Apollonius surfaces, circumscribed spheres of tetrahedra, Menelaus' and Ceva's theorems in $\SXR$ and $\HXR$ geometries
AApollonius surfaces, circumscribed spheres oftetrahedra, Menelaus’ and Ceva’s theorems in S × R and H × R geometries *Jen ˝o Szirmai Budapest University of Technology andEconomics Institute of Mathematics,Department of GeometryBudapest, P. O. Box: 91, [email protected] 14, 2020
Abstract
In the present paper we study S × R and H × R geometries, which arehomogeneous Thurston 3-geometries. We define and determine the gener-alized Apollonius surfaces and with them define the “surface of a geodesictriangle”. Using the above Apollonius surfaces we develop a procedure todetermine the centre and the radius of the circumscribed geodesic sphereof an arbitrary S × R and H × R tetrahedron. Moreover, we generalizethe famous Menelaus’ and Ceva’s theorems for geodesic triangles in bothspaces. In our work we will use the projective model of S × R and H × R geometries described by E. Moln´ar in [6]. * Mathematics Subject Classification 2010: 53A20, 53A35, 52C35, 53B20.Key words and phrases: Thurston geometries, S × R , H × R geometries, geodesic triangles,circumscribed spheres of tetrahedra in S × R , H × R geometries, Menelaus’ and Ceva’s theorems a r X i v : . [ m a t h . M G ] D ec Jen ˝o Szirmai
The classical definition of the Apollonius circle in the Euclidean plane E is theset of all points of E whose distances from two fixed points are in a constantratio λ ∈ R + . This definition can be extended in a natural way to the Thurstongeometries E , S , H , S × R , H × R , Nil , (cid:94) SL R , Sol . Definition 1.1
The Apollonius surface in the Thurston geometry X is the set ofall points of X whose geodesic distances from two fixed points are in a constantratio λ ∈ R + . Remark 1.2
A special case of Apollonius surfaces is the geodesic-like bisector(or equidistant) surface ( λ = 1) of two arbitrary points of X . These surfaceshave an important role in structure of Dirichlet - Voronoi (briefly, D-V) cells.The D-V-cells are relevant in the study of tilings, ball packing and ball cover-ing. E.g. if the point set is the orbit of a point - generated by a discrete isometrygroup of X - then we obtain a monohedral D-V cell decomposition (tiling) of theconsidered space and it is interesting to examine its optimal ball packing and cov-ering. In -dimensional spaces of constant curvature, the D-V cells have beenwidely investigated, but in the other Thurston geometries S × R , H × R , Nil , Sol , (cid:94) SL R there are few results on this topic.In [10], [11], [12] we studied the geodesic-like equidistant surfaces in S × R , H × R and Nil geometries, and in [16], [23] the translation-like equidistantsurfaces in
Sol and
Nil geometries.
In the present paper, we are interested in Apollonius surfaces, geodesic trian-gles and their surfaces, generalized Menelaus’ and Ceva’s theorems in S × R and H × R spaces [14, 22].In Section 2 we describe the projective model and the isometry group of theconsidered geometries, moreover, we give an overview about its geodesic curves.In Section 3 we study the generallized Apollonius surfaces and their properties inthe considered spaces and using them we define the surfaces of geodesic triangles.In Section 4 we generalize and prove the theorems of Menelaus and Ceva forgeodesic triangles in S × R and H × R spaces.The computation and the proof is based on the projective model of S × R and H × R geometries described by E. Moln´ar in [6]. pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . H × R and S × R spaces E. Moln´ar has shown in [6], that the homogeneous 3-spaces have a unified in-terpretation in the projective 3-sphere PS ( V , V , R ) . In our work we shalluse this projective model of S × R and H × R geometries. The Cartesianhomogeneous coordinate simplex is given by E ( e ) , E ∞ ( e ) , E ∞ ( e ) , E ∞ ( e ) , ( { e i } ⊂ V and with the unit point E ( e = e + e + e + e )) . Moreover, y = c x with < c ∈ R (or c ∈ R \ { } ) defines a point ( x ) = ( y ) of theprojective 3-sphere PS (or that of the projective space P where opposite rays ( x ) and ( − x ) are