can be generated by elliptic motions and this one-parameterfamily of scalings (see Remark 5.8). Theorem 6.4.
Any elliptic Laguerre transformation f ∈ PO( n + 1 , can be written as f = Φ S t Ψ , where Φ , Ψ ∈ PO( n + 1 , p are elliptic motions and t ∈ R . p ⊥ LS s s R P n +2 S p ⊥ ’ R P n +1 π p ( s ) π p ( s ) Figure 20.
Left:
The Lie quadric
L ⊂ R P n +2 (depicted in the case n = 1). The choice of p with h p, p i < S ⊂ L . The points s , s ∈ L contained in a common isotropic subspace of L correspond to two oriented hyperspheres in orientedcontact. Right:
The M¨obius quadric
S ⊂ R P n +1 (depicted in the case n = 2). The two points π p ( s ) , π p ( s ) ∈ S + correspond to two hyperspheres in S via polarity. In the chosen normalization,their orientation is encoded in the last component of s , s respectively. M¨obius geometry (signature ( n +1 , n, n + 1 , n, , Liegeometry (signature ( n + 2 , sphere complexes . The groups of M¨obius transformations, Laguerre transformations,and isometries appear as quotients of the group of Lie transformations . S n We first give an intuitive description of Lie (sphere) geometry as the geometry of orientedhyperspheres of the n -dimensional sphere S n and their oriented contact.Thus, let S n = n y ∈ R n +1 (cid:12)(cid:12)(cid:12) y · y = 1 o ⊂ R n +1 , where y · y denotes the standard scalar product on R n +1 . An oriented hypersphere of S n can berepresented by its center c ∈ S n and its signed spherical radius r ∈ R (see Figure 20). Tuples( c, r ) ∈ S n × R represent the same oriented hypersphere if they are related by a sequence of thetransformations ρ : ( c, r ) ( c, r + 2 π ) , ρ : ( c, r ) ( − c, r − π ) . (10)The corresponding hypersphere as a set of points is given by { y ∈ S n | c · y = cos r } , S n into two regions. For r ∈ [0 , π ) consider the region which contains the center c to be the“inside” of the hypersphere, and endow the hypersphere with an orientation by assigning normalvectors pointing towards the other region, the “outside” of the hypersphere. The orientation ofthe hypersphere for other values of r is then obtained by (10). Definition 7.1.
We call ~ S := ( S n × R ) (cid:30) { ρ , ρ } . the space of oriented hyperspheres of S n . Remark . Orientation reversion defines an involution on ~ S , which is given by ρ : ( c, r ) ( c, − r ) . Thus, the space of (non-oriented) hyperspheres of S n may be represented by S := ~ S (cid:30) ρ = ( S n × R ) (cid:30) { ρ, ρ , ρ } . Two oriented hyperspheres ( c , r ) and ( c , r ) are in oriented contact if (see Figure 20) c · c = cos( r − r ) , (11)which is a well-defined relation on ~ S . Upon introducing coordinates ( c, cos r, sin r ) on ~ S thetransformations (10) may be replaced by( c, cos r, sin r ) ( − c, cos( r − π ) , sin( r − π )) = − ( c, cos r, sin r ) , (12)while (11) becomes a bilinear relation, i.e. c · c − cos r cos r − sin r sin r = 0 . (13)This gives rise to a projective model of Lie geometry as described in the following. Definition 7.2. (i) The quadric
L ⊂ R P n +2 corresponding to the standard bilinear form of signature ( n + 1 , h x, y i := n +1 X i =1 x i y i − x n +2 y n +2 − x n +3 y n +3 for x, y ∈ R n +3 , is called the Lie quadric .(ii) Two points s , s ∈ L on the Lie quadric are called Lie orthogonal if h s , s i = 0, orequivalently if the line s ∧ s is isotropic, i.e. is contained in L . An isotropic line is calleda contact element .(iii) The projective transformations of R P n +2 that preserve the Lie quadric L Lie := PO( n + 1 , . are called Lie transformations . 48 roposition 7.1.
The set of oriented hyperspheres ~ S of S n is in one-to-one correspondencewith the Lie quadric L by the map ~S : ~ S → L , ( c, r ) ( c, cos r, sin r ) such that two oriented hyperspheres are in oriented contact if and only if their correspondingpoints on the Lie quadric are Lie orthogonal.Proof. A point s ∈ L can always be represented by s = [ c, cos r, sin r ] with c ∈ S n , r ∈ R . Nowthe statement follows from (12) and (13). spherical geometry Lie geometry point ˆ x ∈ S n [ˆ x, , ∈ L oriented hypersphere with center ˆ s ∈ R n and signed radius r ∈ R [ˆ s, cos r, sin r ] ∈ L Table 5.
Correspondence of hyperspheres of the n -sphere S n and points on the Lie quadric L . This correspondence leads to an embedding of S n into the Lie quadric in the following way.Among all oriented hyperspheres the map ~S distinguishes the set of “points”, or null-spheres ,as the set of oriented hyperspheres with radius r = 0. It turns out that n ~S ( c, (cid:12)(cid:12)(cid:12) c ∈ S n o = { x ∈ L | x n +3 = 0 } = L ∩ p ⊥ , where p := [ e n +3 ] = [0 , . . . , , ∈ R P n +2 . Definition 7.3.
The quadric S := L ∩ p ⊥ is called the point complex . Remark . Every choice of a timelike point p ∈ R P n +2 , i.e. h p, p i <
0, leads to the definition ofa point complex S = L ∩ p ⊥ , all of which are equivalent up to a Lie transformation. The chosenpoint complex S then leads to a correspondence of points on the Lie quadric L and orientedhyperspheres on S ’ S n .The point complex is a quadric of signature ( n + 1 ,
1) which we identify with the M¨obiusquadric (see Section 5.4). The corresponding involution and projection associated with the point p (see Definition 5.1) take the form σ p : [ x , . . . , x n +2 , x n +3 ] [ x , . . . , , x n +2 , − x n +3 ] ,π p : [ x , . . . , x n +2 , x n +3 ] [ x , . . . , , x n +2 , . The image of the Lie quadric L under the projection π p is given by π p ( L ) = S + ∪ S = n s ∈ p ⊥ (cid:12)(cid:12)(cid:12) h s, s i ≥ o . By polarity, each point S + ∪ S corresponds to a hyperplanar section of S . Thus, the Lie quadriccan be seen as a double cover of the set of spheres of M¨obius geometry, encoding their orientation,while, vice versa, the orientation of hyperspheres in Lie geometry vanishes in the projection toM¨obius geometry. 49 roposition 7.2. (i) The involution σ p : L → L corresponds to the orientation reversion on ~ S .(ii) The projection π p : L → S + ∪ S defines a double cover with branch locus S .(iii) The set S of non-oriented hyperspheres of S n (see Remark 7.1) is in one-to-one corre-spondence with S + ∪ S by the map S = π p ◦ ~S : S → S + ∪ S , ( c, r ) ( c, cos r, . (iv) The set of “points” on S ⊂ L lying on an oriented hypersphere s ∈ L , or equivalently lyingon the non-oriented hypersphere π p ( s ) ∈ S + ∪ S is given by s ⊥ ∩ S = π p ( s ) ⊥ ∩ S . (v) The non-oriented hyperspheres corresponding to two points s , s ∈ S + ∪ S touch if andonly if the line s ∧ s connecting them is tangent to S .Thus, the points on the cone of contact C S ( s ) (see Definition 3.1) correspond to all spherestouching the sphere corresponding to s ∈ S + ∪ S .Proof. (i) Note that σ p ( ~S ( c, r )) = ~S ( c, − r ) and compare with Remark 7.1.(ii) See Proposition 5.1 (ii).(iii) Follows from (i) and (ii).(iv) The set s ⊥ ∩ S ⊂ L describes all hyperspheres in oriented contact with s that simulta-neously correspond to “points”, i.e. “points” that lie on the hypersphere. Indeed, with s = [ b s, cos r, sin r ] ∈ L we find for a “point” x = [ b x, , ∈ S that h s, x i = 0 ⇔ h b s, b x i = cos r. (v) This generalizes the statement in (iv) and follows from the fact that the isotropic subspacesof L (contact elements, cf. Definition 7.2) project to tangent lines of S .The subgroup Lie p of Lie transformations that preserve the point complex S , i.e. map“points” to “points”, becomes the group of M¨obius transformations in the projection to p ⊥ Mob = Lie p (cid:30) σ p ’ PO( n + 1 , . A sphere complex in Lie geometry is given by the intersection of the Lie quadric with a hyper-plane of R P n +2 . It may equivalently be described by the polar point of this hyperplane. Twopoints in R P n +2 can be mapped to each other by a Lie transformation if and only if they havethe same signature. Thus, any two sphere complexes of the same signature are Lie equivalent. Definition 7.4.
For a point q ∈ R P n +2 the set of points L ∩ q ⊥ on the Lie quadric as well as the n -parameter family of oriented hyperspheres corresponding tothese points is called a sphere complex . A sphere complex is further called50 elliptic if h q, q i > (cid:73) hyperbolic if h q, q i < (cid:73) parabolic if h q, q i = 0. Remark . (i) We adopted the classical naming convention for sphere complexes here, see e.g. [Bla1929].(ii) The point complex (see Definition 7.3) is a hyperbolic sphere complex.(iii) A non-parabolic sphere complex induces an invariant for pairs of oriented spheres (seeAppendix B). In particular, the invariant induced by the point complex, i.e., the point p , is the signed inversive distance (see Appendix B.2), which generalizes the intersectionangle of spheres. It further allows for a geometric description of sphere complexes (seeAppendix B.3).Laguerre geometry is the geometry of oriented hyperplanes and oriented hyperspheres in acertain space form, and their oriented contact (cf. Section 6). It appears as a subgeometry ofLie geometry by distinguishing the set of “oriented hyperplanes” as a sphere complex amongthe set of oriented hyperspheres.The point complex S = L ∩ p ⊥ , where p ∈ R P n +2 is a timelike point, induces the notionof orientation reversion given by the involution σ p . For another sphere complex L ∩ q ⊥ , where q ∈ R P n +2 , to play the distinguished role of the set of “oriented hyperplanes” on S it must beinvariant under orientation reversion, i.e., σ p ( L ∩ q ⊥ ) = L ∩ q ⊥ , which is equivalent to h p, q i = 0. Definition 7.5.
For a point q ∈ R P n +2 with h p, q i = 0we call the sphere complex B := L ∩ q ⊥ , a plane complex .Up to a Lie transformation that fixes p , i.e. a M¨obius transformation (cf. Section 5.4), wecan set, w.l.o.g., q = [ e n +1 ] = [0 , . . . , , ,
0] if h q, q i > e n +2 ] = [0 , . . . , , ,
0] if h q, q i < e ∞ ] = [0 , . . . , , ,
0] if h q, q i = 0 . Consider the restriction of the Lie quadric to q ⊥ ’ R P n +1 . Then for the non-parabolic cases weidentify each of the plane complexes with one of the Laguerre quadrics which we have introducedin Section 6. The parabolic plane complex corresponds to the classical case of Euclidean Laguerregeometry. Thus, we recover (see Figure 21) (cid:73) hyperbolic Laguerre geometry if h q, q i > (cid:73) elliptic (“spherical”) Laguerre geometry if h q, q i < (cid:73) Euclidean Laguerre geometry if h q, q i = 0 (see Section A.4). Remark . Note that according to the classical naming convention of sphere complexes, whichwe adopted in Definition 7.4, an elliptic sphere complex is associated with hyperbolic Laguerregeometry, while a hyperbolic sphere complex is associated with elliptic Laguerre geometry.51 sq L p ⊥ q ⊥ B BS Figure 21.
