Discrete Koenigs nets and discrete isothermic surfaces
aa r X i v : . [ m a t h . DG ] S e p Discrete Koenigs nets and discrete isothermic surfaces
Alexander I. Bobenko ∗ Yuri B. Suris † May 31, 2018
Abstract.
We discuss discretization of Koenigs nets (conjugate nets with equal Laplace in-variants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilat-erals: two planar quadrilaterals are called dual, if their corresponding sides are parallel, and theirnon-corresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planarquadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets arefound; in particular, two-dimensional discrete Koenigs nets can be characterized by co-planarityof the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; thischaracterization is invariant with respect to projective transformations. Discrete isothermic netsare defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermicsurfaces introduced previously as circular nets with factorized cross-ratios. ∗ Institut f¨ur Mathematik, Technische Universit¨at Berlin, Str. des 17. Juni 136, 10623 Berlin, Germany.E–mail: [email protected] † Zentrum Mathematik, Technische Universit¨at M¨unchen, Boltzmannstr. 3, 85747 Garching beiM¨unchen, Germany. E–mail: [email protected]
Research for this article was supported by the DFG Research Unit “Polyhedral Surfaces”. Introduction
This paper is devoted to the discretization of a geometrically important class of two-dimensional conjugate nets, very popular in the classical differential geometry under thename of nets with equal invariants . With a view towards discretization, we prefer tocall them
Koenigs nets , for the following reason: among various geometric and analyticcharacterizations, the property of having equal Laplace invariants belongs to the minorpart which do not survive by discretization, at least literally. Therefore the term “discretenets with equal invariants” would be misleading. On the other hand, the French geometerG. Koenigs contributed a lot to the study of their properties [K1, K2], see also [Da, E].The term “discrete Koenigs nets” will be suggestive and well justified.Another class of nets, whose discretizations are discussed in the present paper, are isothermic surfaces . Classically, their theory was considered as one of the highest achieve-ments of the local differential geometry, see [Bi, Da, E] and modern studies [CGS, BHPP,KPP, Sch, HJ, Bu].Both classes of nets has been discretized already. Historically, discrete isothermicsurfaces happened to be introduced earlier [BP], as circular nets with factorized cross-ratios. An approach to the discretization of Koenigs nets have been proposed in [D2],based on a characterization of smooth Koenigs nets as conjugate nets possessing a socalled conic of Koenigs in each tangent plane [K2] (a conic of Koenigs has a second ordercontact both with the u tangent line at the corresponding point of the Laplace transform f − and with the u tangent line of f at the corresponding point of the Laplace transform f ).In the present paper, we propose a novel definition of discrete Koenigs nets and discreteisothermic surfaces. This definition is based on one of the characterizations of the Koenigsnets and isothermic surfaces, namely on the notion of duality . We believe that it is thisdefinition that lies in the core of the whole theory and leads most directly to variousother properties. All discretizations we consider belong to the class of Q-nets, or netswith planar elementary quadrilaterals [DS], which are the fundamental objects of discretedifferential geometry (see [BS1, BS2] for a detailed presentation of the current state ofdiscrete differential geometry as well as for historical remarks).Two planar quadrilaterals are said to be dual , if their corresponding sides are paral-lel and their non-corresponding diagonals are also parallel. In [PLWBW], this propertyhas been identified as a characterization of pairs of quadrilaterals with parallel sides andwith the vanishing mixed area, and it has been observed that the corresponding circularquadrilaterals of dual discrete isothermic surfaces possess this property. These observa-tions stimulated the development presented here. Namely, we study here the geometricand analytic properties of nets all of whose quadrilaterals can be dualized simultaneously.A net with planar quadrilaterals admitting a dual net is called a discrete Koenigs net .A discrete Koenigs net with all circular quadrilaterals is called a discrete isothermic net .Discrete surfaces we arrive at are not new. The class of discrete isothermic surfacesturns out to coincide with the original class introduced in [BP], so that we get just a novelcharacterization of the latter. In the case of discrete Koenigs nets, the history is morecomplicated: they first appeared in [S1] (see also [S2, S3]) in the context of infinitesimaldeformations of surfaces, with exactly the same definition as we use (dual quadrilaterals2re called antiparallel there); however, the geometric and analytic properties of these netsremained to a large extent unexplored. Recently, this class has been studied in [D3, BS3],but again some of the crucial properties passed unnoticed. The main novel results of thepresent paper include: • Definition of discrete Koenigs nets as those admitting dual nets (Definition 10). • A characterization of discrete Koenigs nets in terms of a closed multiplicative one-form on diagonals, defined through ratios of diagonal segments (Theorem 12). In-tegrating this closed form, we arrive at the function ν defined at the vertices of adiscrete Koenigs net. This function is a novel and important ingredient of an analyticdescription of discrete Koenigs nets. In particular, the function ν allows us to finda discrete analog of a Laplace equation with equal invariants (equation (30)), anddefines the Moutard representatives of a discrete Koenigs net (Theorem 20). • A novel projective-geometric characterization of two-dimensional Koenigs nets: in-tersection points of diagonals of elementary quadrilaterals of such a net form a netwith planar quadrilaterals (Theorem 14). Interestingly, the net comprised by the in-tersection points of diagonals of quadrilaterals of a discrete Koenigs net in the senseof our definition turns out to satisfy the definition of discrete Koenigs nets from [D2]. • A novel definition of discrete isothermic nets as circular nets admitting dual nets(Definition 21). • A novel understanding of the discrete metric of a discrete isothermic net, as thefunction ν in the circular context (Theorem 28). Definition 1 (Koenigs net)
A map f : R → R N is called a Koenigs net , if it satisfiesa differential equation ∂ ∂ f = ( ∂ log ν ) ∂ f + ( ∂ log ν ) ∂ f (1) with some scalar function ν : R → R ∗ . The following characterization of Koenigs nets will be of a fundamental importance for us.