identified). The dual system { ( e i ) } ⊂ V describes the simplexplanes, especially the plane at infinity ( e ) = E ∞ E ∞ E ∞ , and generally, v = u c defines a plane ( u ) = ( v ) of PS (or that of P ). Thus x u = y v definesthe incidence of point ( x ) = ( y ) and plane ( u ) = ( v ) , as ( x ) I ( u ) also denotes it.Thus S × R can be visualized in the affine 3-space A (so in E ) as well. S × R space In this section we recall the important notions and results from the papers [6],[10], [15], [17], [18].The well-known infinitezimal arc-length square at any point of S × R as fol-lows ( ds ) = ( dx ) + ( dy ) + ( dz ) x + y + z . (2.1)We shall apply the usual geographical coordiantes ( φ, θ ) , ( − π < φ ≤ π, − π ≤ θ ≤ π ) of the sphere with the fibre coordinate t ∈ R . We describe points in theabove coordinate system in our model by the following equations: x = 1 , x = e t cos φ cos θ, x = e t sin φ cos θ, x = e t sin θ. (2.2)Then we have x = x x = x , y = x x = x , z = x x = x , i.e. the usual Cartesiancoordinates. We obtain by [6] that in this parametrization the infinitezimal arc-length square at any point of S × R is the following ( ds ) = ( dt ) + ( dφ ) cos θ + ( dθ ) . (2.3)The geodesic curves of S × R are generally defined as having locally minimal arclength between their any two (near enough) points. The equation systems of theparametrized geodesic curves γ ( t ( τ ) , φ ( τ ) , θ ( τ )) in our model can be determinedby the general theory of Riemann geometry (see [3], [18]). Jen ˝o Szirmai
Then by (2.1-2) we get the equation systems of a geodesic curve in our Eu-clidean model (see [17]): x ( τ ) = e τ sin v cos ( τ cos v ) ,y ( τ ) = e τ sin v sin ( τ cos v ) cos u,z ( τ ) = e τ sin v sin ( τ cos v ) sin u, − π < u ≤ π, − π ≤ v ≤ π . (2.4) Definition 2.1
The distance d S × R ( P , P ) between the points P and P is de-fined by the arc length of the geodesic curve from P to P . Definition 2.2
The geodesic sphere of radius ρ (denoted by S P ( ρ ) ) with centerat the point P is defined as the set of all points P in the space with the condition d S × R ( P , P ) = ρ . Moreover, we require that the geodesic sphere is a simplyconnected surface without selfintersection in S × R . Definition 2.3
The body of the geodesic sphere of centre P and of radius ρ inthe S × R space is called geodesic ball, denoted by B P ( ρ ) , i.e., Q ∈ B P ( ρ ) iff ≤ d ( P , Q ) ≤ ρ . Proposition 2.4
The geodesic sphere and ball of radius ρ exists in the S × R space if and only if ρ ∈ [0 , π ] . H × R geometry In this section we recall the important notions and results from the papers [6],[12], [19].The points of H × R space, forming an open cone solid in the projective space P , are the following: H × R := (cid:8) X ( x = x i e i ) ∈ P : − ( x ) + ( x ) + ( x ) < < x , x (cid:9) . In this context E. Moln´ar [6] has derived the infinitezimal arc-length square at anypoint of H × R as follows ( ds ) = 1( − x + y + z ) · [( x ) + ( y ) + ( z ) ]( dx ) ++2 dxdy ( − xy ) + 2 dxdz ( − xz ) + [( x ) + ( y ) − ( z ) ]( dy ) ++2 dydz (2 yz ) + [( x ) − ( y ) + ( z ) ]( dz ) . (2.5) pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . ( t, r, α ) , ( r ≥ , − π < α ≤ π ) with the fibre coordinate t ∈ R . We describe points in ourmodel by the following equations: x = 1 , x = e t cosh r, x = e t sinh r cos α, x = e t sinh r sin α. (2.6)Then we have x = x x = x , y = x x = x , z = x x = x , i.e. the usual Cartesiancoordinates. We obtain by [6] that in this parametrization the infinitezimal arc-length square by (2.5) at any point of H × R is the following ( ds ) = ( dt ) + ( dr ) + sinh r ( dα ) . (2.7)The geodesic curves of H × R are generally defined as having locally minimal arclength between their any two (near enough) points. The equation systems of theparametrized geodesic curves γ ( t ( τ ) , r ( τ ) , α ( τ )) in our model can be determinedby the general theory of Riemann geometry (see [19]):Then by (2.6-7) we get the equation systems of a geodesic curve in our model[19]: x ( τ ) = e τ sin v cosh ( τ cos v ) ,y ( τ ) = e τ sin v sinh ( τ cos v ) cos u,z ( τ ) = e τ sin v sinh ( τ cos v ) sin u, − π < u ≤ π, − π ≤ v ≤ π . (2.8) Definition 2.5