Laguerre geometry from Lie geometry. The choice of a point q with h p, q i = 0 determinesa plane complex, or Laguerre quadric B ⊂ L . This induces Laguerre geometry on a Cayley-Kleinspace in the base plane B . A point s ∈ B corresponds to an oriented line in that space via polarprojection. The corresponding groups of
Laguerre transformations are induced by the groups of Lie trans-formations that preserve the corresponding Laguerre quadric B , or equivalently the point q , Lie q (cid:30) σ q ’ PO( n,
2) if h q, q i > n + 1 ,
1) if h q, q i < n, ,
1) if h q, q i = 0 , where σ q is the involution associated with the plane complex (cf. Definition 5.1), and we set σ q = id if h q, q i = 0. Remark . In the non-parabolic cases, the condition h p, q i = 0 is equivalent to the conditionthat the two involutions σ p and σ q commute, i.e. σ p ◦ σ q = σ q ◦ σ p . We recognized the different Laguerre quadrics by their signature, which depends on thesignature of the point q only, but is entirely independent of the point p with h p, q i = 0. We yethave to establish the geometric relation to Lie geometry. Definition 7.6.
Given the two points p , q ∈ R P n +2 defining the point complex and the planecomplex respectively, we call the set B := p ⊥ ∩ q ⊥ ’ R P n . the base plane .In the restriction to the hyperplane of the point complex p ⊥ , the point π p ( q ) plays the roleof the point q from the Sections 6.2 and 6.3. Thus, polar projection with respect to this pointyields hyperbolic/elliptic geometry in the base plane B . In the parabolic case, projection with52espect to q should be replaced by stereographic projection, which recovers Euclidean (similarity)geometry (cf. Appendix A.3).On the other hand, in the restriction to the hyperplane of the plane complex q ⊥ , the point π q ( p ) plays the role of the point p from the Sections 6.2 and 6.3. Thus, polar projectionwith respect to this point yields hyperbolic/elliptic geometry in the base plane B , while in theparabolic case projection with respect to p leads to dual Euclidean (similarity) geometry (cf.Appendix A.4).On the level of the transformation group this can be described in the following way. Considerthe Lie transformations Lie p , q that fix all “points” and “planes”, i.e. the point complex S and theplane complex B , or equivalently, the two points p , q ∈ R P n +2 . These transformations naturallyfactor to Lie p , q / { σ p , σ q } , where again we set σ q = id if h q, q i = 0. Their action is well-defined onthe quotient space S / σ q , due to Remark 7.5. The quotient space S / σ q can be embedded into thebase plane B using the projection π q , which should be replaced by stereographic projection inthe Euclidean case (cf. Appendix A.4). Thus, we may equivalently consider the action of theseLie transformations on the base plane B , on which they act as lower dimensional projectiveorthogonal groups again: Lie p , q (cid:30) { σ p , σ q } ’ PO( n + 1 , (cid:30) σ q ’ PO( n, , if h q, q i > n + 1) , if h q, q i < n, , , if h q, q i = 0 . We recognize PO( n,
1) and PO( n + 1) as the isometry groups of hyperbolic and elliptic space (cf.Sections 4.4 and 4.5), while PO( n, ,
1) corresponds to the group of dual similarity transforma-tions, i.e. the group of dual transformations PO( n, , ∗ corresponds to isometries and scalingsof Euclidean space (cf. Appendix A.2). Remark . We end up with two models of the space form associated to each Laguerre geometry(see Figure 21). One is represented by the point complex
S ⊂ p ⊥ ’ R P n +1 , with opposite pointswith respect to σ q identified, which we refer to as the sphere model (see Figures 26, 27, and 28,top). In this model the oriented hyperspheres that correspond to sections of S with hyperplanesthat contain the point π p ( q ) are the distinguished “oriented hyperplanes”Another model is obtained by its projection π q ( S ) onto the base plane B ’ R P n , which werefer to as the projective model (see Figures 26, 27, and 28, left). In this model the “orientedhyperplanes” become (oriented) projective hyperplanes. Proposition 7.3. (i) In the non-Euclidean cases of Laguerre geometry, i.e. h q, q i 6 = 0 , the point complex S maybe identified with hyperbolic/elliptic space respectively, after taking the quotient with respectto σ q , or equivalently, projection onto the base plane S (cid:30) σ q ’ π q ( S ) ⊂ B ’ R P n . The Lie transformations that fix the point complex and the plane complex act on π q ( S ) ⊂ B as the corresponding isometry group.(ii) In the case of Euclidean Laguerre geometry, i.e. h q, q i = 0 , the point complex S may beidentified with Euclidean space upon stereographic projection. The Lie transformations thatfix the point complex and the plane complex act on B as dual similarity transformations.Remark . In Laguerre geometry the hyperplanar sections correspond to oriented spheres,which, in the non-Euclidean cases, can be identified with their polar points. In elliptic Laguerregeometry the Lie quadric projects to the “outside” of the elliptic Laguerre quadric π q ( L ) = B +ell ∪ B ell π q ( L ) = B − hyp ∪ B ell , while the poles of hyperplanar sections are the whole space (8). The Lie quadric only projectsto the points corresponding to hyperbolic spheres/distance hypersurfaces/horospheres, and notto points representing deSitter spheres (cf. Remark 6.6 (i)). Vice versa, Laguerre spheres thatare deSitter spheres do not possess a (real) lift to the Lie quadric. Choosing different signatures for the points p and q , i.e. different signatures for the point complexand plane complex, we recover different subgeometries of Lie geometry (see Table 6).Fixing both points in the Lie group induces (a quadruple covering of) the correspondingisometry group. We call the group obtained by fixing only p (a double cover of) the correspond-ing M¨obius group , and the group obtained by fixing only q (a double cover of) the corresponding Laguerre group . For each isometry group the corresponding M¨obius group describes the trans-formations that map points in the space form to points while preserving spheres, while theLaguerre group describes the transformations that map hyperplanes to hyperplanes while pre-serving spheres. For this to hold, the transformations either have to be considered locally, oracting on the set of oriented points/oriented hyperplanes respectively (see Theorem 5.1, Remark5.6 (iii), Remark 5.10 (iii), Remark 5.11 (iii), Section 6.2.1, Section 6.3.1).
Remark . Note that certain geometries have the same transformation group. In particular, n -dimensional Lie geometry has the same transformation group as ( n + 1)-dimensional hyperbolicLaguerre geometry. Geometrically this is due to the fact that one can identify the orientedhyperspheres of S with the oriented hyperbolic hyperplanes of the inside H = S − . space form isometry grp. M¨obius grp. Laguerre grp. sign. p , q elliptic space PO( n + 1) PO( n + 1 ,
1) PO( n + 1 ,
1) ( − ) ( − )hyperbolic space PO( n,
1) PO( n + 1 ,
1) PO( n,
2) ( − ) (+)deSitter space PO( n,
1) PO( n,
2) PO( n + 1 ,
1) (+) ( − )(dual) Euclidean space PO( n, ,
1) PO( n + 1 ,
1) PO( n, ,
1) ( − ) (0)(dual) Minkowski space PO( n − , ,
1) PO( n,
2) PO( n, ,
1) (+) (0)
Table 6.
Isometry group, M¨obius group, and Laguerre group for different space forms, and thesignatures of the points p and q defining the corresponding point complex and plane complex inLie geometry respectively. In the degenerate cases of Euclidean and Minkowski geometry, the given“isometry group” is actually the group of similarity transformations represented on the dual space. igure 22. Lie-circumscribed quadrilaterals.
In this section, as an application of two-dimensional Lie and Laguerre geometry, we present newresearch results. While incircular nets and their Laguerre geometric generalization to checker-board incircular nets have been studied in great detail [B¨oh1970, AB2018, BST2018], we in-troduce their generalization to Lie geometry, and show that they may be classified in terms ofcheckerboard incircular nets in hyperbolic/elliptic/Euclidean Laguerre geometry. We prove in-cidence theorems of Miquel type, show that all lines of a checkerboard incircular net are tangentto a hypercycle, and give explicit formulas in terms of Jacobi elliptic functions. This generalizesthe results from [BST2018] and leads to a unified treatment of checkerboard incircular nets inall space forms.
To investigate configurations of oriented circles and their oriented contact on the two-sphere, weidentify oriented circles with points on the Lie quadric
L ⊂ R P , which is a quadric of signature(+ + + − − ), as described in Section 7. Definition 8.1 (Lie quadrilateral) . A Lie quadrilateral is a quadruple of oriented circles, called edge circles . Remark . Two edge circles of a Lie quadrilateral do not necessarily intersect. Thus, e.g., bothquadrilaterals shown in Figure 22 are admissible Lie quadrilaterals.
Definition 8.2 (Lie circumscribed) . A Lie quadrilateral is called circumscribed if the fourpoints on the Lie quadric corresponding to its four oriented edge circles are coplanar. Wecall the signature of the plane in which these points lie the signature of the circumscribed Liequadrilateral.To justify the term “circumscribed” consider a plane U ⊂ R P of signature (+ + − ). Thenaccording to Lemma 3.1 its polar line has signature (+ − ), and thus, U ⊥ ∩ L = { c , c } consistsof exactly two points. The one parameter family of circles corresponding to the points in U ∩ L are the circles in oriented contact with the two circles corresponding to c and c . Therefore,a circumscribed Lie quadrilateral of signature (+ + − ) is in oriented contact with exactly twocircles (see Figure 22).To characterize all possible cases of circumscribed Lie quadrilaterals we need to distinguishall possible signatures of the plane U . 55 m ‘ ‘ m ‘ m ‘ m m m m ‘ ‘ ‘ ‘ Figure 23.
Lie geometric version of Miquel’s theorem.
Left:
Combinatorial picture.
Right:
Geo-metric picture.
Proposition 8.1.
For a plane U ⊂ R P the family of oriented circles corresponding to U ∩ L isexactly one of the following depending on the signature of U with respect to the Lie quadric L . (cid:73) (+ + +) Empty family. (cid:73) (+ + − ) One parameter family of circles in oriented contact with the two oriented circlesgiven by U ∩ L . (cid:73) (+ − − ) Circles from the intersection of two hyperbolic circle complexes (cf. Definition 7.4and Appendix B.3). (cid:73) (+ − Two contact elements (see Definition 7.2) with a common circle. (cid:73) (+ + 0)
One circle. (cid:73) (+00)
One contact element.Proof.
The Lie quadric has signature (+ + + − − ). Thus, the listed signatures are all possiblecases that can occur. A plane with signature (+ + +) does not intersect the Lie quadric. Thecase (++ − ) was already discussed before the proposition. For the case (+ −− ) the polar line hassignature (++). Thus, we may view U as the intersection of two hyperbolic circle complexes.The cases (+ − Remark . For a generic circumscribed Lie quadrilateral, i.e., no three of the four points onthe Lie quadric are collinear, only the signatures (+ + − ), (+ − − ), and (+ −
0) can occur.The definition of Lie circumscribility via planarity in the Lie quadric immediately impliesa Lie geometric version of the classical Miquel’s theorem. To see this, we employ the followingstatement of projective geometry about the eight intersection points of three quadrics in space,see, e.g., [BS2008, Theorem 3.12].
Lemma 8.1 (Associated points) . Given eight distinct points which are the set of intersectionsof three quadrics in R P , all quadrics through any seven of those points must pass through theeighth point. Theorem 8.1 (Miquel’s theorem in Lie geometry) . Let ‘ , ‘ , ‘ , ‘ , m , m , m , m be eightgeneric oriented circles on the sphere such that the five Lie quadrilaterals ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) are circumscribed, then so isthe Lie quadrilateral ( ‘ , ‘ , m , m ) (see Figure 23). ‘ ‘ ‘ m m m m ‘ ‘ Figure 24.
On the combinatorics of adjacent “cubes” of a checkerboard incircular net.
Remark . A sufficient genericity condition for the eight points on the Lie quadric is that nofive points are coplanar.
Proof.