Theorem 2 (Dual Koenigs net)
A conjugate net f : R → R N is a Koenigs net, if andonly if there exists a scalar function ν : R → R such that the differential one-form df ∗ defined by ∂ f ∗ = ∂ fν , ∂ f ∗ = − ∂ fν (2) is closed. In this case the map f ∗ : R → R N , defined (up to a translation) by theintegration of this one-form, is also a Koenigs net, called dual to f . ∂ f ∗ k ∂ f, ∂ f ∗ k ∂ f, ( ∂ + ∂ ) f ∗ k ( ∂ − ∂ ) f, ( ∂ − ∂ ) f ∗ k ( ∂ + ∂ ) f. (3) Definition 3 (Isothermic surface)
A curvature line parametrized surface f : R → R N is called an isothermic surface , if its first fundamental form is conformal, possibly upon are-parametrization of independent variables u i ϕ i ( u i ) ( i = 1 , , i.e., if at every point u ∈ R of the definition domain there holds | ∂ f | / | ∂ f | = α ( u ) /α ( u ) . In other words, isothermic surfaces are characterized by the relations ∂ ∂ f ∈ span( ∂ f, ∂ f )and h ∂ f, ∂ f i = 0 , | ∂ f | = α s , | ∂ f | = α s , (4)with some s : R → R + and with the functions α i depending on u i only ( i = 1 , ∂ ∂ f = ( ∂ log s ) ∂ f + ( ∂ log s ) ∂ f, h ∂ f, ∂ f i = 0 . (5)Comparison with eq. (1) shows that isothermic surfaces are nothing but orthogonal Koenigsnets , the role of the function ν being played by the metric s .In the case of isothermic surfaces the duality is specialized as follows. Theorem 4 (Dual isothermic surface)
Let f : R → R N be an isothermic surface.Then the R N -valued one-form df ∗ defined by ∂ f ∗ = α ∂ f | ∂ f | = ∂ fs , ∂ f ∗ = − α ∂ f | ∂ f | = − ∂ fs (6) is closed. The surface f ∗ : R → R N , defined (up to a translation) by the integration ofthis one-form, is isothermic, with h ∂ f ∗ , ∂ f ∗ i = 0 , | ∂ f ∗ | = α s − , | ∂ f ∗ | = α s − . (7) The surface f ∗ is called dual to, or the Christoffel transform of the surface f . Remarkably, the defining property (1) turns out to be invariant under projective transfor-mations of R N , so that the notion of Koenigs nets actually belongs to projective geometry.If one considers the ambient space R N of a Koenigs net as an affine part of RP N , then thereis an important choice of representatives for f ∼ ( f,
1) in the space R N +1 of homogeneouscoordinates, namely y = ν − ( f, . (8)Indeed, a straightforward computation shows that the representatives (8) satisfy the fol-lowing simple differential equation: ∂ ∂ y = qy (9)4ith the scalar function q = ν∂ ∂ ( ν − ). Differential equation (9) is known as the Moutardequation . Accordingly, we call a map y : R → R N +1 a Moutard net , if it satisfies theMoutard equation (9) with some q : R → R . Theorem 5 (Koenigs nets = Moutard nets in homogeneous coordinates)
For aKoenigs net f : R → R N , the lift (8) is a Moutard net. Conversely, given a Moutard net y : R → R N +1 with a non-vanishing last component ν − : R → R ∗ , define f : R → R N by eq. (8), then f is a Koenigs net. More generally, for a given Moutard net y in R N +1 , it is not difficult to figure out thecondition for a scalar function ν : R → R ∗ , under which ˜ f = νy satisfies equation of theLaplace type: ν − has to be a solution of the same Moutard equation (9) (not necessarilyone of the components of the vector y ), and then there holds ∂ ∂ ˜ f = ( ∂ log ν ) ∂ ˜ f + ( ∂ log ν ) ∂ ˜ f . Of course, Moutard nets can be considered also on their own rights, i.e., one does nothave to regard the ambient space R N +1 of a Moutard net as the space of homogeneouscoordinates for RP N . Nevertheless, such an interpretation is useful in the most cases.In application to isothermic surfaces, the construction of Moutard representatives canbe performed within the projective model of M¨obius geometry. Recall that, although con-ditions (4) are formulated in Euclidean terms, they are invariant not only with respect toEuclidean motions and dilations in R N , but also with respect to the inversion f → f / h f, f i .Therefore, the notion of isothermic surfaces belongs to M¨obius differential geometry.Recall (see, e.g., [HJ] or [Bu]) that the basic space of the projective model of M¨obiusgeometry in R N is the projectivization P ( R N +1 , ) of the Minkowski space R N +1 , . Thelatter is the space spanned by N +2 linearly independent vectors e , . . . , e N +2 and equippedwith the Minkowski scalar product h e i , e j i = , i = j ∈ { , . . . , N + 1 } , − , i = j = N + 2 , , i = j. It is convenient to introduce two isotropic vectors e = ( e N +2 − e N +1 ), e ∞ = ( e N +2 + e N +1 ), satisfying h e , e ∞ i = − .A point f ∈ R N is modelled in the space P ( R N +1 , ) by the element with homogeneouscoordinates ˆ f = f + e + | f | e ∞ . Thus, points f ∈ R N ∪ {∞} are in a one-to-onecorrespondence with points of the projectivized light cone P ( L N +1 , ), where L N +1 , = (cid:8) ξ ∈ R N +1 , : h ξ, ξ i = 0 (cid:9) . (10)A surface f : R → R N is curvature lines parametrized, if and only if its lift ˆ f : R → L N +1 , into the light cone is a conjugate net. In particular, eqs. (5) are equivalent to ∂ ∂ ˆ f = ( ∂ log s ) ∂ ˆ f + ( ∂ log s ) ∂ ˆ f . Thus, the following result by Darboux [Da] holds:5 heorem 6 (Isothermic surfaces = Moutard nets in the light cone)