The distance d H × R ( P , P ) between the points P and P is de-fined by the arc length of the geodesic curve from P to P . Definition 2.6
The geodesic sphere of radius ρ (denoted by S H × R P ( ρ ) ) with centreat the point P is defined as the set of all points P in the space with the condition d H × R ( P , P ) = ρ . Moreover, we require that the geodesic sphere is a simplyconnected surface without selfintersection in H × R space. Remark 2.7
In this paper we consider only the usual spheres with ”proper cen-tre”, i.e. P ∈ H × R . If the centre of a ”sphere” lie on the absolute quadric orlie out of our model the notion of the ”sphere” (similarly to the hyperbolic space),can be defined, but that cases we shall study in a forthcoming work. Definition 2.8
The body of the geodesic sphere of centre P and of radius ρ in H × R space is called geodesic ball, denoted by B P ( ρ ) , i.e. Q ∈ B P ( ρ ) iff ≤ d ( P , Q ) ≤ ρ . Jen ˝o Szirmai
Proposition 2.9 S ( ρ ) is a simply connected surface in E for ρ > . Remark 2.10 S × R and H × R are affine metric spaces (affine-projective spaces– in the sense of the unified formulation of [6]). Therefore their linear, affine,unimodular, etc. transformations are defined as those of the embedding affinespace. S × R and H × R geome-tries The generalization of the classical definition of the Apollonius circle of the Eu-clidean plane E to Thurston geometries is the following Definition 3.1
The Apollonius surface AS XP P ( λ ) in the Thurston geometry X isthe set of all points of X whose geodesic distances from two fixed points are in aconstant ratio λ ∈ R +0 where X ∈ E , S , H , S × R , H × R , Nil , (cid:94) SL R , Sol . i.e. AS XP P ( λ ) of two arbitrary points P , P ∈ X consists of all points P (cid:48) ∈ X ,for which d X ( P , P (cid:48) ) = λ · d X ( P (cid:48) , P ) ( λ ∈ [0 , ∞ ) where d X is the correspondingdistance function of X . If λ = 0 , then AS XP P (0) := P and it is clear, that incase λ → ∞ then d ( P (cid:48) , P ) → therefore we say AS XP P ( ∞ ) := P . Now, we consider only the S × R and H × R geometries and can be assumedby the homogeneity of the above geometries that the starting point of a givengeodesic curve segment in both geometries is P = (1 , , , . The other endpointwill be given by its homogeneous coordinates P = (1 , x, y, z ) . In order to obtainthe Apollonius surfaces of two given points P , P to a given costant ratio λ ∈ R + we consider a geodesic curve segment g XP P ( X ∈ { S × R , H × R } ) and determineits parameters ( u, v, τ ) expressed by the real coordinates x , y , z of P .We obtain directly by equation system (2.4) and (2.8) the following lemmas(see PSSz11-2,PSSz10,[19, 21]): Lemma 3.1
Let (1 , x, y, z ) ( x, y, z ∈ R , x + y + z (cid:54) = 0) be the homogeneouscoordinates of the point P ∈ S × R . The paramerters of the correspondinggeodesic curve g S × R ( A , P ) are the following: pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . y, z ∈ R \ { } and x + y + z (cid:54) = 1 ; v = arctan (cid:16) log (cid:112) x + y + z arccos x √ x + y + z (cid:17) , u = arctan (cid:16) zy (cid:17) ,τ = log (cid:112) x + y + z sin v , where − π < u ≤ π, − π/ ≤ v ≤ π/ , τ ∈ R + . (3.1) y = 0 , z (cid:54) = 0 and x + z (cid:54) = 1 ; u = π , v = arctan (cid:16) log √ x + z arccos x √ x + z (cid:17) ,τ = log √ x + z sin v , where − π/ ≤ v ≤ π/ , τ ∈ R + . (3.2) y = 0 , z (cid:54) = 0 and x + z = 1 ; u = π , v = 0 , τ = arccos( x ) , τ ∈ R + . (3.3) y, z = 0 ; u = 0 , v = π , τ = log (cid:112) x + y + z , τ ∈ R + . (3.4) x = 0 , y = 0 and z (cid:54) = 1 ; u = π , v = arctan 2 log | z | π , τ = log | z | sin v , − π/ ≤ v ≤ π/ , τ ∈ R + . (3.5) (cid:3) Lemma 3.2