Consider the eight points on the Lie quadric as the vertices of a combinatorial cube (seeFigure 23). Coplanarity of the bottom and side faces corresponds to the assumed circumscribility.Thus, we have to show that the top face is planar as well.As a first step we show that all eight vertices of the cube are contained in a three-dimensionalprojective subspace. Indeed, let V be the subspace spanned by ‘ , ‘ , m , m . Then the assumedcircumscribility implies that for instance ‘ lies in a plane with ‘ , m , m and therefore ‘ ∈ V .Similarly, ‘ , m ∈ V , and finally m ∈ V .A three-dimensional subspace intersects the Lie quadric in a (at most once degenerate) two-dimensional quadric ˜ L . Consider the two degenerate quadrics Q , Q consisting of two oppositeface planes of the cube, respectively. Then, due to the genericity condition, the eight pointsof the cube are the intersection points of ˜ L , Q , Q . Now consider the degenerate quadric Q consisting of the bottom plane of the cube and the plane spanned by ‘ , ‘ , m . Then Q contains seven of the eight points, and therefore, according to Lemma 8.1, also the eighth point m . Since m may not lie in the bottom plane, we conclude that the quadrilateral ( ‘ , ‘ , m , m )is circumscribed.We now introduce nets consisting of two families of oriented circles such that every secondLie quadrilateral (in a checkerboard-manner) is circumscribed. Definition 8.3 (Lie checkerboard incircular nets) . Two families ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z of orientedcircles on the sphere are called a Lie checkerboard incircular net if for every i, j ∈ Z with even i + j the Lie quadrilateral ( ‘ i , ‘ i +1 , m j , m j +1 ) is circumscribed.In the following we will always assume generic Lie checkerboard incircular nets in the senseof Remark 8.3. As an immediate consequence of Theorem 8.1 we find that Lie checkerboardincircular nets have many more circumscribed Lie quadrilaterals than introduced in its definition. Corollary 8.1.
Let ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z be the oriented circles of a Lie checkerboard incircularnet. Then for every i, j, k ∈ Z with even i + j the Lie quadrilateral ( ‘ i , m j , ‘ i +2 k +1 , m j +2 k +1 ) iscircumscribed. Similar to the argument in the proof of Theorem 8.1 (or as a consequence thereof), we findthat the points on the Lie quadric corresponding to a Lie checkerboard incircular net can notspan the entire space. 57 igure 25.
Inscribed quadrilaterals in the hyperbolic plane. The right most case is degenerate andconsists of four oriented lines “touching” an oriented line at its points at infinity.
Theorem 8.2.
The points on the Lie quadric
L ⊂ R P corresponding to the oriented circles ofa Lie checkerboard incircular net lie in a common hyperplane of R P .Proof. Consider “adjacent” cubes ( ‘ , ‘ , ‘ , ‘ , m , m , m , m ) and ( ‘ , ‘ , ‘ , ‘ , m , m , m , m )from the Lie checkerboard incircular net with vertices on the Lie quadric (see Figure 24). Eachof these cubes lies in a three-dimensional subspace of R P , and they coincide in six of its eightvertices. Thus, both cubes, and by induction the whole net, lie in the same three-dimensionalsubspace.As we have seen in Section 7, depending on its signature, a three-dimensional subspaceof the Lie quadric induces one of the three types of Laguerre geometry. Thus, our study ofLie checkerboard incircular nets may be reduced to the study of its three Laguerre geometriccounterparts as we will see in the next section. Two dimensional hyperbolic/elliptic/Euclidean Laguerre geometry is the geometry of orientedlines in the hyperbolic/elliptic/Euclidean plane and their oriented contact to oriented circles(Laguerre circles) in the respective space form. We identify oriented lines with points on, andoriented circles with planar sections of, the corresponding Laguerre quadric B , which is a quadricof signature (+ + −− ), (+ + + − ), (+ + −
0) respectively (see Sections 6.2, 6.3, A.4).Similar to the condition for Lie circumscribility, four oriented lines touch a common orientedcircle if and only if the corresponding points on the Laguerre quadric are coplanar. On theother hand, all three Laguerre geometries are subgeometries of Lie geometry, by restrictingthe Lie quadric to a three-dimensional subspace. Thus, in this restriction, a Lie circumscribedquadrilateral turns into four lines touching a common oriented circle. Accordingly one obtainsthe following Laguerre geometric version of Theorem 8.1.
Theorem 8.3 (Miquel’s theorem in Laguerre geometry) . Let ‘ , ‘ , ‘ , ‘ , m , m , m , m beeight generic oriented lines in the hyperbolic/elliptic/Euclidean plane such that the five quadri-laterals ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) are cir-cumscribed (each touches a common oriented circle), then so is the quadrilateral ( ‘ , ‘ , m , m ) (cf. Figure 24). The Laguerre geometric version of checkerboard incircular nets (see Definition 8.3) is giventhe following definition [AB2018]. Examples of checkerboard incircular nets in the elliptic andhyperbolic plane are shown in Figures 26, 27, and 28 (see also [DGDGallery]).
Definition 8.4 (Laguerre checkerboard incircular nets) . Two families ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z of ori-ented lines in the hyperbolic/elliptic/Euclidean plane are called a (hyperbolic/elliptic/Euclidean)checkerboard incircular net if for every i, j ∈ Z with even i + j the four lines ‘ i , ‘ i +1 , m j , m j +1 touch a common oriented circle (Laguerre circle).58 emark . From Corollary 8.1, or Theorem 8.3, we find that, same as in the Lie geometriccase, every quadrilateral ( ‘ i , m j , ‘ i +2 k +1 , m j +2 k +1 ), i, j, k ∈ Z of a checkerboard incircular netwith even i + j is circumscribed.Now we can formulate the following classification result for Lie checkerboard incircular nets. Theorem 8.4 (classification of Lie checkerboard incircular nets) . Every Lie checkerboard in-circular net is given by a Lie transformation of a hyperbolic, elliptic, or Euclidean checkerboardincircular net.Proof.
According to Theorem 8.2 every Lie checkerboard incircular net lies in a three-dimensionalsubspace of R P . This subspace can only have one of the signatures (+++ − ), (++ −− ), (++ − (cid:73) hyperbolic Laguerre geometry (+ + −− ) (see Remark 6.6 (i) and Figure 25) • (++ − ): Four lines touching a common oriented hyperbolic circle/distance curve/horocircle. • (+ − − ): Four lines touching a common deSitter circle. • (+ − (cid:73) elliptic Laguerre geometry (+ + + − ): • (+ + − ): Four lines touching a common oriented elliptic circle. (cid:73) Euclidean Laguerre geometry (+ + − • (+ + − ): Four lines touching a common oriented Euclidean circle. • (+ − igure 26. Top:
Checkerboard incircular net tangent to an ellipse in the sphere model of the ellipticplane.
Bottom-left:
Central projection to the projective model of the elliptic plane.
Bottom-right:
Stereographic projection to a conformal model of the elliptic plane. igure 27. Top:
Checkerboard incircular net tangent to an ellipse in the sphere model of thehyperbolic plane. Two copies of the hyperbolic plane are realized as half-spheres.
Middle-left:
Orthogonal projection to the Klein-Beltrami disk model.
Middle-right:
Stereographic projection tothe Poincar´e disk model.
Bottom:
Stereographic projection to the Poincar´e half-plane model. igure 28. Top:
Checkerboard incircular net tangent to a hyperbola in the sphere model ofthe hyperbolic plane. Two copies of the hyperbolic plane are realized as half-spheres.
Middle-left:
Orthogonal projection to the Klein-Beltrami disk model.
Middle-right:
Stereographic projection tothe Poincar´e disk model.
Bottom:
Stereographic projection to the Poincar´e half-plane model. igure 29. Hypercycle in the Euclidean plane.
In Laguerre geometry the oriented lines, and not the points, of a given space form are invariantobjects. Thus, in Laguerre geometry, it is natural to describe an (oriented) curve in the (hy-perbolic/elliptic/Euclidean) plane by its (oriented) tangent lines. Conversely, we say that everycurve on the Laguerre quadric corresponds to a curve in the plane. Note that in the case of thehyperbolic plane the envelope of such a “curve” might lie partially (or even entirely) “outside”the hyperbolic plane. We still consider this to be an admissible (non-empty)
Laguerre curve . Definition 8.5.
The one-parameter family of oriented lines (in the hyperbolic/elliptic/Euclideanplane) corresponding to a curve on the Laguerre quadric B is called a (hyperbolic/elliptic/Euclidean)Laguerre curve .We have noted that planar sections of the Laguerre quadric correspond to Laguerre circles,also called (generalized) cycles in the two-dimensional case. Consequently, the next higher orderintersections with the Laguerre quadric are called hypercycles [Bla1910]. Definition 8.6.
A (hyperbolic/elliptic/Euclidean) Laguerre curve corresponding to the inter-section of the Laguerre quadric with another quadric is called a (hyperbolic/elliptic/Euclidean)hypercycle . The corresponding curve on the Laguerre quadric is called hypercycle base curve . Example 8.1.
In every space form a conic endowed with both orientations, joined together asthe two components of a single oriented curve (see Section 8.4) is a hypercycle. A more generalexample is shown in Figure 29.The intersection curve of two quadrics ( base curve ) is contained in all quadrics of the pencilspanned by the two quadrics. Thus, a hypercycle, through its hypercycle base curve, correspondsnot just to one quadric but the whole pencil of quadrics spanned by it and the Laguerre quadric.We call a hypercycle non-degenerate if its hypercycle base curve contains at least 8 points ingeneral position. In this case the hypercycle can be uniquely identified with the correspondingpencil of quadrics. In the following we assume all hypercycles to be non-degenerate.The following theorem establishes a relation between a checkerboard incircular net and ahypercycle, as well as two certain hyperboloids in the pencil of quadrics corresponding to itshypercycle base curve. In the Euclidean case this was shown in [BST2018, Theorem 3.4] as partof an incidence theorem for checkerboard incircular nets (see Theorem 8.8).
Theorem 8.5.
The lines of a (hyperbolic/elliptic/Euclidean) checkerboard incircular net are inoriented contact with a common hypercycle (see Figure 29).Moreover, the corresponding pencil of quadrics, which contains the hypercycle base curve,contains two unique hyperboloids Q , e Q distinguished in the following way (see Figure 30). Let ‘ ‘ ‘ ‘ ‘ m m m m m m L L L L L M M M M M S S S S S S S S S S S S S m ‘ ‘ m m ‘ ‘ m Q e Q m ‘ m ‘ ‘ m ‘ m Figure 30.
Left:
Combinatorial picture of the lines of a checkerboard incircular net.
Middle/Right:
The two hyperboloids in the pencil of quadrics through the hypercycle base curve associated with acheckerboard incircular net in the elliptic plane. ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z be the points on the Laguerre quadric B ⊂ R P corresponding to the orientedlines of the checkerboard incircular net. Consider the lines L i := ‘ i ∧ ‘ i +1 , M i := m i ∧ m i +1 . Then, all lines L k , M l lie on a common hyperboloid Q ⊂ R P , and similarly, all lines L k +1 , M l +1 lie on a common hyperboloid e Q ⊂ R P .Proof. Due to the inscribability property of checkerboard incircular nets every line L k intersectsevery line M l , and vice versa. Thus, all lines L k , M l generically lie on a common hyperboloid Q . Similarly, all lines L k +1 , M l +1 lie on a common hyperboloid e Q . We now show that bothhyperboloids Q , e Q intersect the Laguerre quadric B in the same curve, that is, they belongto the same pencil of quadrics. Indeed, according to Lemma 8.2, for each line L k +1 , thereexists a unique quadric in the pencil spanned by B and Q containing L k +1 . Same for each line M l +1 . Since the lines L k +1 and M l +1 pairwise intersect, again according to Lemma 8.2, thecorresponding quadrics coincide with each other and eventually with e Q . Thus, all points ‘ i , m j lie on the intersection B ∩ Q = B ∩ e Q . Lemma 8.2.
Let x , x be two points which belong to all members of a pencil of quadrics Q λ . Then, there exists a unique quadric Q λ from the pencil which contains the whole line L = x ∧ x .If the line L = x ∧ x associated with another pair of base points x , x intersects the line L then the two quadrics Q λ and Q λ coincide.Proof. Let q , q be two quadratic forms generating the pencil with the quadratic form q λ = q + λq . The points x , x belong to all quadrics of the pencil if and only if q ( p ) = q ( p ) = q ( p ) = q ( p ) = 0 . The line L = x ∧ x belongs to the quadric determined by q λ if and only if q λ ( p , p ) = 0so that t = − q ( p , p ) q ( p , p ) . Vanishing of the denominator is the case when the line lies on the quadric determined by q .Moreover, if the line L = x ∧ x passing through another pair of common points x , x intersects the line L then the point of intersection and x , x belong to the quadric Q λ .Accordingly, the line L is contained in Q λ so that Q λ = Q λ .64 igure 31. Incidence theorem for eight lines touching a hypercycle.