For anisothermic surface f : R → R N , with the conformal metric s : R → R + , define itslift y : R → L N +1 , into the light cone by y = s − ( f + e + | f | e ∞ ) . (11) Then y satisfies the Moutard equation (9) with q = s∂ ∂ ( s − ) .Conversely, given a Moutard net y : R → L N +1 , in the light cone, define s : R → R ∗ and f : R → R N by eq. (11), so that s − is the e -component, and s − f is the R N -partof y in the basis e , . . . , e N , e , e ∞ . Then f is an isothermic surface, and the definition(4) holds with the functions α i = h ∂ i y, ∂ i y i depending on u i only. Note that in the second part of the theorem we can always assume that s : R → R + ,changing y to − y , if necessary. Definition 7 (Dual quadrilaterals, see [S1, S2, S3, PLWBW])
Two quadrilaterals ( A, B, C, D ) and ( A ∗ , B ∗ , C ∗ , D ∗ ) in a plane are called dual , if their corresponding sidesare parallel: ( A ∗ B ∗ ) k ( AB ) , ( B ∗ C ∗ ) k ( BC ) , ( C ∗ D ∗ ) k ( CD ) , ( D ∗ A ∗ ) k ( DA ) , (12) and the non-corresponding diagonals are parallel: ( A ∗ C ∗ ) k ( BD ) , ( B ∗ D ∗ ) k ( AC ) . (13)PSfrag replacements AB CD A ∗ B ∗ C ∗ D ∗ M M ∗ Figure 1: Dual quadrilaterals
Lemma 8 (Existence and uniqueness of dual quadrilateral)
For any planar quadri-lateral ( A, B, C, D ) a dual one exists and is unique up to scaling and translation. roof. Uniqueness of the form of the dual quadrilateral can be argued as follows. Denotethe intersection point of the diagonals of (
A, B, C, D ) by M = ( AC ) ∩ ( BD ). Take anarbitrary point M ∗ in the plane as the designated intersection point of the diagonals ofthe dual quadrilateral. Draw two lines ℓ and ℓ through M ∗ parallel to ( AC ) and ( BD ),respectively, and choose an arbitrary point on ℓ to be A ∗ . Then the rest of constructionis unique: draw the line through A ∗ parallel to ( AB ), its intersection point with ℓ willbe B ∗ ; draw the line through B ∗ parallel to ( BC ), its intersection point with ℓ will be C ∗ ; draw the line through C ∗ parallel to ( CD ), its intersection point with ℓ will be D ∗ .It remains to see that this construction closes, namely that the line through D ∗ parallelto ( DA ) intersects ℓ at A ∗ . Clearly, this property does not depend on the initial choiceof A ∗ on ℓ , since this choice only affects the scaling of the dual picture. Therefore, it isenough to demonstrate the closing property for some choice of A ∗ , or, in other words, toshow the existence of one dual quadrilateral. This can be done as follows.Denote by e and e some vectors along the diagonals, and introduce the coefficients α, . . . , δ by −−→ M A = αe , −−→ M B = βe , −−→ M C = γe , −−→ M D = δe , (14)so that −−→ AB = βe − αe , −−→ BC = γe − βe , −−→ CD = δe − γe , −−→ DA = αe − δe . (15)Construct a quadrilateral ( A ∗ , B ∗ , C ∗ , D ∗ ) by setting −−−−→ M ∗ A ∗ = − e α , −−−−→ M ∗ B ∗ = − e β , −−−−→ M ∗ C ∗ = − e γ , −−−−→ M ∗ D ∗ = − e δ . (16)Its diagonals are parallel to the non-corresponding diagonals of the original quadrilateral,by construction. The corresponding sides are parallel as well: −−−→ A ∗ B ∗ = − β e + 1 α e = 1 αβ −−→ AB, −−−→ B ∗ C ∗ = − γ e + 1 β e = 1 βγ −−→ BC, −−−→ C ∗ D ∗ = − δ e + 1 γ e = 1 γδ −−→ CD, −−−→ D ∗ A ∗ = − α e + 1 δ e = 1 δα −−→ DA.
Thus, the quadrilateral ( A ∗ , B ∗ , C ∗ , D ∗ ) is dual to ( A, B, C, D ). (cid:3) Note that the quantities α, . . . , δ in eq. (14) are not well defined by the geometry ofthe quadrilateral (
A, B, C, D ), since they depend on the choice of the vectors e , e . Welldefined are their ratios, which can be viewed also as ratios of the directed lengths of thecorresponding segments of diagonals, say γ : α = l ( M, C ) : l ( M, A ) and δ : β = l ( M, D ) : l ( M, B ). It is natural to associate these ratios with directed diagonals:
Definition 9 (Ratio of diagonal segments)
Given a quadrilateral ( A, B, C, D ) , withthe intersection point of diagonals M = ( AC ) ∩ ( BD ) , we set q ( −→ AC ) = l ( M, C ) l ( M, A ) , q ( −−→ BD ) = l ( M, D ) l ( M, B ) . (17)7 hanging the direction of a diagonal corresponds to inverting the associated quantity q . Note that (
A, B, C, D ) convex ⇔ q ( −→ AC ) < q ( −−→ BD ) < . (18) In dealing with discrete nets f : Z m → R N we will use the usual notations: τ i f ( u ) = f ( u + e i ) , δ i f ( u ) = f ( u + e i ) − f ( u ) , where e i stands for the unit vector of the i th coordinate direction. Moreover, we oftenabbreviate f ( u ), τ i f ( u ), τ i τ j f ( u ) to f , f i , f ij , respectively. The following definition is thefundamental one for the present paper. Definition 10 (Discrete Koenigs net)
A Q-net f : Z m → R N is called a discreteKoenigs net , if it admits a dual net , i.e., a Q-net f ∗ : Z m → R N such that all elementaryquadrilaterals of the net f ∗ are dual to the corresponding quadrilaterals of f : δ f ∗ k δ f, δ f ∗ k δ f,f ∗ − f ∗ k f − f , f ∗ − f ∗ k f − f. (19)This definition can be seen as a discretization of conditions (3).In order to understand restrictions imposed on a Q-net by this definition, we start withthe following construction. Each lattice Z m is bi-partite: one can color its vertices blackand white so that each edge connects a black vertex with a white one (for instance, onecan call vertices u = ( u , . . . , u m ) with an even value of | u | = u + . . . + u m black and thosewith an odd value of | u | white). Each elementary quadrilateral has a black diagonal (theone connecting two black vertices) and a white one. One can introduce the black graph Z m even with the set of vertices consisting of the white vertices of Z m and the set of edgesconsisting of black diagonals of all elementary quadrilaterals of Z m , and the analogous white graph Z m odd . The geometry of the elementary quadrilaterals of a Q-net f : Z m → R N induces, according to Definition 9, the quantities q (ratios of directed lengths of diagonalsegments) on all directed diagonals, white and black. Definition 11 (Multiplicative one-form)