Let (1 , x, y, z ) ( x, y, z ∈ R , x − y − z ≥ , x ≥ be thehomogeneous coordinates of the point P ∈ H × R . The paramerters of thecorresponding geodesic curve g H × R ( A , P ) are the following: Jen ˝o Szirmai y, z ∈ R \ { } and x − y − z (cid:54) = 1 ; v = arctan (cid:16) log (cid:112) x − y − z arccosh x √ x − y − z (cid:17) , u = arctan (cid:16) zy (cid:17) ,τ = log (cid:112) x − y − z sin v , where − π < u ≤ π, − π/ ≤ v ≤ π/ , τ ∈ R + . (3.6) y = 0 , z (cid:54) = 0 and x − z (cid:54) = 1 ; u = π , v = arctan (cid:16) log √ x − z arccosh x √ x − z (cid:17) ,τ = log √ x − z sin v , where − π/ ≤ v ≤ π/ , τ ∈ R + . (3.7) y = 0 , z (cid:54) = 0 and x − z = 1 ; u = π , v = 0 , τ = arccosh( x ) , τ ∈ R + . (3.8) y, z = 0 ; u = 0 , v = π , τ = log( x ) , τ ∈ R + . (3.9) (cid:3) It is clear that Y ∈ AS XP P ( λ ) iff d X ( P , Y ) = λ · d X ( Y, P ) and d ( Y, P ) = d ( Y F , P F ) , where F is a composition of isometries of geometry X which maps Y onto A (1 , , , ( X ∈ { S × R , H × R } ). Therefore, apply the Lemmas 3.1-2the length of a given geodesic segment (i.e. the distance between the two points)and thus it is comparable to d X ( P , Y ) .This method clearly leads to the following implicit equation of the Apolloniussurfaces AS XP P ( λ ) of two proper points P (1 , a, b, c ) and P (1 , d, e, f ) with givenratio λ ∈ R +0 , in X geometry: Theorem 3.2
The implicit equation of the Apollonius surfaces AS XP P ( λ ) of twoproper points P (1 , a, b, c ) and P (1 , d, e, f ) with given ratio λ ∈ R +0 , in X ge- pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . ometry: AS XP P ( λ )( x, y, z ) ⇒ ω X (cid:32) ax ± by ± cz √ a ± b ± c (cid:112) x ± y ± z (cid:33) + log (cid:16) a ± b ± c x ± y ± z (cid:17) == λ (cid:104) ω X (cid:32) dx ± ey ± f z (cid:112) d ± e ± f (cid:112) x ± y ± z (cid:33) + log (cid:16) d ± e ± f x ± y ± z (cid:17)(cid:105) , where if X = S × R then all ± signs are + , ω X ( x ) = arccos( x ) and if X = H × R then the all ± signs are − , ω X ( x ) = arccosh( x ) . (cid:3) P P P P Figure 1: The Apollonius surface AS S × R P P ( λ ) where P = (1 , , , , P =(1 , , , , λ = 2 (left) and λ = 1 (right, equidistance surface)We will use the statements of the following lemma Lemma 3.3 (J. Sz, [21])
Let P be an arbitrary point and g X ( P , P ) ( X ∈ { S × R , H × R } , P = (1 , , , ) is a geodesic curve in the considered model of X geometry. The points of the geodesic curve g X ( P , P ) and the centre of the model E = (1 , , , lie in a plane in Euclidean sense. (cid:3) We consider points A , A , A in the projective model of X space (see Section2) ( X ∈ { S × R , H × R } ) . The geodesic segments a k connecting the points A i and A j ( i < j, i, j, k ∈ { , , } , k (cid:54) = i, j ) are called sides of the geodesic triangle with vertices A , A , A .0 Jen ˝o Szirmai P P P P Figure 2: The Apollonius surface AS H × R P P ( λ ) where P = (1 , , , , P =(1 , / , / , − / , λ = 2 (left) and λ = 1 (right, equidistance surface)However, defining the surface of a geodetic triangle in X space is not straight-forward. The usual geodesic triangle surface definition in these geometries is notpossible because the geodesic curves starting from different vertices and ending atpoints of the corresponding opposite edges define different surfaces, i.e. geodesicsstarting from different vertices and ending at points on the corresponding oppo-site side usually do not intersect (Fig. 3 illustrates it in S × R space) . Therefore,we introduce a new definition of the surface S A A A of the geodesic triangle bythe following steps: Definition 3.3
1. We consider the geodesic triangle A A A in the projectivemodel of X space ( X ∈ { S × R , H × R } ) and consider the Apolloniussurfaces AS XA A ( λ ) and AS XA A ( λ ) ( λ , λ ∈ [0 , ∞ ) , λ + λ > ). It isclear, that if Y ∈ C ( λ , λ ) := AS XA A ( λ ) ∩ AS XA A ( λ ) then d X ( A ,Y ) d X ( Y,A ) = λ and d X ( A ,Y ) d X ( Y,A ) = λ ⇒ d X ( A ,Y ) d X ( Y,A ) = λ · λ for parameters λ , λ ∈ (0 , ∞ ) and if λ = 0 then C ( λ , λ ) = A , if λ = 0 then C ( λ , λ ) = A P X ( λ , λ ) := { P ∈ X | P ∈ C ( λ , λ ) and d X ( P, A ) = min Q ∈C ( λ ,λ ) ( d X ( Q, A )) with given real parameters λ , λ ∈ [0 , ∞ ) , λ + λ > } (3.10) pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . A A A A A F F F F A Figure 3: A A A is a geodesic triangle in S × R space with vertices A =(1 , , , , A = (1 , − , − , , A = (1 , , , and F is the midpoint of thegeodesic segment A A , F is the midpont of geodesic segment A A The rightfigure shows that the geodesic segments A F and A F do not intersect eachother.