The oriented circles of a checkerboard incircular net correspond to the planes spanned bypairs of lines L k , M l or L k +1 , M l +1 , i.e. they correspond to tangent planes of the two hy-perboloids Q , e Q , respectively. We identify each circle with its polar point with respect to theLaguerre quadric B , or in the Euclidean case with a point in the cyclographic model (cf. Ap-pendix A.4). Corollary 8.2. (i) The polar points corresponding to the oriented circles of a hyperbolic/elliptic checkerboardincircular net lie on two quadrics, the polar pencil of which contains the Laguerre quadric(polar with respect to the Laguerre quadric).(ii) The points in the cyclographic model corresponding to the oriented circles of a Euclideancheckerboard incircular net lie on two quadrics, the dual pencil of which contains the ab-solute quadric (i.e. the two quadrics are Minkowski confocal quadrics).Proof. (i) Under polarization in the Laguerre quadric B the tangent planes of Q become points onthe polar quadric Q ⊥ . Similarly, the tangent planes of e Q become points on the polarquadric e Q ⊥ . Since Q , e Q , and B are contained in a common pencil of quadrics, their polarimages Q ⊥ , e Q ⊥ , and B ⊥ ∼ = B are contained in the polar pencil of quadrics.(ii) For the Euclidean case a similar argument holds by dualization to the cyclographic model.We conclude this section on hypercycles by stating an incidence result concerning eight linestouching a hypercycle, which is similar to Theorem 8.3. Theorem 8.6.
Let ‘ , ‘ , ‘ , ‘ , m , m , m , m be eight generic lines touching a hypercycle. Ifthe three quadrilaterals ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) , ( ‘ , ‘ , m , m ) are circumscribed, thenso is the quadrilateral ( ‘ , ‘ , m , m ) (see Figure 31).Proof. We identify the eight oriented lines with its corresponding points on the Laguerre quadric.The hypercycle base curve is the intersection of two quadrics. Define the degenerate quadricgiven by the two planes through ‘ , ‘ , m , m and ‘ , ‘ , m , m respectively. Then the giveneight points on the Laguerre quadric are the intersection of those three quadrics. According toLemma 8.1 every quadric through seven of those points must pass through the eighth. Considerthe degenerate quadric given by the two planes through ‘ , ‘ , m , m and ‘ , ‘ , m respectively.Then this quadric must also pass through m . Since no five points may lie in a plane we canconclude that ‘ , ‘ , m , m lie in a common plane, and thus, that the corresponding quadrilateralis circumscribed. 65 .4 Conics and incircular nets Towards the parametrization of checkerboard incircular nets it turns out to be useful to considercertain normal forms of hypercycles, one of which are conics. In [BST2018] it is demonstratedthat in the Euclidean case a generic hypercycle can be mapped to a conic by a Laguerre transfor-mation if and only if the corresponding pencil of quadrics is diagonalizable. In the non-Euclideancases diagonalizable hypercycles are still a subset of hypercycles that can be mapped to conics.
Definition 8.7 (Conics in spaceforms) . In the projective model of the hyperbolic/elliptic/Euclideanplane embedded into R P a (hyperbolic/elliptic/Euclidean) conic is a projective conic in R P . Remark . (i) From this projective definition of conics one recovers the familiar metric properties ofconics in the different space forms, see, e.g., [Cha1841, Sto1883, Izm2017].(ii) In hyperbolic geometry a conic might lie “outside” the hyperbolic plane and be considereda “deSitter conic”. These cases are still relevant in our Laguerre geometric considerationsas long as they possess hyperbolic tangent lines.Recall that in Laguerre geometry reflection in the special point p corresponds to orientationreversion (see Section 6). We use this point in the following way to characterize conics in theset of hypercycles. Lemma 8.3.
A hypercycle in the hyperbolic/elliptic/Euclidean plane is a conic (doubly coveredwith opposite orientation) if and only if its hypercycle base curve is given by the intersection ofthe Laguerre quadric with a cone with vertex p .Proof. In hyperbolic and elliptic Laguerre geometry p is the polar point of the base plane of theprojective model of the corresponding space form. The polar of a cone with vertex p is thereforea conic contained in this base plane. Thus, the tangent planes to the hypercycle base curve arethe planes tangent to a conic, if and only if p is the vertex of a cone intersecting the Laguerrequadric in the hypercycle base curve. In that case corresponding oriented lines envelop the conic(twice with opposite orientation).In Euclidean Laguerre geometry a similar argument holds upon dualization and consideringthe cyclographic model. Remark . A generic hypercycle for which the corresponding pencil of quadrics is in diagonalform is a conic. Vice versa, in elliptic and Euclidean geometry a generic conic (excluding thenon-generic case of parabolas) can be brought into diagonal form by an isometry (a Laguerretransformation fixing the point p ). In hyperbolic geometry there also exist non-diagonalizablegeneric conics (semihyperbolas, cf. [Izm2017]). Thus, by considering conics up to Laguerretransformations, we are restricting the class of hypercycles to (a subclass of) diagonalizablehypercycles.We now give the definition for incircular nets [B¨oh1970, AB2018]. Examples of incircular netsin the elliptic and hyperbolic plane are shown in Figures 32, 33, and 34 (see also [DGDGallery]). Definition 8.8 (Incircular nets) . Two families ( ‘ k ) k ∈ Z , ( m l ) l ∈ Z of (non-oriented) lines in thehyperbolic/elliptic/Euclidean plane are called a (hyperbolic/elliptic/Euclidean) incircular net (IC-net) if for every k, l ∈ Z the four lines ‘ k , ‘ k +1 , m l , m l +1 touch a common circle (non-orientedLaguerre circle) S kl . Remark . While checkerboard incircular nets are instances of the corresponding (hyper-bolic/elliptic/ Euclidean) Laguerre geometry, incircular nets are a notion of the correspondingmetric geometry, i.e. only invariant under isometries (Laguerre transformations that fix p ).66n the limit of a checkerboard incircular net ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z in which all incircles of thequadrilaterals ‘ k , ‘ k +1 , m l , m l +1 collapse to a point, the pairs of lines ‘ k , ‘ k +1 as well as thepairs of lines m l , m l +1 coincide respectively up to their orientation. Such a pair of oriented linesmay be regarded as a non-oriented line. The points on the Laguerre quadric corresponding totwo lines that agree up to their orientation are connected by a line that goes through the point p .Considering the associated hyperboloids of a checkerboard incircular net from Theorem 8.5 wefind that the generator lines L k , M l all go through the point p and the hyperboloid Q becomesa cone with vertex at p . In this limit a checkerboard incircular net becomes an “ordinary”incircular net. Remark . An incircular net obtained from a checkerboard incircular net as the special casedescribed above possesses the following additional regularity property : The line through thecenters of S kl , S k +1 ,l +1 and the line through the centers of S k +1 ,l , S k,l +1 are the distinct anglebisectors of the lines ‘ k +1 and m l +1 . Thus, it is convenient to append this property to thedefinition of incircular nets.By Lemma 8.3 incircular nets are now characterized as special checkerboard incircular netsin terms of its associated hyperboloids (see Theorem 8.5). Theorem 8.7.
A checkerboard incircular net is an incircular net, if one of its two associatedhyperboloids is a cone with vertex at p . Together with Theorem 8.5 and Lemma 8.3 we obtain that for incircular nets the tangenthypercycle is a conic [B¨oh1970], [AB2018].
Corollary 8.3.
All lines of a (hyperbolic/elliptic/Euclidean) incircular net touch a commonconic.Remark . By the classical Graves-Chasles theorem incircular nets are closely related to con-figurations of confocal conics (see [B¨oh1970] for the Euclidean case, [AB2018] for the Euclideanand hyperbolic case, and [Izm2017] for a treatment in all space forms). A relation to discreteconfocal conics is given in [BSST2016, BSST2018].67 igure 32.
Top:
Incircular net tangent to an ellipse in the sphere model of the elliptic plane.
Bottom-left:
Central projection to the projective model of the elliptic plane.
Bottom-right:
Stereo-graphic projection to a conformal model of the elliptic plane. igure 33. Top:
Incircular net tangent to an ellipse in the sphere model of the hyperbolic plane.Two copies of the hyperbolic plane are realized as half-spheres.
Middle-left:
Orthogonal projection tothe Klein-Beltrami disk model.
Middle-right:
Stereographic projection to the Poincar´e disk model.
Bottom:
Stereographic projection to the Poincar´e half-plane model. igure 34. Top:
Incircular net tangent to a hyperbola in the sphere model of the hyperbolic plane.Two copies of the hyperbolic plane are realized as half-spheres.
Middle-left:
Orthogonal projection tothe Klein-Beltrami disk model.
Middle-right:
Stereographic projection to the Poincar´e disk model.
Bottom:
Stereographic projection to the Poincar´e half-plane model. .5 Construction and parametrization of checkerboard incircular nets The elementary construction of a checkerboard incircular net from a small patch (line by line,while ensuring the incircle constraint) is guaranteed to work due to the following incidencetheorem (see Figure 30, left) [AB2018, BST2018]. This construction has 12 real degrees offreedom.
Theorem 8.8.
Let ‘ , . . . , ‘ , m , . . . , m be 12 oriented lines in the hyperbolic/elliptic/Euclideanplane which are in oriented contact with 12 oriented circles S , . . . , S , in a checkerboard man-ner, as shown in Figure 30, left. In particular, the lines ‘ , ‘ , m , m are in oriented contactwith the circle S , the lines ‘ , ‘ , m , m are in oriented contact with the circle S etc. Then,the 13th checkerboard quadrilateral also has an inscribed circle, i.e., the lines ‘ , ‘ , m , m havea common circle S in oriented contact.Remark . This incidence theorem holds in all three Laguerre geometries with literally thesame proof as given in [BST2018] for the Euclidean case.Though possible in principle, the elementary construction from, e.g., 6 lines as initial data,which only describes the local behavior, is not stable, and thus impractical for the construction oflarge checkerboard incircular nets. Yet, by Theorem 8.5, we find that a checkerboard incircularnet can equivalently be prescribed by (cid:73) choosing a hypercycle (8 degrees of freedom), (cid:73) choosing two hyperboloids Q , ˜ Q in the pencil of quadrics corresponding to the hypercyclebase curve (2 degrees of freedom), (cid:73) and choosing two initial lines tangent to the hypercycle, one from each of the m - and ‘ -family(2 degrees of freedom).Then further lines of, say, the ‘ -family are obtained by alternately going along a chosen familyof rulings of Q and ˜ Q from one point of the base curve to the next (see Figure 30, middle/right).Similarly for the m -family of lines, while using the respective other families of rulings of the twohyperboloids. The intersection of two rulings from the two different families of the same hyper-boloid implies the coplanarity of the four intersection points with the base curve, which, in turn,corresponds to the existence of an incircle. We demonstrate for certain classes of checkerboardincircular nets how the parametrization of the hypercycle base curve in terms of Jacobi ellipticfunctions leads to explicit formulas for the net, in which the free parameters determine the globalbehavior. They can be further constraint to obtain periodic and “embedded” solutions. Remark . Note the resemblance to a “confocal billiards” type construction and a Ponceletporism type statement in the periodic case.In the following we derive explicit formulas for checkerboard incircular nets tangent to certaintypes of diagonalizable conics (see Remark 8.6). We treat the hyperbolic/elliptic/Euclidean casessimultaneously by considering the standard bilinear form of signature (+ + ε − ) in R , i.e., h x, y i = x y + x y + εx y − x y for x, y ∈ R , which defines the corresponding Laguerre quadric B ∈ R P . The hyperbolic caseis given by ε = −
1, the elliptic case by ε = 1, and the Euclidean case by ε = 0 (see Sec-tions 6.3, 6.2, A.4). By Lemma 8.3, a hypercycle that corresponds to a conic is given by theintersection curve of B with a cone with vertex p = [0 , , , (cid:73) ellipses in all space forms, (cid:73) hyperbolas in the Euclidean plane, and convex hyperbolas in the hyperbolic plane,excluding concave hyperbolas, deSitter hyperbolas and (the non-diagonalizable) semihyperbolasin the hyperbolic plane (cf. Remark 8.6), as well as all further non-diagonalizable hypercycles.71 hyp C B ell
C B euc C Figure 35.