Given a graph G with the set of vertices V and with the set of directed edges ~E , the function q : ~E → R ∗ is called a multiplicativeone-form on G , if for any directed edge e ∈ ~E there holds q ( − e ) = 1 /q ( e ) . Such a form iscalled closed , if for any cycle of directed edges the product of values of q along this cycleis equal to one. Thus, any Q-net yields a multiplicative one-form q (or, better, two multiplicative one-forms) on both the black and the white graphs of Z m . Theorem 12 (Algebraic characterization of discrete Koenigs nets)
A Q-net f : Z m → R N is a Koenigs net, if and only if the multiplicative one-form q is closed on both Z m even and Z m odd . roof. For a given Q-net, one can try to construct a dual net, applying Lemma 8, startingwith an arbitrary quadrilateral. It is easy to realize that obstructions in extending thisconstruction to the whole net may appear when running along closed chains of elementaryquadrilaterals in which any two subsequent quadrilaterals share an edge. m = The basic example of a closed chain of quadrilaterals in this case is given byfour elementary quadrilaterals attached to a (black, say) vertex f . Let the diagonals ofPSfrag replacements α β γ δ α β γ δ α β γ δ α β γ δ Figure 2: Four quadrilaterals around a vertex of a two-dimensional neteach quadrilateral be divided by their intersection point in the relations γ k : α k and δ k : β k ( k = 1 , . . . , λ k ( k = 1 , . . . , λ α δ = λ α β ⇔ λ λ = α δ α β . Similarly, we find: λ λ = α δ α β , λ λ = α δ α β , λ λ = α δ α β . All four edges adjacent to f can be matched, if and only if the cyclic product of expressionsfor the quotients of scaling factors is equal to one. This condition reads: α δ α β · α δ α β · α δ α β · α δ α β = 1 , or δ β · δ β · δ β · δ β = 1 . (20)9his is nothing but the closeness condition of the form q for an elementary quadrilateralof the white graph. All other white and black cycles are products of elementary ones,therefore (20) for all elementary white and black cycles are necessary and sufficient for thecloseness of the form q . But it is easy to see that if the closeness condition is fulfilled for allwhite and black cycles, then no closed chain of quadrilaterals can lead to an obstructionby the construction of the dual net. m = In this case the most elementary closed chain of quadrilaterals is given by threefaces of any elementary hexahedron of the net, sharing a (black, for definiteness) vertex f ,see Fig. 3. The further arguments are completely analogous to the two-dimensional case.PSfrag replacements α β γ δ α β γ δ α β γ δ Figure 3: Three quadrilaterals around a vertex of a three-dimensional netMatching the edges shared by the dual quadrilaterals 1 and 2, by the dual quadrilaterals 2and 3, and by the dual quadrilaterals 3 and 1, we find the relations between their scalingfactors: λ λ = α δ α β , λ λ = α δ α β , λ λ = α δ α β . All three edges adjacent to f can be matched simultaneously, if and only if the cyclicproduct of expressions for the quotients of scaling factors is equal to one, which conditionafter cancellations reads: δ β · δ β · δ β = 1 . (21)This is nothing but the closeness condition for the elementary cycle of the white graph ofthe lattice Z , which is a triangle. All cycles of the white and of the black graphs (includingthose encountered in the m = 2 case, i.e., the squares of the two-dimensional slices of thewhite and the black graphs of Z ) are products of elementary triangles. Again, closenesscondition for all white and black cycles guarantees that no closed chain of quadrilateralsleads to an obstruction. 10 ≥ Also in this case any white or black cycle is a product of elementary triangles,as for m = 3, therefore no additional conditions appear. (cid:3) The definition of discrete Koenigs nets obviously belongs to affine geometry, since it relieson the notion of parallelism. It turns out however that the class of discrete Koenigs netsis projectively invariant (it has been pointed out already in [S1, S2]). The proof of thecorresponding projectively invariant characterizations will rely on the generalized Menelaustheorem [Bo, BN] which has a similar flavor: its conditions are of affine-geometric nature,while its conclusions are projectively invariant.
Theorem 13 (Generalized Menelaus theorem)
Let P , ..., P n +1 be n + 1 points ingeneral position in R n , so that the affine space through the points P i is n -dimensional. Let P i,i +1 be some points on the lines ( P i P i +1 ) (indices are read modulo n +1 ). The n +1 points P i,i +1 lie in an ( n − -dimensional affine subspace, if and only if the following relation forthe ratios of the directed lengths holds: n +1 Y i =1 l ( P i , P i,i +1 ) l ( P i,i +1 , P i +1 ) = ( − n +1 . Proof.