3. The surface S XA A A of the geodesic triangle A A A is S XA A A := { P X ( λ , λ ) ∈ X, where λ , λ ∈ [0 , ∞ ) , λ + λ > } . (3.11)We introduce the following notations:
1. If the surface of the geodesic triangle is a plane in Euclidean sense whichcontains the centre of the model E = (1 , , , , then it is called fibre type trian-gle.2. In other cases the triangle is in general type. Definition 3.4
Let S XA A A be the surface of the geodesic triangle A A A and P , P ∈ S XA A A two given point.1. If the geodesic triangle is in fibre type, then the connecting curve G S XA A A P P ⊂S XA A A is a geodesic curve in X space.2. In other cases the connecting curve G S XA A A P P is the image of the geodesiccurve g XP P into the surface S XA A A by a central projection with centre E =(1 , , , . Jen ˝o Szirmai A A A A A A C ( ) C ( ) ( ) C ( ) ( ) ( ) ( ) C ( ) Figure 4: Intersection curves C ( λ , λ ) := AS S × R A A ( λ ) ∩ AS S × R A A ( λ ) related togeodesic triangle A A A in S × R geometry. We consider points A , A , A , A in the projective model of X space (seeSection 2, X ∈ { S × R , H × R } ) . These points are the vertices of a geodesictetrahedron in the X space if any two geodesic segments connecting the points A i and A j ( i < j, i, j ∈ { , , , } ) do not have common inner points and anythree vertices do not lie in a same geodesic curve. Now, the geodesic segments A i A j are called edges of the geodesic tetrahedron A A A A . The circumscribedsphere of a geodesic tetrahedron is a geodesic sphere (see Definitions 2.2, 2.6, andFig. 6,7,8,9) that touches each of the tetrahedron’s vertices. As in the Euclideancase the radius of a geodesic sphere circumscribed around a tetrahedron T is calledthe circumradius of T , and the center point of this sphere is called the circumcenterof T . S × R space The next Lemma follows directly from the properties of the geodesic distancefunction of S × R space (see Definition 2.1 and (2.14)): Lemma 3.4
For any S × R geodesic tetrahedron there exists uniquely a geodesicsurface on which all four vertices lie. If its radius less or equal to π then the abovesurface is a geodesic sphere (called circumscibed sphere, see Definition 2.2). (cid:3) pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . A A A A A A C ( ) C ( ) C ( ) C 3/5,1 ( ) C ( ) ( ) C ( ) C 3/5,1 ( )
Figure 5: Intersection curves C ( λ , λ ) = AS H × R A A ( λ ) ∩ AS H × R A A ( λ ) related togeodesic triangle A A A in H × R geometry.The procedure to determine the radius and the circumcenter of a given geodesic S × R tetrahedron is the folowing:The circumcenter C = (1 , x, y, z ) of a given translation tetrahedron A A A A ( A i = (1 , x i , y i , z i ) , i ∈ { , , , } ) has to hold the following system of equation: d S × R ( A , C ) = d S × R ( A , C ) = d S × R ( A , C ) = d S × R ( A , C ) , (3.12)therefore it lies on the geodesic-like bisector surfaces AS S × R A i ,A j ( i < j, i, j ∈{ , , , } ) which equations are determined in Lemma 3.3. The coordinates x, y, z of the circumcenter of the circumscribed sphere around the tetrahedron A A A A are obtained by the system of equation derived from the facts: C ∈ AS S × R A A (1) , AS S × R A A (1) , AS S × R A A (1) . (3.13)Finally, we get the circumradius r as the geodesic distance e.g. r = d S × R ( A , C ) .We applied the above procedure to two tetrahedra determined their centres andthe radii of their circumscribed balls that are described in Fig. 6 and 7. (cid:3) H × R space In the H × R geometry the procedure to determine the radius and the circumcenterof a given geodesic H × R tetrahedron is similar to the S × R case but not thesame. The circumcenter C and its circumradius r of the circumscribed spherearound the tetrahedron A A A A is obtained by the system of equation derivedfrom the bisector surfaces of the edges of the given tetrahedron: C ∈ AS H × R A A (1) , AS H × R A A (1) , AS H × R A A (1) , (3.14)4 Jen ˝o Szirmai A A A A A A A C Figure 6: Geodesic S × R tetrahedron with vertices A = (1 , , , , A =(1 , − , − / , , A = (1 , , , , A = (1 , , − , and its circumscibed sphereof radius r ≈ . with circumcenter C = (1 , ≈ . , ≈ . , ≈ . .but the above surfaces do not always intersect at a proper point of H × R space.It can be an outer point (relating to the model) or a point at the infinity (a point onthe boundary of the model), similarly to the hyperbolic spaces. Remark 3.5
If the common point of the above bisectors lies at the infinity then thevertices of tetrahedron lie on a horosphere-like surface and if the common pointis an outer point then the vertices of the tetrahedron are on a hypershere-likesurface. These surfaces will be examined in detail in a forthcoming paper.