Hypercycle base curve
B ∩ C for an ellipse in hyperbolic ( left ), elliptic ( middle ), andEuclidean ( right ) Laguerre geometry.
Consider a cone C given by α x + β x − x = 0 . (14)with α > β > , εα , εβ > . (15)It intersects the Laguerre quadric B given by x + x + εx − x = 0 (16)in the hypercycle base curve B ∩ C (see Figure 35).
Proposition 8.2.
The hypercycle base curve
B ∩ C corresponds to the (oriented) tangent linesof an ellipse given in homogeneous coordinates of the hyperbolic/elliptic/Euclidean plane by x α + x β − x = 0 . (17) Proof.
The hyperbolic and elliptic planes are naturally embedded into p ⊥ . The projection ofthe intersection curve onto p ⊥ is a conic with equation (14). Its polar conic is given by (17).For the Euclidean case, see [BST2018]. Proposition 8.3.
The hypercycle base curve
B ∩ C consists of two components which areparametrized in terms of Jacobi elliptic functions by v ± ( u ) = " √ εα cn( u, k ) , p εβ sn( u, k ) , α √ εα dn( u, k ) , ± , (18) for u ∈ R , where the modulus k is given by k = 1 − β (1 + εα ) α (1 + εβ ) . Alternatively, v ± (ˆ u ) = " α cn(ˆ u, ˆ k ) , β sn(ˆ u, ˆ k ) , , ± √ εα α dn(ˆ u, ˆ k ) , (19) for ˆ u ∈ R , where the modulus ˆ k is given by ˆ k = 1 − α (1 + εβ ) β (1 + εα ) . (20)72 roof. Using the elementary identities [NIST, WW1927]cn + sn = 1 , dn + k sn = 1 , one easily checks that, e.g., the parametrization (19) with (20) satisfies the two equations (14)and (16). The two parametrizations are related by the real Jacobi transformationscd( u, k ) = cn(ˆ u, ˆ k ) , sd( u, k ) = 1 √ − k sn(ˆ u, ˆ k ) , nd( u, k ) = dn(ˆ u, ˆ k ) . where ˆ u = p − k u, ˆ k = k k − , cd = cndn , sd = sndn , nd = 1dn . Remark . (i) From (15) we find 0 < k <
1, or equivalently ˆ k <
0, and thus v ± attains real values for u, ˆ u ∈ R .(ii) Over the complex numbers the intersection curve B ∩ C is connected and constitutes anembedding of an elliptic curve, i.e., a torus. The two real components are related by v ± ( u ) = v ∓ (2 i K ( k ) − u ) , (21)where K ( k ) and K ( k ) = K ( √ − k ) are the quarter periods of the Jacobi elliptic functions.(iii) The signs in the parametrizations of the two compontents are chosen such that points onthe different components with the same argument u represent the same line with oppositeorientation v ± ( u ) = σ p ( v ∓ ( u )) . (iv) The hypercycle base curves treated in Section 8.5.1 and 8.5.2 are all projectively equivalentfor different values of ε . Thus, their parametrizations may all be obtained from, e.g., (18)with ε = 0 by reinterpreting another quadric of the pencil as the Laguerre quadric andapplying a suitable projective transformation.This parametrization features the following remarkable property which is related to theaddition on elliptic curves (cf. [Hus1987]). Proposition 8.4. (i) Let u, ˜ u, s ∈ R . Then the four points v + ( u ) , v − ( u + s ) , v − (˜ u ) , v + (˜ u + s ) are coplanar (seeFigure 36, left).(ii) Let s ∈ R . Then the lines v + ( u ) ∧ v − ( u + s ) with u ∈ R constitute one family of rulingsof a common hyperboloid in the pencil B ∧ C (1 + λα ) x + (1 + λβ ) x + ( ε − λ ) x − x = 0 (22) given by λ ( s ) = 1 β cs ( s , k ) + ε ns ( s , k ) , where cs = cnsn , ns = 1sn . (23) The second family of rulings is given by the lines v + ( u ) ∧ v − ( u − s ) with u ∈ R . C v + ( u ) v + (˜ u + s ) v − (˜ u ) v − ( u + s ) λ C , ∞ − α − β ε ( − + + − )( − − + − )(+ + −− ) (+ + + − ) Figure 36.
Left:
Four coplanar points on a hypercycle base curve
B ∩ C . The two lines are rulingsfrom a common hyperboloid in the pencil corresponding to
B ∩ C . Right:
The parameter λ for thepencil B ∧ C as given by (22). The four values − β , − α , ε, ∞ correspond to the degenerate quadricsin the pencil. In between, the signature of the quadrics from the pencil are given. The function (23)takes values in [ ε, ∞ ] and corresponds to hyperboloids whose rulings intersect both components ofthe base curve. Proof. (i) By (21) we obtaindet ( v + ( u ) , v − ( u + s ) , v − (˜ u ) , v + (˜ u + s ))= det (cid:0) v + ( u ) , v + (2 i K ( k ) − u − s ) , v + ( − i K ( k ) − ˜ u ) , v + (˜ u + s ) (cid:1) which is zero due to the following addition theorem for Jacobi elliptic functions [NIST]: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) cn( z , k ) sn( z , k ) dn( z , k ) 1cn( z , k ) sn( z , k ) dn( z , k ) 1cn( z , k ) sn( z , k ) dn( z , k ) 1cn( z , k ) sn( z , k ) dn( z , k ) 1 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = 0 , z + z + z + z = 0 . (24)(ii) By Lemma 8.2, there exists a unique hyperboloid Q in the pencil B ∧ C containing the line v + ( u ) ∧ v − ( u + s ). If we denote by h· , ·i C the symmetric bilinear form corresponding tothe quadratic form (14) of the cone C , the parameter λ corresponding to Q is given by h v + ( u ) , v − ( u + s ) i + λ h v + ( u ) , v − ( u + s ) i C = 0 , which is equivalent to (25) with ρ cn = 1 + λα εα , ρ sn = 1 + λβ εβ , ρ dn = ( ε − λ ) α εα , ρ = 1 . Thus, by Lemma 8.4, one obtains(1 + λα ) cn( s ) + ( ε − λ ) α dn( s ) + 1 + εα = 0 , (1 + λβ ) cn( s ) + ( ε − λ ) β + (1 + εβ ) dn( s ) = 0 . These two equations for λ are equivalent and give λ = 1 β cn( s ) + dn( s )1 − cn( s ) + ε s )1 − cn( s ) = 1 β cs ( s ) + ε ns ( s ) . s and − s correspond to the same hyperboloid since λ ( s ) = λ ( − s ). Given the line L = v + ( u ) ∧ v − ( u + s ), all lines v + (˜ u ) ∧ v − (˜ u − s ) with ˜ u ∈ R intersect the line L by (i), andthus belong to the respective other (and therefore the same) family of rulings of Q . Remark . (i) The alternative expression for λ in terms of ˆ s = √ − k s and ˆ k = k k − is given by λ (ˆ s ) = 1 α cs ( ˆ s , ˆ k ) + ε ns ( ˆ s , ˆ k ) . (ii) The four degenerate quadrics from the pencil (22) are given by the values λ = − β , − α , ε, ∞ ,where λ = ∞ corresponds to the cone C (see Figure 36, right). By construction, the hyper-boloids obtained by (23) have rulings connecting the two components of the base curve,which corresponds to the fact that ε ≤ λ ( s ) ≤ ∞ for s ∈ R with λ (0) = ∞ and λ (2 K ( k )) = ε . Lemma 8.4.
Let s ∈ R and ρ cn , ρ sn , ρ dn , ρ ∈ R such that ρ cn cn( u ) cn( u + s ) + ρ sn sn( u ) sn( u + s ) + ρ dn dn( u ) dn( u + s ) + ρ = 0 (25) for all u ∈ R . Then ρ cn cn( s ) + ρ dn dn( s ) + ρ = 0 ,ρ sn cn( s ) + ρ dn (1 − k ) + ρ dn( s ) = 0 . (26) Proof.
Applying the addition theorems for Jacobi elliptic functions [NIST] to (25), the resultingequation can be written as a sum of three independent functions, say, sn ( u ), sn( u ) cn( u ) dn( u ),and 1. The vanishing of the coefficients of these three functions leads to three linear equationsin ρ cn , ρ sn , ρ dn , ρ , which constitute a rank 2 linear system equivalent to (26).This allows to parametrize a checkerboard incircular net tangent to a given ellipse in thefollowing way (see Figures 37, 38). Theorem 8.9.
Let ε ∈ {− , , } . Then for α > β > with εα , εβ > , s, ˜ s ∈ R , and u ‘ , u m ∈ R the two families of lines ( ‘ i ) i ∈ Z and ( m j ) j ∈ Z given by ‘ k = v + ( u ‘ + k ( s + ˜ s )) ‘ k +1 = v − ( u ‘ + k ( s + ˜ s ) + s ) m l = v − ( u m + l ( s + ˜ s )) m l +1 = v + ( u m + l ( s + ˜ s ) + s ) constitute a hyperbolic/elliptic/Euclidean checkerboard incircular net (according to the value of ε )tangent to the ellipse x α + x β − x = 0 . Proof.
The existence of incircles follows from Proposition 8.4, while tangency to the given ellipsefollows from Proposition 8.2.The choice of (cid:73) α and β determines the ellipse, 75 s and ˜ s determines two hyperboloids in the pencil of quadrics, and further allows to distin-guish the two families of rulings on each of them, (cid:73) u v0 , u h0 determines one initial line tangent to the ellipse in each of the two families of lines.Note that for s = 0 the corresponding hyperboloid degenerates to the cone C . In this case, ‘ k = ‘ k +1 , m l = m l +1 , and thus, ( ‘ i ) i ∈ Z and ( m j ) j ∈ Z constitutes an “ordinary” incircular net.Periodicity can be achieved by setting s + ˜ s = 4 K ( k ) N .
To achieve “embeddedness” of the checkerboard incircular nets one has to additionally demandthat the two different families of lines agree (up to their orientation), e.g., ‘ i = σ p ( m i ) , which is obtained by setting u ‘ = u m . Figure 37.
Periodic checkerboard incircular net in elliptic geometry ( ε = 1) tangent to an ellipsewith α = 0 . β = 0 . N = 11, and s = 0 .
23 ( top ) / s = 0 ( bottom ). Represented on the Laguerrequadric ( left ) and on the sphere model of elliptic geometry ( right ). igure 38. Periodic checkerboard incircular net in hyperbolic geometry ( ε = −
1) tangent to anellipse with α = 0 . β = 0 . N = 11, and s = 0 .
23 ( top ) / s = 0 ( bottom ). Represented on theLaguerre quadric ( left ) and on the Poincar´e disk model of hyperbolic geometry ( right ). Consider a cone C given by α x − β x − x = 0 (27)with α, β > , εα , − εβ > . It intersects the Laguerre quadric B given by x + x + εx − x = 0in the hypercycle base curve B ∩ C (see Figure 39).
Remark . In the elliptic plane all generic conics are ellipses. Correspondingly, for ε = 1 thecase (27) is equivalent to (14). Proposition 8.5.
The hypercycle base curve
B ∩ C corresponds to the (oriented) tangent linesof a hyperbola given in homogeneous coordinates of the hyperbolic/Euclidean plane by x α − x β − x = 0 . hyp C B euc C Figure 39.
Hypercycle base curve
B ∩ C for a hyperbola in hyperbolic ( left ), and Euclidean ( right )Laguerre geometry.
Proposition 8.6.