The points P i,i +1 lie in an ( n − n +1 X i =1 µ i P i,i +1 = 0 with n +1 X i =1 µ i = 0 . Substituting P i,i +1 = (1 − ξ i ) P i + ξ i P i +1 , and taking into account the general positioncondition, which can be read as linear independence of the vectors −−→ P P i , we come to ahomogeneous system of n + 1 linear equations for n + 1 coefficients µ i : ξ i µ i + (1 − ξ i +1 ) µ i +1 = 0 , i = 1 , . . . , n + 1(where indices are understood modulo n + 1). Clearly it admits a non-trivial solution ifand only if n +1 Y i =1 ξ i − ξ i = n +1 Y i =1 l ( P i , P i,i +1 ) l ( P i,i +1 , P i +1 ) = ( − n +1 . (Menelaus theorem corresponds to n = 2.) (cid:3) In the following considerations, we use the negative indices − − τ − , τ − . Consider four elementary quadrilaterals of a Q-net adjacentto the point f = f ( u ), i.e., the quadrilaterals ( f, f i , f ij , f j ) with ( i, j ) ∈ { ( ± , ± } . Weassume that the vertex f is non-planar, i.e., that there is no plane containing these fourquadrilaterals (or, what is the same, there is no plane containing f and its four neighbors f i , i ∈ {± , ± } ). Recall that we always assume that the dimension of the ambient spaceis N ≥
3. 11 heorem 14 (Discrete 2d Koenigs nets; characterization in terms of intersec-tion points of diagonals)
A two-dimensional Q-net f : Z → R N with non-planarvertices is a discrete Koenigs net, if and only if for every point f = f ( u ) the intersectionpoints of diagonals of the four quadrilaterals adjacent to f lie in a two-dimensional plane. Proof.
This is an immediate consequence of eq. (20) and the n = 3 case of the generalizedMenelaus theorem (Theorem 13). (cid:3) Remark.
Thus, intersection points of diagonals of elementary quadrilaterals of a two-dimensional Koenigs net comprise a Q-net. Such Q-nets are not generic; it turns out thatthey can be characterized as discrete Koenigs nets in the sense of [D2].
Theorem 15 (Discrete 2d Koenigs nets; characterization in terms of vertices)
1) Let f : Z → R N be the a Q-net in the space of dimension N ≥ . Then f is adiscrete Koenigs net, if and only if for every u ∈ Z the five points f and f ± , ± lie in athree-dimensional subspace V = V ( u ) ⊂ R N , not containing some (and then any) of thefour points f ± , f ± .2) Let f : Z → R be a Q-net in the space of dimension N = 3 . Then f is a discreteKoenigs net, if and only if for every u ∈ Z the three planes Π (up) = ( f f f − , ) , Π (down) = ( f f , − f − , − ) , Π (1) = ( f f f − ) have a common line ℓ (1) , or, equivalently, the three planes Π (left) = ( f f − , f − , − ) , Π (right) = ( f f , f , − ) , Π (2) = ( f f f − ) have a common line ℓ (2) . Proof.
1) If the net f satisfies the property of Theorem 14, then the space V through f and f ± , ± is clearly three-dimensional. Conversely, let this space be three-dimensional. Thefour quadrilaterals ( f, f i , f ij , f j ) lie in a four-dimensional space through f , f ± , f ± . Theintersection points of their diagonals lie in the intersection of V with the three-dimensionalspace through f ± , f ± . The intersection of two three-dimensional subspaces of a four-dimensional space is generically a plane.2) Let M ij denote intersection point of diagonals of the quadrilateral ( f, f i , f ij , f j ), with( i, j ) ∈ { ( ± , ± } . Co-planarity of the four points M ij is equivalent to the statement thatthe lines ( M , M − , ) and ( M , − M − , − ) intersect. These two lines lie in the planes( f f f − ), ( f f − f − ), respectively, therefore their intersection point has to belong to theintersection of these planes, i.e., to the line ( f f − ). Thus, coplanarity of the points M ij is equivalent to the fact that three lines ( M , M − , ), ( M , − M − , − ), and ( f f − ) have acommon point L (1) , see Fig. 4. Now the planes Π (up) , Π (down) and Π (1) can be viewed asthe planes through the point f and the lines ( M , M − , ), ( M , − M − , − ), and ( f f − ),respectively. Therefore their intersection is the line ℓ (1) through f and L (1) . (cid:3) Remark 1.
It is not difficult to see that in the dimension N ≥ N ≥ f , f ± , f ± and f ± , ± lie generically in afour-dimensional subspace of RP N . In this subspace one can consider, along with the12Sfrag replacements f f f f − f − L (1) Figure 4: Four quadrilaterals around a vertex, once morethree-dimensional subspace V , the three-dimensional subspaces V (up) containing the twoquadrilaterals ( f, f , f , f ), ( f, f − , f − , , f ), and V (down) containing the quadrilaterals( f, f , f , − , f − ), ( f, f − , f − , − , f − ). Obviously, one has:Π (up) = V (up) ∩ V, Π (down) = V (down) ∩ V, Π (1) = V (up) ∩ V (down) . Generically, three three-dimensional subspaces V , V (up) and V (down) of a four-dimensionalspace intersect along a line ℓ (1) . Remark 2.
The equivalence of two conditions in part 2) of Theorem 15 follows, ofcourse, from the fact that in the notion of discrete Koenigs nets there is no asymmetrybetween the coordinate directions 1 and 2. However, it might be worthwhile to give anadditional illustration of this equivalence. For this aim, consider a central projection ofthe whole picture from the point f to some plane not containing f . In this projection, theplanarity of elementary quadrilaterals ( f, f i , f ij , f j ) turns into collinearity of the triples ofpoints f i , f j and f ij . The traces of the planes Π (up) , Π (down) and Π (1) on the projectionplane are the lines ( f f − , ), ( f , − f − , − ), and ( f f − ), respectively, and the first versionof the condition of part 2) of Theorem 15 turns into the requirement for these three linesto meet in a point. Similarly, the traces of the planes Π (left) , Π (right) and Π (2) on theprojection plane are the lines ( f − , f − , − ), ( f , f , − ), and ( f f − ), respectively. Therequirement for the latter three lines to meet in a point is equivalent to the previous one– this is the statement of the famous Desargues theorem, see Fig. 5. Theorem 16 (Discrete 3d Koenigs nets; characterization in terms of intersec-tion points of diagonals)
A three-dimensional Q-net f : Z → R N is a discrete Koenigsnet, if and only if for every point f = f ( u ) and for every elementary hexahedron with a f f f − f − f f , − f − , f − , − ℓ (1) ℓ (2) Figure 5: Desargues theorem vertex f , the intersection points of diagonals of the three hexahedron faces adjacent to f are collinear. Proof.