The next Lemma follows directly from the properties of the geodesic distancefunction of H × R space (see Definition 2.6 and (2.10)): Lemma 3.5
For any H × R geodesic tetrahedron there exists uniquely a geodesicsurface on which all four vertices lie. If its centre is a proper point of H × R space then the above surface is a geodesic sphere (called circumscibed sphere,see Definition 2.6). (cid:3) We applied the above procedure to two tetrahedra determined their centres andthe radii of their circumscribed balls that are described in Fig. 8 and 9. (cid:3) pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . A A A A A A A C Figure 7: Geodesic S × R tetrahedron with vertices A = (1 , , , , A =(1 , , , , A = (1 , , , , A = (1 , , − , and its circumscibed sphereof radius r ≈ . with circumcenter C = (1 , ≈ . , ≈ − . , ≈ . . S × R and H × R spaces First we recall the definition of simply ratios in spherical S and the hyperbolic H planes. The models of the above plane geometries of constant curvature areembedded in the models of the previously described geometries S × R and H × R as “base planes” and are used hereinafter for our discussions.A spherical triangle is the space included by arcs of great circles on the surfaceof a sphere, subject to the limitation that these arcs and the further circle arcs in thefollowing items in the spherical plane are always less or equal than a semicircle. Definition 4.1 If A , B and P are distinct points on a line in the Y ∈ { H , S } space, then their simply ratio is s Y ( A, P, B ) = w Y ( d Y ( A, P )) /w Y ( d Y ( P, B )) ,if P is between A and B , and s Y ( A, P, B ) = − w Y ( d Y ( A, P )) /w Y ( d Y ( P, B )) ,otherwise where w Y ( x ) := sin ( x ) if Y = S and w Y ( x ) := sinh ( x ) if Y = H . Remark 4.2
Basic properties of simply ratio:1. s Y ( A, P, B ) = − s Y ( B, P, A ) ,2. if P is between A and B , then s Y ( A, P, B ) ∈ (0 , ,3. if P is on AB , beyond B , then s Y ( A, P, B ) ∈ ( −∞ , − , Jen ˝o Szirmai A A A A A A A A Figure 8: Geodesic H × R tetrahedron with vertices A = (1 , , , , A =(1 , / , , − , A = (1 , , / , , A = (1 , , / , / and its circum-scibed sphere of radius r ≈ . with circumcenter C = (1 , ≈ . , ≈− . , ≈ − . .