The intersection curve
B∩C consists of two components which are parametrizedin terms of Jacobi elliptic functions by v ± ( u ) = " √ εα dn( u, k ) , α p α + β sn( u, k ) , α √ εα cn( u, k ) , ± , for u ∈ R , where the modulus is given by k = α (1 − εβ ) α + β . Alternatively v ± (ˆ u ) = " α nc(ˆ u, ˆ k ) , β sc(ˆ u, ˆ k ) , , ± √ εα α dc(ˆ u, ˆ k ) , for ˆ u ∈ R , where the modulus is given by ˆ k = − α (1 − εβ ) β (1 + εα ) . Remark . (i) The two parametrizations are related by the real Jacobi transformationsdc( u, k ) = nc(ˆ u, ˆ k ) , sc( u, k ) = 1 √ − k sc(ˆ u, ˆ k ) , nc( u, k ) = dc(ˆ u, ˆ k ) . where ˆ u = p − k u, ˆ k = k k − , dc = dncn , sc = sncn , nc = 1cn . (ii) All points from Remark 8.12 also apply to this parametrization. Proposition 8.7. (i) Let u, ˜ u, s ∈ R . Then the four points v + ( u ) , v − ( u + s ) , v − (˜ u ) , v + (˜ u + s ) are coplanar (seeFigure 40, left).(ii) Let s ∈ R . Then the lines v + ( u ) ∧ v − ( u + s ) with u ∈ R constitute one family of rulingsof a common hyperboloid in the pencil B ∧ C (1 + λα ) x + (1 − λβ ) x + ( ε − λ ) x − x = 0 (28) given by λ ( s ) = − β cs ( s , k ) − α ns ( s , k ) , where cs = cnsn , ns = 1sn . (29) The second family of rulings is given by the lines v + ( u ) ∧ v − ( u − s ) with u ∈ R . C v + ( u ) v + (˜ u + s ) v − (˜ u ) v − ( u + s ) λ C , ∞ ε − α β − )( − + + − )(+ − −− ) (+ + −− ) Figure 40.
Left:
Four coplanar points on a hypercycle base curve
B ∩ C . The two lines are rulingsfrom a common hyperboloid in the pencil corresponding to
B ∩ C . Right:
The parameter λ for thepencil B ∧ C as given by (28). The four values − α , ε, β , ∞ correspond to the degenerate quadricsin the pencil. In between, the signature of the quadrics from the pencil are given. The function (29)takes values in [ −∞ , − α ] and corresponds to hyperboloids whose rulings intersect both componentsof the base curve. Remark . (i) The alternative expression for λ in terms of ˆ s = √ − k s and ˆ k = k k − is given by λ (ˆ s ) = − ε cs ( ˆ s , ˆ k ) − α ns ( ˆ s , ˆ k ) . (ii) The four degenerate quadrics from the pencil (28) are given by the values λ = − α , ε, β , ∞ ,where λ = ∞ corresponds to the cone C (see Figure 40, right). By construction, the hyper-boloids obtained by (29) have rulings connecting the two components of the base curve,which corresponds to the fact that −∞ ≤ λ ( s ) ≤ − α for s ∈ R with λ (0) = ∞ and λ (2 K ( k )) = − α .This allows to parametrize a checkerboard incircular net tangent to a given hyperbola in thefollowing way (see Figure 41). Theorem 8.10.
Let ε ∈ {− , } . Then for α, β > with εα , − εβ > , s, ˜ s ∈ R , and u ‘ , u m ∈ R the two families of lines ( ‘ i ) i ∈ Z and ( m j ) j ∈ Z given by ‘ k = v + ( u ‘ + k ( s + ˜ s )) ‘ k +1 = v − ( u ‘ + k ( s + ˜ s ) + s ) m l = v − ( u m + l ( s + ˜ s )) m l +1 = v + ( u m + l ( s + ˜ s ) + s ) constitute a hyperbolic/Euclidean checkerboard incircular net (according to the value of ε ) tangentto the hyperbola x α − x β − x = 0 . emark . Periodicity and “embeddedness” are achieved as described in Section 8.5.1.
Figure 41.
Periodic checkerboard incircular net in hyperbolic geometry ( ε = −
1) tangent to ahyperbola with α = 0 . β = 0 . N = 11, and s = 0 .
23 ( top ) / s = 0 ( bottom ). Represented onthe Laguerre quadric ( left ) and on the Poincar´e disk model of hyperbolic geometry ( right ). According to Remark 8.4 a checkerboard incircular net possesses more incircles than immediatefrom its definition. The full symmetry of such a net is revealed when considering all these circlesand dividing its two families of lines ( ‘ i ) i ∈ Z , ( m j ) j ∈ Z into four families ν (1) k = ‘ k , ν (2) k = ‘ − k +1 , ν (3) k = m k , ν (4) k = m − k +1 , for k , k , k , k ∈ Z . Remark . The decomposition into four families of lines also seems natural after consideringthe formulas for checkerboard incircular nets given in Theorems 8.9 and 8.10, and more basicallythe identity (24).
Proposition 8.8.
For a checkerboard incircular net each quadrilateral ( ν (1) k , ν (2) k , ν (3) k , ν (4) k ) with k + k + k + k = 0 , k , k , k , k ∈ Z (30) is circumscribed.Proof. This is a reformulation of the statement given in Remark 8.4, which describes the wholecollection of incircles of a checkerboard incircular net.80 igure 42.
Left:
Two adjacent octahedra from an octahedral grid of planes. These correspondto the geometric configuration shown in Figure 24.
Right:
Octahedral grid of planes correspondingto an Euclidean incircular net in the cyclographic model. All planes are tangent to the red conics,which are the degenerate dual quadrics in a dual pencil.
From (30) we find that the collection of incircles of a checkerboard incircular net is naturallyassigned to the points of an A root-system (vertices of a tetrahedral-octahedral honeycomblattice , see Figure 42, left), where A = n ( k , k , k , k ) ∈ Z (cid:12)(cid:12)(cid:12) k + k + k + k = 0 o . This correspondence can also be made geometric. To this end we identifying the four families oflines ( ν ( i ) k i ) k i ∈ Z , i = 1 , , , B and denoteits polar planes by P ( i ) k i = ( ν ( i ) k i ) ⊥ (or, in the Euclidean case, its dual planes by P ( i ) k i = ( ν ( i ) k i ) ∗ inthe cyclographic model). Proposition 8.9.
The four families of planes ( P ( i ) k i ) k i ∈ Z , i = 1 , , , corresponding to a checker-board incircular net constitute an octahedral grid of planes , i.e., for each k + k + k + k = 0 , k , k , k , k ∈ Z the four planes P (1) k , P (2) k , P (3) k , P (4) k intersect in a point (see Figure 42, right, and cf. [ABST2019]).Remark . Generally, octahedral grids of planes have the property that all its planes aretangent to all quadrics of a dual pencil, or equivalently, to a certain developable surface (cf.[Bla1928, Sau1925]). In the case of checkerboard incircular nets this property is polar (or dual)to the property, that all the points ν ( i ) k i lie on the hypercycle base curve. For the Euclidean caseof incircular nets this fact was already employed by B¨ohm in [B¨oh1970].Denote the intersection points of the octahedral grid of planes by c a = P (1) k ∩ P (2) k ∩ P (3) k ∩ P (4) k , a = ( k , k , k , k ) ∈ A . By polarity (or duality) the points c a correspond to the incircles of the checkerboard incircularnet. We may now extend the statement from Corollary 8.2 to all “diagonal surfaces” of A . Proposition 8.10.
For an octahedral grid of planes corresponding to a checkerboard incircularnet, the points of intersection c a , c a , c a , c a with a + a + a + a = 0 , a , a , a , a ∈ A lie on a quadric from the dual pencil of quadrics which is polar (or dual in the Euclidean case)to the pencil of quadrics corresponding to the hypercycle base curve. emark . In the case of an “ordinary” incircular net this implies that the intersection pointsof its lines lie on conics which are confocal with the touching conic [B¨oh1970], [AB2018].82
Euclidean cases
The cases of Euclidean geometry and Euclidean Laguerre geometry, which we have excluded fromour general discussion, are induced by degenerate quadrics, see, e.g., [Kle1928, Bla1929, Gie1982].For a degenerate quadric
Q ⊂ R P n , polarity (see Section 3.3) does no longer define a bijectionbetween the set of points and the set of hyperplanes. Instead one can apply the concept of duality . A.1 Duality
The n -dimensional dual real projective space is given by( R P n ) ∗ := P (cid:16) ( R n +1 ) ∗ (cid:17) , where (cid:0) R n +1 (cid:1) ∗ is the space of linear functionals on R n +1 . We identify ( R P n ) ∗∗ = R P n in thecanonical way, and obtain a bijection between projective subspaces U = P( U ) ⊂ R P n and their dual projective subspaces U ∗ := { y ∈ ( R P n ) ∗ | y ( x ) = 0 for all x ∈ U } , satisfying dim U + dim U ∗ = n − . Every projective transformation f : R P n → R P n ∈ PGL( n + 1) induces a dual projectivetransformation f ∗ : ( R P n ) ∗ → ( R P n ) ∗ ∈ PGL( n + 1) ∗ such that f ( U ) ∗ = f ∗ ( U ∗ )for every projective subspace U ⊂ R P n . Introduce a basis on R n +1 , say the conical basis, andits dual basis on ( R n +1 ) ∗ . Then, if F ∈ R ( n +1) × ( n +1) is a matrix representing the transformation f = [ F ], a matrix F ∗ ∈ R ( n +1) × ( n +1) representing the dual transformation f ∗ = [ F ∗ ] is given by F ∗ := F − (cid:124) . (31)For a quadric Q ⊂ R P n its dual quadric Q ∗ ⊂ ( R P n ) ∗ may be defined as the set of pointsdual to the tangent hyperplanes of Q . Example A.1. (i) For a non-degenerate quadric
Q ⊂ R P n of signature ( r, s ) its dual quadric Q ∗ ⊂ ( R P n ) ∗ is non-degenerate with the same signature.(ii) For a cone Q ⊂ R P n of signature ( r, s,
1) with vertex v ∈ Q , its dual quadric Q ∗ ⊂ ( R P n ) ∗ consists of the set of points on a lower dimensional quadric of signature ( r, s ) contained inthe hyperplane v ∗ ⊂ ( R P n ) ∗ . A.2 Euclidean geometry
Let h· , ·i be the standard degenerate bilinear form of signature ( n, , h x, y i := x y + . . . + x n y n for x, y ∈ R n +1 . The corresponding quadric C is an imaginary cone (cf. Example 3.1 (iv)). Itsreal part consisting only of one point, the vertex of the cone: m ∞ ∈ R P n , m ∞ := e n +1 = (0 , . . . , , . C − = ∅ is empty, the set E ∗ := C + = R P n \ { m ∞ } consists of the whole projective space but one point, which we identify with the n -dimensional dual Euclidean space , i.e., the space of Euclidean hyperplanes.While in the projective models of hyperbolic/elliptic geometry, we were able to identifycertain points with hyperplanes in the same projective space by polarity, this is not possible inthe projective model of Euclidean geometry due to the degeneracy of the absolute quadric C .Instead, by duality, every point m ∈ E ∗ corresponds to a hyperplane m ∗ ⊂ E in E := ( R P n ) ∗ \ ( m ∞ ) ∗ ’ R n , which we identify with the n -dimensional Euclidean space . The hyperplane ( m ∞ ) ∗ is called the hyperplane at infinity .For two points k , m ∈ E ∗ one always has 0 ≤ K C ( k , m ) ≤
1, and the Euclidean angle α , orequivalently its conjugate angle π − α , between the two hyperplanes k ∗ , m ∗ ⊂ E is given by K C ( k , m ) = cos α ( k ∗ , m ∗ ) . The two planes are parallel if the line k ∧ m contains the point m ∞ .The dual quadric C ∗ of the absolute cone can be identified with an imaginary quadric inthe hyperplane at infinity ( m ∞ ) ∗ of signature ( n,
0) (cf. Example A.1 (ii)). Since this does notinduce a bilinear form on ( R P n ) ∗ , the Cayley-Klein distance is not well-defined on E . Yet theEuclidean distance may still be recovered in this setup, e.g., as the limit of the Cayley-Kleindistance of hyperbolic/elliptic space [Kle1928, Gun2011]. One may avoid these difficulties bytreating Euclidean geometry as a subgeometry of M¨obius geometry (see Section A.3).We employ the following normalization for the dual Euclidean space( E n ) ∗ := n m ∈ R n +1 (cid:12)(cid:12)(cid:12) h m, m i = 1 o = n ( b m, − d ) ∈ R n +1 (cid:12)(cid:12)(cid:12) b m ∈ R n , d ∈ R , b m · b m = 1 o , where b m · b m denotes the standard scalar product on R n . Upon the (non-canonical) identification( R n +1 ) ∗ ’ R n +1 , by identifying the canonical basis of ( R n +1 ) ∗ with the dual basis of the canonicalbasis of R n +1 , we introduce the following normalization for the Euclidean space. E n := n x ∈ ( R n +1 ) ∗ (cid:12)(cid:12)(cid:12) x ( m ∞ ) = 1 o ’ n ( b x, ∈ R n +1 (cid:12)(cid:12)(cid:12) b x ∈ R n o . Then P( E n ) = E is an embedding and P(( E n ) ∗ ) = E ∗ a double cover. The double cover may beused to encode the orientation of the corresponding Euclidean plane.In this normalization the Euclidean distance of two points x , y ∈ E , x, y ∈ E n is given by | x − y | = d ( x , y ) . The Euclidean hyperplane corresponding to a point m ∈ E ∗ , m = ( b m, − d ) ∈ ( E n ) ∗ is given by { x ∈ E | h m, x i = 0 } = P ( { ( b x, ∈ E n | b m · b x } = d ) , while the formula for the angle between two Euclidean planes k ∈ E ∗ , k = ( b k, − c ) ∈ ( E n ) ∗ and m ∈ E ∗ , m = ( b m, − d ) ∈ ( E n ) ∗ becomes h k, m i = b k · b m = cos α ( k ∗ , m ∗ ) , where the intersection angle and its conjugate angle can be distinguished now. Finally, thesigned distance of a point x ∈ E , x = ( b x, ∈ E n and a plane m ∈ E ∗ , ( b m, − d ) ∈ ( E n ) ∗ is givenby h m, x i = b m · b x − d = d ( x , m ∗ )84 B = b ⊥ q ⊥ q = [ e ∞ ] k b [ e ] σ q , b ( x ) x ‘ ∞ Figure 43.