This is nothing but the re-formulation of eq. (21) in terms of Menelaus theorem( n = 2 case of Theorem 13). (cid:3) Theorem 17 (Discrete 3d Koenigs nets; characterization in terms of vertices)
A Q-net f : Z → R N is a discrete Koenigs net, if and only if for every elementaryhexahedron of the net its four white vertices are co-planar, or its four black vertices areco-planar (each one of these conditions implies another one). Proof.
Consider an elementary hexahedron with the vertices f , f i , f ij , f . Denote theintersection points of diagonals of the quadrilaterals ( f, f i , f ij , f j ) by M ij , and the intersec-tion points of diagonals of the quadrilaterals( f k , f ik , f , f jk ) by Q ij . Clearly, if the points M ij are collinear, then the four points f and f ij (the black ones) are co-planar. We show next that the co-planarity of the fourblack points yield the co-planarity of the four white points, as well.Suppose that the four black points f , f ij lie in a plane Π . Let Π be the plane throughthe three points f , f , f . Set ℓ = Π ∩ Π . Then the intersection points M ij of diagonalsof the quadrilaterals ( f, f i , f ij , f j ) belong to ℓ . Denote by O ij intersection points of thelines ( f ik f jk ) ⊂ Π with ℓ . Then the three lines ( f k O ij ) ⊂ Π intersect in one point,which is clearly f ∈ Π , so that the four points f i , f are co-planar. This claim isnothing but the classical Pappus theorem illustrated on Fig. 6. This incidence theorem ofprojective geometry is not to be confused with another Pappus theorem, the latter beinga particular case of the Pascal hexagon theorem, when a conic section degenerates into apair of lines. The former characterizes a quadrilateral set of points on a line ℓ which can bedefined as consisting of intersection points of this line with the six lines connecting all pairsamong four points in some plane containing ℓ . Quadrilateral sets admit several equivalent14Sfrag replacements ff f f f f f f M M M O O O Figure 6: Pappus theoremcharacterizations: a multi-ratio of such a set is equal to 1; in other words, the points of aquadrilateral set always build three point pairs of a projective involutive self-map of ℓ .Now we can finish the proof of Theorem 17 as follows. Suppose that the black verticesof an elementary hexahedron of a Q-net are co-planar. Then also the white vertices of thishexahedron are co-planar. Then the intersection points of diagonals of all six faces of thehexahedron are collinear (they belong to the common line of the “black” and the “white”planes). According to the characterization of Theorem 16, the net is Koenigs. (cid:3) Remark.
The characterizations of Theorems 15, 17 coincide with the definitions ofB-quadrilateral nets in [D3] and of discrete Moutard nets in [BS3]. Thus, the point wemake here is a new property of these nets, fixed as Definition 10 and put in the base of thewhole theory. A novel derivation and understanding of the Moutard property of discreteKoenigs nets will be given below, in Sect. 3.6.
We start with the following statement which is a direct consequence of the algebraic char-acterization of discrete Koenigs nets given in Theorem 12. Indeed, in our local setting, dueto the simple-connectedness of the underlying graphs, the closeness of the multiplicativeone-form q is equivalent to its exactness: Corollary 18 (Function ν for a discrete Koenigs net) A Q-net f : Z m → R N is adiscrete Koenigs net, if and only if there exists a real-valued function ν : Z m → R ∗ withthe following property: for every elementary quadrilateral ( f, f i , f ij , f j ) there holds: ν ij ν = q ( −−→ f f ij ) = l ( M, f ij ) l ( M, f ) , ν j ν i = q ( −−→ f i f j ) = l ( M, f j ) l ( M, f i ) , (22) where M = ( f f ij ) ∩ ( f i f j ) is the intersection point of diagonals.
15n both the black and the white graphs of Z m such a function ν is defined up to amultiplicative constant. This freedom is fixed by prescribing values of ν arbitrarily at oneblack and at one white point.Eq. (22) is equivalent to1 ν ij −−−→ M f ij = 1 ν −−→ M f , ν i −−→ M f i = 1 ν j −−→ M f j , (23)which can be re-written also as f ij ν ij − fν = (cid:18) ν ij − ν (cid:19) M, f i ν i − f j ν j = (cid:18) ν i − ν j (cid:19) M. (24)There follows: (cid:16) ν j − ν i (cid:17)(cid:16) f ij ν ij − fν (cid:17) = (cid:16) ν ij − ν (cid:17)(cid:16) f j ν j − f i ν i (cid:17) . (25)This formula can be used for an elegant representation of the dual Koenigs net for f . Theorem 19 (Dual Koenigs net)
Let f : Z m → R N be a discrete Koenigs net, and let ν : Z m → R ∗ be the function defined by the property (22). Then the R N -valued discreteone-form δf ∗ defined by δ i f ∗ = δ i fνν i (26) is closed. Its integration defines (up to a translation) the dual Koenigs net f ∗ : Z m → R N . Proof.