4. if P is on AB , beyond A , then s Y ( A, P, B ) ∈ ( − , . Note that the value of s Y ( A, P, B ) determines the position of Y relative to A and B . With this definition, the corresponding sine rule of geometry Y leads to Menelaus’sand Ceva’s theorems [4, 13]: Theorem 4.3 (Menelaus’s Theorem for triangles in Y plane) If is a l line notthrough any vertex of an triangle ABC such that l meets BC in Q , AC in R , and AB in P , then s Y ( A, P, B ) s Y ( B, Q, C ) s Y ( C, R, A ) = − . (cid:3) Theorem 4.4 (Ceva’s Theorem for triangles in Y plane) If T is a point not onany side of a triangle ABC such that AT and BC meet in Q , BX and AC in R ,and CX and AB in P , then s Y ( A, P, B ) s Y ( B, Q, C ) s Y ( C, R, A ) = 1 . (cid:3) pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . A A A A A A CA A A A A A Figure 9: Geodesic H × R tetrahedron with vertices A = (1 , , , , A =(1 , / , / , − / , A = (1 , / , / , , A = (1 , / , − / , / andits circumscibed sphere of radius r ≈ . with circumcenter C = (1 , ≈ . , ≈ . , ≈ . . First we consider a general location geodesic triangle A A A in the projectivemodel of X space (see Section 3.1) ( X ∈ { S × R , H × R } ) . Without limitinggenerality, we can assume that A = (1 , , , and A lies in the coordinateplane [ x, y ] . The geodesic lines that contain the sides A A and A A of the giventriangle can be characterized directly by the corresponding parameters v and u (see (2.4) and (2.8)).The geodesic curve including the side segment A A is also determined byone of its endpoints and its parameters but in order to determine the correspondingparameters of this geodesic line we use orientation preserving isometric transfor- Jen ˝o Szirmai mations T X ( A ) , as elements of the isometry group of X geometry, that maps the A = (1 , x , y , onto A = (1 , , , (up to a positive determinant factor). Remark 4.5
I note here, that this orientation preserving isometry is not unique,but in all cases the v parameters of the “image geodesics” are equal (of coursethe u parameters may be different) and in the further derivation, only the valuesof the parameter v will be needed. We extend the definition of the simply ratio to the X ∈ { S × R , H × R } spaces.If X = S × R then is clear that the space contains its “base sphere” (unit spherecentred in E ) which is a geodesic surface. Therefore, similarly to the sphericalspaces we assume that the geodesic arcs in the following items are always less orequal than a semicircle. Definition 4.6 If A , B and P be distinct points on a non-fibrum-like geodeticcurve in the X ∈ { S × R , H × R } space , then their simply ratio is s Xg ( A, P, B ) = w X (cid:16) d X ( A, P )cos( v ) (cid:17) /w X (cid:16) d X ( P, B )cos( v ) (cid:17) , if P is between A and B , and s Xg ( A, P, B ) = − w X (cid:16) d X ( A, P )cos( v ) (cid:17) /w X (cid:16) d X ( P, B )cos( v ) (cid:17) , otherwise where w X ( x ) := sin ( x ) if X = S × R , w Y ( x ) := sinh ( x ) if X = H × R and v is the parameter of the geodesic curve containing points A, B, P (see Fig. 10).
Theorem 4.7 (Ceva’s Theorem for triangles in general location) If T is a pointnot on any side of a geodesic triangle A A A in X ∈ { S × R , H × R } such thatthe curves A T and g XA A meet in Q , A T and g XA A in R , and A T and g XA A in P , ( A T, A T, A T ⊂ S XA A A ) then s Xg ( A , P, A ) s Xg ( A , Q, A ) s Xg ( A , R, A ) = 1 . Proof:
Let A , B and P be distinct points on a non-fibrum-like geodetic curve in the X and let A ∗ , B ∗ and P ∗ their centrally projected images from the origin E intothe base plane of X (see Fig. 10). We assume without loss of genarality, that A = A ∗ = (1 , , , and B lies in the [ x, y ] coordinate plan. Therefore, the pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . A E x yBBP ** P A E x yBBP ** P Figure 10: Simply ratios in S × R and H × R spacespoints of the geodesic curve segment g XAB also are included in it (see Lemma 3.3).Let the parameters of g XAB be ( t, u = 0 , v ) (see (2.4), (2.8)) and the geodesic curve g XAP are ( t p , u p = 0 , v p = v ) . Their images g XA ∗ P ∗ and g XA ∗ B ∗ are determined byparameters ( t p · cos v, u p = 0 , v p = 0) and ( t · cos v, u = 0 , v = 0) : d X ( A ∗ , P ∗ ) = d X ( A, P ) · cos v, d X ( P ∗ , B ∗ ) = d X ( P, B ) · cos v. Let A ∗ A ∗ A ∗ ( A ∗ = A ) be the centrally projected image of the geodesic triangle A A A from the origin E into the base plane of X (see Fig. 10).Moreover, let T is a point not on any side of a geodesic triangle A A A suchthat the curves A T and g XA A meet in Q , A T and g XA A in R , and A T and g XA A in P , ( A T, A T, A T ⊂ S XA A A ) .The centrally projected images A ∗ Q ∗ , A ∗ R ∗ and A ∗ P ∗ of the curves A Q , A R and A P from E are geodesic curves in the corresponding “base plane” of X . T ∗ and T lie in the planes A QE , A RE and A P E (see Lemma 3.3).Therefore, this theorem is direct consequence of the Definition 4.6 of the simplyratio and the corresponding Ceva’s theorem in spherical and hyperbolic planes(Theorem 4.4). (cid:3) Theorem 4.8 (Menelaus’s theorem for triangles in general location)