Stereographic projection from B to S through the point q . Every point k ∈ q ⊥ corresponds to a hyperplane in B . The transformation group induced by the absolute quadric C on the dual Euclidean space E ∗ is given by PO( n, , A ] = b A b a (cid:124) ε ∈ PO( n, , , where b A ∈ O ( n ) , b a ∈ R n , ε = 0. Thus, its dual transformations, see (31), are given by[ A − (cid:124) ] = b A − b A b a ε − ∈ PO( n, , ∗ . They act on E as the group of similarity transformations , i.e., Euclidean motions and scalings. A.3 Euclidean geometry from M¨obius geometry
In Section 5 we have excluded the choice of a point q ∈ Q on the quadric, since the projection π q (see Definition 5.1) to the polar hyperplane q ⊥ is not well-defined in that case. Yet most ofthe constructions described still apply if we project to any other hyperplane instead. We showthis in the example of recovering Euclidean (similarity) geometry from M¨obius geometry.Thus, let h· , ·i be the standard non-degenerate bilinear form of signature ( n + 1 , h x, y i := x y + . . . + x n +1 y n +1 − x n +2 y n +2 for x, y ∈ R n +2 , and denote by S ⊂ R P n +1 the corresponding M¨obius quadric . Let q ∈ S be apoint on the M¨obius quadric, w.l.o.g., q := [ e ∞ ] , e ∞ := ( e n +1 + e n +2 ) = (0 , . . . , , , ) . While q ⊥ is the tangent plane of S at q , we choose a different plane B for the projection, w.l.o.g., B := b ⊥ , b := e n +1 = (0 , . . . , , , , B to S through the point q , which is also called stereographic projection (see Figure 43). To this end, denote by [ e ] the intersection point of theline q ∧ b with S , where e := ( e n +2 − e n +1 ) = (0 , . . . , , − , ) . Then we have h e , e i = h e ∞ , e ∞ i = 0 , h e , e ∞ i = − , and h e , e i i = h e ∞ , e i i = 0 for i = 1 , . . . , n , and the vectors e , . . . , e n , e , e ∞ constitute a basisof R n +1 , . Proposition A.1.
Let ‘ ∞ := B ∩ q ⊥ . The stereographic projection from B \ ‘ ∞ to S \ q throughthe point q is given by the map σ q , b : x = [˜ x + e − e ∞ ] [˜ x + e + | ˜ x | e ∞ ] , where ˜ x ∈ span { e , . . . , e n } .Proof. First note that a point in x ∈ B \ ‘ ∞ may be normalized to x = ˜ x + e − e ∞ The (second)intersection point of the line q ∧ x with Q is then given by − h x, e ∞ i x + h x, x i e ∞ = x + ( | ˜ x | − e ∞ = ˜ x + e + | ˜ x | e ∞ . Now the Euclidean metric on B may be recovered from the bilinear form corresponding to S by observing that h x, y i = D ˜ x + e + | ˜ y | e ∞ , ˜ y + e + | ˜ y | e ∞ E = − | ˜ x − ˜ y | . Remark
A.1 . To obtain the Euclidean metric in a projectively well-defined way one can start byconsidering the quantity h x, y ih e ∞ , x i h e ∞ , y i , similar to Definition B.1. Though not being invariant under different choices of homogeneouscoordinate vectors for the point q = [ e ∞ ], the quotient of two such expressions is. This fits thefact that it is not actually Euclidean geometry that we are recovering but similarity geometry.The restriction of the M¨obius quadric S to the tangent hyperplane q ⊥ yields a quadric ofsignature ( n, , S , every point k ∈ q ⊥ correspondsto a hyperplanar section of S containing the point q , i.e., an S -sphere through q , which is, inturn, mapped to a hyperplane of B by stereographic projection. The Cayley-Klein distance oftwo points in the tangent hyperplane yields the Euclidean angle between the two correspondinghyperplanes of B . The group of M¨obius transformations fixing the point q induces the group ofdual similarity transformations on BMob q = PO( n + 1 , q ’ PO( n, , . P n E ∗ B euc p x π p ( x ) σ p ( x ) m ∞ G ( c ) ( R P n ) ∗ E B ∗ euc ‘ c G ( c ) ∗ x ∗ Figure 44.
Euclidean Laguerre geometry. The corresponding Laguerre quadric B euc is a cone(“Blaschke cylinder”) with its vertex corresponding to the point m ∞ that represents the line atinfinity in the dual Euclidean plane E ∗ . Under dualization the Laguerre quadric becomes a conic B ∗ euc in the cyclographic model of Laguerre geometry. A point x ∈ B euc represents an oriented line ‘ in the Euclidean plane E . By dualization the point becomes a plane x ∗ that touches the conic B ∗ euc and intersects E in the line ‘ . A planar section G ( c ) of B euc represents an oriented circle c . Bydualization it becomes a cone G ( c ) ∗ that contains the conic B ∗ euc and intersects E in the circle c . A.4 Euclidean Laguerre geometry
In the spirit of Sections 5 and 6 the absolute quadric
C ⊂ R P n of the dual Euclidean (similarity)space with signature ( n, ,
1) can be lifted to a quadric B euc ⊂ R P n +1 of signature ( n, , Euclidean Laguerre quadric , or classically the
Blaschke cylinder . The groupof
Euclidean Laguerre transformations is given by
Lag euc = PO( n, , . For a point p with h p, p i <
1, w.l.o.g., p := [0 , · · · , , , σ p and projection π p (see Definition 5.1) are still well-defined, and the quotient( Lag euc ) p (cid:30) σ p ’ PO( n, , π p restricted to B euc realizes a double cover of the dual Euclidean space C + = E ∗ , which may be interpreted as carrying the information of the orientation of thecorresponding hyperplanes in E . The involution σ p plays again the role of orientation reversion(see Figure 44, left).The hyperplanar sections of B euc ⊂ R P n +1 correspond to (the tangent hyperplanes) of aEuclidean sphere in E . Yet due to the degeneracy of B euc they cannot be identified with (polar)points in the same space. Instead they can be identified with points in the dual space ( R P n +1 ) ∗ ,which is classically called the cyclographic model of Laguerre geometry (see Figure 44, right).The dual quadric B ∗ euc is given by a lower dimensional quadric of signature ( n,
1) contained in thehyperplane m ∗∞ . Thus, the cyclographic model is isomorphic to ( n + 1)-dimensional Minkowskispace . 87 .5 Lie geometry in Euclidean space
A Euclidean model of Lie geometry is obtained by stereographic projection of the point complex
S ⊂ L (cf. Section 7).We write the bilinear form corresponding to the Lie quadric as h x, y i := b x · b y − x n +2 y n +2 − x n +3 y n +3 = n +1 X i =1 x i y i − x n +2 y n +2 − x n +3 y n +3 for x, y ∈ R n +3 , where b · : R n +3 → R n +1 , ( x , . . . , x n +3 ) ( x , . . . , x n +1 ) . The point complex S is projectively equivalent to S n . Indeed, p ⊥ = (cid:8) x ∈ R P n +2 (cid:12)(cid:12) x n +3 = 0 (cid:9) ’ R P n +1 , and for a point x = [ b x, , ∈ p ⊥ we find that in affine coordinates ( x n +2 = 1) h x, x i = 0 ⇔ b x · b x = 1 . Thus, we obtain the identification S = n x ∈ p ⊥ (cid:12)(cid:12)(cid:12) h x, x i = 0 o ’ nb x ∈ R n +1 (cid:12)(cid:12)(cid:12) b x · b x = 1 o = S n . We embed the sphere S n into the light cone L n +1 , = n x ∈ R n +3 (cid:12)(cid:12)(cid:12) h x, x i = 0 o in the following way σ S N : S n , → L n +1 , , b x b x + e n +2 + 0 · e n +3 . Then we have S = P( σ S N ( S n )), where P acts one-to-one on the image of σ S N .Denote e ∞ := ( e n +2 + e n +1 ) , e := ( e n +2 − e n +1 ) , which are homogeneous coordinate vectors for the north pole and south pole of S ’ S n respec-tively. They satisfy h e , e i = h e ∞ , e ∞ i = 0 , h e , e ∞ i = − , and h e , e i i = h e ∞ , e i i = 0 for i = 1 , . . . , n, n + 3. The vectors e , . . . , e n , e , e ∞ , e n +3 constitutea basis of R n +1 , . We define an embedding of R n into the light cone L n +1 , by the map σ R N : R n , → L n +1 , , ˜ x ˜ x + e + | ˜ x | e ∞ + 0 · e n +3 and recognize that upon renormalizing the ( n + 2)-nd coordinate to 1 this is nothing but stere-ographic projection from R n onto the sphere S n , i.e.( σ S N ) − ◦ σ R N : R n → S n , ˜ x x | ˜ x | + 1 , − | ˜ x | | ˜ x | ! , and S = P ( σ S N ( S n )) = P( σ R N ( R n )) ∪ { [ e ∞ ] } . Every point s ∈ L with s = 0 can be representedby s = ˜ s + e + ( | ˜ s | − r ) e ∞ + re n +3 with ˜ s ∈ R n and r ∈ R . Then for x = ˜ x + e + | ˜ x | e ∞ we find h s, x i = 0 ⇔ | ˜ s − ˜ x | = r . Thus, we may identify the point s with the oriented Euclidean hypersphere of R n with center ˜ s and signed radius r ∈ R . Analogously a point n ∈ L with n = 0 may be represented by n = ˜ n + 0 · e + 2 de ∞ + e n +3 and identified with the oriented hyperplane of R n with normal ˜ n ∈ S n − and signed distance d ∈ R to the origin. 88 uclidean geometry Lie geometry point ˜ x ∈ R n [˜ x + e + | x | e ∞ + 0 · e n +3 ]= h ˜ x, −| ˜ x | , | ˜ x | , i ∈ L oriented hypersphere with center ˜ s ∈ R n and signed radius r ∈ R [˜ s + e + ( | ˜ s | − r ) e ∞ + re n +3 ]= h ˜ s, −| ˜ s | + r , | ˜ s | − r , r i ∈ L oriented hyperplane h ˜ n, ˜ x i = d ,with normal ˜ n ∈ S n − and signed distance d ∈ R [˜ n + 0 · e + 2 de ∞ + e n +3 ]= [˜ n, − d, d, ∈ L Table 7.
Correspondence between the geometric objects of Lie geometry in Euclidean space andpoints on the Lie quadric.
Proposition A.2.