Eq. (25) can be equivalently re-written as f ij − f i ν i ν ij + f i − fνν i = f ij − f j ν j ν ij + f j − fνν i . (27)This is equivalent to the closeness of the discrete form δf ∗ . Note that eq. (26) says thatthe corresponding sides of elementary quadrilaterals of the nets f and f ∗ are parallel. Itremains to show that the non-corresponding diagonals of elementary quadrilaterals of f and f ∗ are also parallel, so that these quadrilaterals are dual in the sense of Definitions 7.For this aim we demonstrate the following two formulas: f ∗ ij − f ∗ = a ij f j − f i ν i ν j , f ∗ j − f ∗ i = 1 a ij f ij − fνν ij , (28)where a ij = (cid:16) ν ij − ν (cid:17).(cid:16) ν j − ν i (cid:17) . (29)Indeed, upon using eqs. (25) and (29) we find: f ∗ ij − f ∗ = ( f ∗ ij − f ∗ i ) + ( f ∗ i − f ∗ ) = f ij − f i ν i ν ij + f i − fνν i = 1 ν i (cid:16) f ij ν ij − fν (cid:17) − f i ν i (cid:16) ν ij − ν (cid:17) = a ij ν i (cid:16) f j ν j − f i ν i (cid:17) − a ij f i ν i (cid:16) ν j − ν i (cid:17) = a ij f j − f i ν i ν j , f ∗ j − f ∗ i = ( f ∗ ij − f ∗ i ) − ( f ∗ ij − f ∗ j ) = f ij − f i ν i ν ij − f ij − f j ν j ν ij = 1 ν ij (cid:16) f j ν j − f i ν i (cid:17) − f ij ν ij (cid:16) ν j − ν i (cid:17) = 1 a ij ν ij (cid:16) f ij ν ij − fν (cid:17) − f ij a ij ν ij (cid:16) ν ij − ν (cid:17) = 1 a ij f ij − fν ij ν . Theorem 19 is completely proven. (cid:3)
For future reference, we note here that after some manipulations formula (25) can betransformed into δ i δ j f = ν j ν ij − νν i ν ( ν i − ν j ) δ i f + ν i ν ij − νν j ν ( ν j − ν i ) δ j f . (30) Constructions of the previous subsection (functions ν and a ij for a given Koenigs net) canbe used also in a different spirit. Theorem 20 (Discrete Koenigs nets = discrete Moutard nets in homogeneouscoordinates)
A Q-net f : Z m → R N is a discrete Koenigs net, if and only if there existsa function ν : Z m → R ∗ such that the points y : Z m → R N +1 , y = ν − ( f, , (31) satisfy the Moutard equation with minus signs τ i τ j y − y = a ij ( τ j y − τ i y ) (32) with a ij ∈ R given by eq. (29). The net y = ν − ( f, , considered as a special lift of f tothe space of homogeneous coordinates for RP N , will be called the Moutard representative of the discrete Koenigs net f . Proof.
First let f : Z m → R N be a discrete Koenigs net. Define the function ν : Z m → R ∗ ,according to Corollary 18. Then eq. (24) holds, with M being the intersection point ofdiagonals of the quadrilateral ( f, f i , f ij , f j ). Denoting y = ν − ( f, a ij defined by eq. (29).Note that the quantities a ij are naturally assigned to elementary quadrilaterals of Z m parallel to the coordinate plane B ij .Conversely, given a solution y : Z m → R N +1 of the Moutard equation (32) in R N +1 ,define ν : Z m → R and f : Z m → R N by y = ν − ( f, ν − denote thelast component of y , and let f be the vector in R N obtained by multiplying the first N components of y by ν . Then, inverting the previous arguments, it is easy to show that f is a discrete Koenigs net. Indeed, one finds immediately expression (29) for the coefficient a ij of the Moutard equation, then from y ij − y = a ij ( y j − y i )17here follows eq. (25). This allows to define the point M by eq. (24). The latter equationis equivalent to (23), therefore M is nothing but the intersection point of diagonals of( f, f i , f ij , f j ). There holds eq. (22), so by Corollary 18 f is a Koenigs net. (cid:3) In the context of discrete integrable systems the discrete Moutard equation (32) hasbeen introduced in [DJM], its importance for discrete differential geometry has beenre-iterated in [NSch], based on the fact that this equation expresses the permutabilityproperties of the so called Moutard transformation for the differential Moutard equation[M, Bi, GT, NSch]. The role played by the discrete Moutard equation in the discretedifferential geometry turns out to be manifold. In particular, the so called Lelieuvre rep-resentation of discrete asymptotic nets involves discrete Moutard nets in R [KP, D1]. Forthe multidimensional consistency of discrete Moutard nets, which lies in the basis of thetransformation theory for discrete Koenigs nets, the reader is referred to [BS1, BS3, D3]. In order for a Q-net to admit a continuous limit, all its quadrilaterals should be of a rea-sonable shape. Anyway, they should be convex. As mentioned in subsection 3.2, diagonalsof convex quadrilaterals carry negative quantities q (ratios of segments of diagonals). The-orem 12 shows that a discrete Koenigs net cannot consist of convex quadrilaterals (andthus cannot admit a continuous limit) for m ≥
3. However, there are no obstructions incase m = 2. This is in a good agreement with the existence of two-dimensional smoothKoenigs nets only.Eq. (22) shows that in case m = 2 with all convex quadrilaterals we can assume,without losing generality, that the sign of ν ( u ) at u = ( u , u ) ∈ Z is either ( − u or( − u . Clearly, such a wildly oscillating function cannot have a well-behaved continuouslimit. However, upon re-defining ν ( u ) ( − u ν ( u ) , resp . ν ( u ) ( − u ν ( u ) (33)we get a positive function, which turns out to be a proper discrete analog of the function ν for smooth Koenigs nets. Note that this re-definition is equivalent to changing eq. (22)to ν ν = l ( f , M ) l ( M, f ) , ν ν = l ( f , M ) l ( M, f ) . (34)We mention also that eq. (30) with the re-defined ν changes its shape into δ δ f = ν ν − νν ν ( ν + ν ) δ f + ν ν − νν ν ( ν + ν ) δ f , (35)with eq. (1) as a continuous limit. Likewise, formulas (26) turn into δ f ∗ = δ fνν , δ f ∗ = − δ fνν , (36)where the second re-definition of ν in eq. (33) has