If is a l linenot through any vertex of an geodesic triangle A A A lying in its surface S XA A A in X ∈ { S × R , H × R } geometry such that l meets geodesic curves g XA A in Q , g XA A in R , and g XA A in P , then s Xg ( A , P, A ) s Xg ( A , Q, A s Xg ( A , R, A ) = − . Jen ˝o Szirmai
Proof:
Similarly to the above proof, this theorem is follows by the Definition4.6 of the simply ratio and the corresponding Ceva’s theorem in spherical andhyperbolic planes (Theorem 4.4). (cid:3)
We consider a fibre type geodesic triangle A A A in the projective model of X space. Without limiting generality, we can assume that A = (1 , , , , A = (1 , x , y , and A = (1 , x , y , lies in the coordinate plane [ x, y ] . Thegeodesic lines that contain the sides A A and A A of the given triangle can becharacterized directly by the corresponding parameters v and u = 0 similarly tothe above case.We extend the definition of the simply ratio to the X ∈ { S × R , H × R } spaces.If X = S × R is clear that the S × R space contains its “base sphere” (unitsphere centred in E ) which is a geodesic surface. Therefore, similarly to thespherical spaces we assume that the geodesic arcs in the following items are al-ways less or equal than a semicircle. Definition 4.9 If A , B and P be distinct points on fibrum-like geodetic curve inthe X ∈ { S × R , H × R } space , then their simply ratio is s Xf ( A, P, B ) = d X ( A, P ) /d X ( P, B ) if P is between A and B , and s Xf ( A, P, B ) = − d X ( A, P ) /d X ( P, B ) (see Fig. 10). Theorem 4.10 (Ceva’s Theorem in X geometry for triangles in fibre types) If T is a point not on any side of a geodesic triangle A A A in X ∈ { S × R , H × R } such that the geodesic curves g XA T and g XA A meet in Q , g XA T and g XA A in R , and g XA T and g XA A in P , ( g XA T , g XA T , g XA T ⊂ S XA A A then s Xf ( A , P, A ) s Xf ( A , Q, A ) s Xf ( A , R, A ) = 1 . Theorem 4.11 (Menelaus’s theorem in X geometry for triangles in fibre types) If is a l line not through any vertex of an geodesic triangle A A A lying in its pollonius surfaces, circumscribed spheres of tetrahedra, Menelaus’ and Ceva’s theorems . . . surface S XA A A in X ∈ { S × R , H × R } geometry such that l meets geodesiccurves g XA A in Q , g XA A in R , and g XA A in P , then s Xf ( A , P, A ) s Xf ( A , Q, A s Xf ( A , R, A ) = − . Proof of both theorems:
Let P = (1 , a, b, c ) and P = (1 , d, e, f ) be proper points of X space. Theirdistance d X ( P , P )) is (see Theorem 3.3): d X ( P , P )) = (cid:118)(cid:117)(cid:117)(cid:116) ω X (cid:32) ax ± by ± cz √ a ± b ± c (cid:112) d ± e ± f (cid:33) + log (cid:16) √ a ± b ± c (cid:112) d ± e ± f (cid:17) , where if X = S × R then all ± signs are + , ω X ( x ) = arccos( x ) and if X = H × R then the all ± signs are − , ω X ( x ) = arccosh( x ) . Without loss of generality wecan assume, that P = A = (1 , , , and P = B = (1 , d, e, f ) (see Fig. 10) and A ∗ and B ∗ are their centrally projected images from the origin E into the baseplane of X . From the structure of geometries X follows, that d X ( P , P )) = (cid:112) ( d X ( A ∗ , B ∗ )) + ( d X ( B ∗ , B )) where ( d X ( A ∗ , B ∗ )) = ω X (cid:18) d (cid:113) √ d ± e ± f (cid:19) and ( d X ( B ∗ , B )) = log (cid:16) √ d ± e ± f (cid:17) .Therefore, each right angled fibre-like triangle in X space can be uniquelyassigned to an Euclidean righ angled triangle with the same side lenghts and it isclear, that the same is true for any fibre-like geodesic triangle.Thus, we can formulate similar theorems for the fibre-like geodesic triangleas for the corresponding Euclidean triangles therefore the Ceva’s and Menelaus’theorems in X geometry follows from the well-known corresponding Euclideanones. (cid:3) Similar problems in other homogeneous Thurston geometries represent an-other huge class of open mathematical problems. For
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