Under the aforementioned identification two oriented hyperspheres/hyperplanesof Euclidean space are in oriented contact if and only if the corresponding points on the Liequadric are Lie orthogonal.Proof.
For, e.g., two oriented hyperspheres of R n represented by homogeneous coordinate vectors s i = ˜ s i + e + ( | ˜ s i | − r i ) e ∞ + r i e n +3 , i = 1 , h s , s i = 0 ⇔ | ˜ s − ˜ s | = ( r − r ) . The condition n = 0, which characterizes the oriented hyperplanes among all oriented hy-perspheres, is equivalent to h n, e ∞ i = 0. Thus, we can interpret oriented hyperplanes as orientedhyperspheres containing the point q := [ e ∞ ]. Similar to the point complex (see Definition 7.3),we may introduce the Euclidean plane complex (cf. Definition 7.5)
L ∩ q ⊥ ’ B euc (32)representing all oriented hyperplanes of R n . The Euclidean plane complex is a parabolic spherecomplex (see Definition 7.4). Its signature is given by ( n, , q ⊥ of all Lie transformations that fixthe point q Lie q ’ Lag euc . Generalized signed inversive distance
While two points x , y ∈ Q on a quadric Q ⊂ R P n +1 with h x, y i 6 = 0 possess no projectiveinvariant, the additional choice of a fixed point q ∈ R P n +1 \ Q allows for the definition of suchan invariant. It is closely related to the Cayley-Klein distance under the projection from thepoint q .A special case is given by a signed version of the classical inversive distance introducedCoxeter [Cox1971], which generalizes the intersection angle of spheres. It can be used for ageometric description of sphere complexes in Lie geometry. B.1 Invariant on a quadric induced by a point
Definition B.1.
Let q ∈ R P n +1 \ Q . Then we call I Q , q ( x , y ) := 1 − h x, y i h q, q ih x, q i h y, q i . the q -distance for any two points x , y ∈ Q . Remark
B.1 . Although we are interested in the q -distance of points on the quadric for now,it can be extended to all of R P n +1 \ q ⊥ . Then the relation between the q -distance and theCayley-Klein distance induced by Q is given by K Q ( x , y ) = (1 − I Q , q ( x , y )) (1 − I Q , q ( x , x ))(1 − I Q , q ( y , y ))for x , y ∈ R P n +1 \ ( Q ∪ q ⊥ ).The q -distance is projectively well-defined, in the sense that it does not depend on the choiceof homogeneous coordinate vectors for the points q , x , and y , and it is invariant under the actionof the group PO( r, s, t ) q : Proposition B.1.
Let q ∈ R P n +1 \ Q . Then the q -distance is invariant under all projectivetransformations that preserve the quadric Q and fix the point q , i.e. I Q , q ( f ( x ) , f ( y )) = I Q , q ( x , x ) for f ∈ PO( r, s, t ) q and x , y ∈ Q . Applying the involution σ q to only one of the arguments of the q -distance results in a changeof sign. Proposition B.2.
Let q ∈ R P n +1 \ Q . Then the q -distance satisfies I Q , q ( σ q ( x ) , y ) = I Q , q ( x , σ q ( y )) = − I Q , q ( x , y ) . for all x , y ∈ Q .Proof. Using Definitions B.1 and 5.1 we obtain I Q , q ( σ q ( x ) , y ) = 1 − h σ q ( x ) , y i h q, q ih σ q ( x ) , q i h y, q i = 1 − h x, y i h q, q i − h x, q i h y, q i− h x, q i h y, q i = h x, y i h q, q ih x, q i h y, q i − − I Q , q ( x , y ) . q -distance is well-defined on the quotient Q / σ q , which,according to Proposition 5.1, can be identified with its projection π q ( Q ) to the plane q ⊥ . In thisprojection the square of the q -distance becomes the Cayley-Klein distance induced by e Q = Q∩ q ⊥ (see Proposition 5.3) I Q , q ( x , y ) = K e Q ( π q ( x ) , π q ( y )) . Hypersurfaces of Q of constant q -distance to a point on Q are hyperplanar sections of Q , i.e.the Q -spheres (see Definition 5.2). Proposition B.3.
The hypersurface in Q of constant q -distance ν ∈ R to a point ˜ x ∈ Q isgiven by the intersection with the polar hyperplane of the point x ∈ R P n +1 , x := h q, q i ˜ x + ( ν − h ˜ x, q i q, i.e. { y ∈ Q | I Q , q ( ˜ x , y ) = ν } = x ⊥ ∩ Q . Proof.
The equation I Q , q ( ˜ x , y ) = 1 − h ˜ x, y i h q, q ih ˜ x, q i h y, q i = ν is equivalent to h x, y i = h q, q i h ˜ x, y i + ( ν − h ˜ x, q i h q, y i = 0 . But are all hyperplanar sections of Q such hypersurfaces (cf. Proposition 5.4)? FollowingProposition B.3 the potential centers of a given planar section x ⊥ ∩ Q are given by the pointsof intersection of the line q ∧ x with the quadric Q . Yet such lines do not always intersect thequadric in real points. Proposition B.4.
Denote by ∆ q ( x ) := h x, q i − h x, x i h q, q i = − h q, q i h x, x i q the quadratic form of the cone of contact C Q ( q ) . Let x ∈ R P n +1 such that x ⊥ ∩ Q 6 = ∅ . (cid:73) If ∆ q ( x ) > , then the line q ∧ x intersects the quadric Q in two (real) points, and x ⊥ ∩ Q = { y ∈ Q | I Q , q ( x ± , y ) = ν ± } with x ± = h q, q i x + (cid:16) − h x, q i ± √ ∆ (cid:17) q, ν ± := ± h x, q i√ ∆ . (cid:73) If ∆ q ( x ) < , then the line q ∧ x intersects the quadric Q in two complex conjugate points,and x ⊥ ∩ Q = { y ∈ Q | I Q , q ( x ± , y ) = ν ± } with x ± = h q, q i x + (cid:16) − h x, q i ± i √− ∆ (cid:17) q, ν ± := ± h x, q i i √− ∆ . Proof.
The first equality for the quadratic form of the cone of contact follows from Lemma 3.3,while the second equality immediately follows from substituting x = αq + π q ( x ).In the case ∆ q ( x ) = 0 the form of the intersection points x ± follows from Lemma 3.2.Substituting into the q -distance gives, e.g., in the case ∆ q ( x ) > I Q , q ( x ± , y ) = 1 − ( − h x, q i ± √ ∆) h q, q ih x ± , q i = ± h x, q i√ ∆ , where we used h x, q i = ±√ ∆ h q, q i . 91 emark B.2 . The q -distance of two points ˜ x , ˜ y ∈ R P n +1 with x ⊥ ∩ Q 6 = ∅ , which represent two q -spheres with centers x , y ∈ Q and q -radii ν , ν is given by I Q , q ( ˜ x , ˜ y ) = I Q , q ( x , y ) ν ν . Note that the change of the representing center and radius, e.g. x → σ q ( x ), ν → − ν , leavesthe resulting quantity invariant. B.2 Signed inversive distance
We first give a Euclidean definition for the signed inversive distance.
Definition B.2.
The signed inversive distance of two oriented hyperspheres in R n with centers˜ s , ˜ s ∈ R n and signed radii r , r ∈ R is given by I := r + r − | ˜ s − ˜ s | r r . In particular, if the two spheres intersect, it is the cosine of their intersection angle, by the cosinelaw for Euclidean triangles.
Remark
B.3 . This classical invariant is usually given in its unsigned version, which was intro-duced by Coxeter [Cox1971] as a M¨obius invariant.
Proposition B.5.
The signed inversive distance I satisfies (cid:73) I ∈ ( − , ⇔ the two oriented hyperspheres intersect. In this case I = cos ϕ where ϕ ∈ [0 , π ] is the angle between the two oriented hyperspheres. (cid:73) I = 1 ⇔ the two oriented hyperspheres touch with matching orientation (Lie incidence). (cid:73) I = − ⇔ the two oriented hyperspheres touch with opposite orientation. (cid:73) I ∈ ( ∞ , − ∪ (1 , ∞ ) ⇔ the two oriented hyperspheres are disjoint. The signed inversive distance is nothing but the p -distance (see Definition B.1) associatedwith the point complex S ⊂ L in Lie geometry (see Definition 7.3), where p = [0 , · · · , , ∈ R P n +2 . Proposition B.6.
For two oriented hyperspheres represented by s i = [˜ s i + e + ( | ˜ s i | − r i ) e ∞ + r i e n +3 ] , i = 1 , with Euclidean centers ˜ s , ˜ s ∈ R n and signed radii r , r = 0 the p -distance associated with thepoint complex S is equal to the signed inversive distance, i.e. I L , p ( s , s ) = r + r − | ˜ s − ˜ s | r r . Proof.
With the given representation of the hyperspheres we find I L , p ( s , s ) = 1 − h s , s i h p, p ih s , p i h s , p i = 1 + ( r + r − r r ) − | ˜ s − ˜ s | r r = r + r − | ˜ s − ˜ s | r r . Remark
B.4 . Since we have expressed the signed inversive distance in terms of the p -distanceit follows that it is similarly well-defined for two oriented hyperspheres of S n . Furthermore,the signed inversive distance is invariant under all Lie transformations that preserve the pointcomplex S , i.e. all M¨obius transformations. In particular, the intersection angle of spheresis a M¨obius invariant. As follows from Proposition 5.3 the Cayley-Klein distance of M¨obiusgeometry, i.e. the Cayley-Klein distance induced by S onto p ⊥ is the squared inversive distance.92 .3 Geometric interpretation for sphere complexes We now use the inversive distance to give a geometric interpretation for most sphere complexesin Lie geometry (see Definition 7.4). Let again p = [0 , · · · , , ∈ R P n +2 , which distinguishes the point complex S = L ∩ p ⊥ . Proposition B.7.
Let q ∈ R P n +2 , q = p such that the line p ∧ q through p and q intersectsthe Lie quadric in two points, i.e. p ∧ q has signature (+ − ) . Denote by { q + , q − } := ( p ∧ q ) ∩ L the two intersection points of this line with the Lie quadric (the oriented hyperspheres corre-sponding to q + and q − only differ in their orientation).Then the sphere complex corresponding to the point q is given by the set of oriented hy-perspheres that have some fixed constant inversive distance I L , p to the oriented hyperspherecorresponding to q + , or equivalently, fixed constant inversive distance − I L , p to the orientedhypersphere corresponding to q − .In particular, in this case the sphere complex is (cid:73) elliptic if I L , p ∈ ( − , , (cid:73) hyperbolic if I L , p ∈ ( −∞ , − ∪ (1 , ∞ ) , (cid:73) parabolic if I L , p ∈ {− , } .Proof. The two points q ± may be represented by q ± = ˜ q + e + (cid:16) | ˜ q | − R (cid:17) e ∞ ± Re n +3 , with some R = 0, where we assumed that the e -component of q does not vanish. The case with h q, e ∞ i = 0, which corresponds to q ± being planes, may be treated analogously.Now the point q may be represented by q = ˜ q + e + (cid:16) | ˜ q | − R (cid:17) e ∞ + κe n +3 with some κ ∈ R . For any point s ∈ L represented by s = ˜ q + e + (cid:16) | ˜ q | − r (cid:17) e ∞ + re n +3 , we find that the condition to lie on the sphere complex is given by h q, s i = 0 ⇔ h q, s i p = rκ. Thus, the signed inversive distance of q + and s is given by I p ( q + , s ) = 1 − h q + , s i h p, p ih q + , p i h s, p i = h s, q i p rR = κR . The change q + → q − is equivalent to R → − R which leads to I → − I .The distinction of the three types of sphere complexes in terms of the value of the inversivedistance is obtained by observing that h q, q i > , if κ < R , h q, q i < , if κ > R , h q, q i = 0 , if κ = R . emark B.5 . For an elliptic sphere complex the line p ∧ q always has signature (+ − ). Fur-thermore, in this case we have I p ∈ ( − , p ∧ q can have signature (+ − ), ( −− ), or ( − −− ) is given by q = [0 , sin R, cos R ], which describes all oriented hyperspheres of S n with spherical radius R .An example with signature ( −
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