been used, for definiteness (the first onewould result in changing signs of both fractions).18or the Moutard representative y : Z → R N +1 of a two-dimensional discrete Koenigsnet the change (33) leads to y ( u ) ( − u y ( u ) , resp . y ( u ) ( − u y ( u ) . (37)These points satisfy the Moutard equation with the plus signs : τ τ y + y = a ( τ y + τ y ) , (38)or, equivalently, δ δ f = q ( τ f + τ f ) , (39)with some a = 1 + q : Z → R . Clearly, the latter equation has eq. (9) as continuouslimit. Definition 21 (Discrete isothermic net) A discrete isothermic net is a circular Koenigsnet, i.e., a circular net f : Z m → R N admitting a dual net f ∗ : Z m → R N in the sense ofDefinition 10. We can use characterizations of Koenigs net derived in Sect. 3 in order to find char-acterizations of discrete isothermic nets. For this aim, we use the fact that for a circularnet f : Z m → R N its lift ˆ f = f + e + | f | e ∞ into the light cone L N +1 , satisfies the sameequation of the Laplace type as the net f itself. In particular, a circular net f in R N isdiscrete Koenigs, if and only if ˆ f is a discrete Koenigs net in R N +1 , .Projectively invariant characterizations of Koenigs nets ˆ f in R N +1 , immediately trans-late into M¨obius-geometric characterizations of isothermic nets f in R N . Thereby condi-tions like “points ˆ f lie in a d -dimensional space” should be understood as “vectors ˆ f spana ( d + 1)-dimensional linear subspace”, and this is translated as “points f belong to a( d − f in R N +1 , , into the language of M¨obius geometry in R N , we come to the following statement. Theorem 22 (Central spheres for a discrete isothermic surface) A two-dimensional circularnet f : Z → R N not lying in a two-sphere is discrete isothermic, if and only if for every u ∈ Z the five points f and f ± , ± lie on a two-sphere not containing some (and thenany) of the four points f ± , f ± . (Discrete isothermic net on a sphere) A two-dimensional circular net f : Z → S ⊂ R N in a two-sphere is discrete isothermic, if and only if for every u ∈ Z the threecircles through f , C (up) = circle( f, f , f − , ) , C (down) = circle( f, f , − , f − , − ) ,C (1) = circle( f, f , f − ) , ave one additional point in common, or, equivalently, the three circles through f , C (left) = circle( f, f − , , f − , − ) , C (right) = circle( f, f , , f , − ) ,C (2) = circle( f, f , f − ) , have one additional point in common. Figure 7: Four circles of a generic discrete isothermic surface, with a central sphere.PSfrag replacements f f f f − f − f f , − f − , f − , − C (up) C (down) C (1) Figure 8: Four circles of a planar (or spherical) discrete isothermic net.The cases 1), 2) of Theorem 22 are illustrated on Figs. 7, 8, respectively.Similarly, translating Theorem 17, applied to a multidimensional Koenigs net ˆ f in R N +1 , , into the language of M¨obius-geometric properties of the net f in R N , we get thefollowing statement. 20 heorem 23 (Multidimensional discrete isothermic nets) A circular net f : Z m → R N is discrete isothermic, if and only if for any elementary hexahedron of the net its fourwhite vertices are concircular, and its four black vertices are concircular (each one of theseconditions implies another one). Another characterization of discrete isothermic surfaces can be given in terms of the cross-ratios. Recall that for any four concircular points a, b, c, d ∈ R N their (real-valued) cross-ratio is defined by q ( a, b, c, d ) = ( a − b )( b − c ) − ( c − d )( d − a ) − , (40)with the Clifford multiplication in the Clifford algebra C ℓ ( R N ). The Clifford productof x, y ∈ R N satisfies xy + yx = − h x, y i , and the inverse element of x ∈ R N in theClifford algebra is given by x − = − x/ | x | . Alternatively, one can identify the plane of thequadrilateral ( a, b, c, d ) with the complex plane C , and then multiplication in eq. (40) canbe interpreted as the complex multiplication. An important property of the cross-ratio isits invariance under M¨obius transformations.For discrete isothermic surfaces Theorem 22 yields the following characterization. Theorem 24 (Cross-ratios of four adjacent quadrilaterals)
A two-dimensional cir-cular net f : Z → R N is a discrete isothermic surface, if and only if the cross-ratios q = q ( f, f , f , f ) of its elementary quadrilaterals satisfy the following condition: q · q − , − = q − · q − . (41) Here, as usual, the negative indices − i denote the backward shifts τ − i , so that, e.g., q − = q ( f − , f, f , f − , ) , see Fig. 9. Proof.
Perform a M¨obius transformation sending f to ∞ . Under such a transformation,the four adjacent circles through f turn into four straight lines ( f ± f ± ), containing thecorresponding points f ± , ± . The cross-ratios turn into ratios of directed lengths, e.g., q ( f, f , f , , f ) = − l ( f , f , ) l ( f , , f ) . If the affine space through the points f ± , f ± is three-dimesnional, then, according topart 1) of Theorem 22, the four points f ± , ± lie in a plane (a sphere through f = ∞ ).Generalized Menelaus theorem (Theorem 13) provides us with the following necessary andsufficient condition for this, which reads: l ( f , f , ) l ( f , , f ) · l ( f , f , − ) l ( f , − , f − ) · l ( f − , f − , − ) l ( f − , − , f − ) · l ( f − , f − , ) l ( f − , , f ) = 1 . (42)This is equivalent to eq. (41) with f = ∞ .If, on the contrary, the four points f ± , f ± are co-planar, then, according to part 2)of Theorem 22, both lines ( f − , f , ) and ( f − , − f , − ) meet the line ( f − f ) at the same21Sfrag replacements f f f f − f − f , f − , f − , − f , − qq − q − q − , − Figure 9: Four adjacent quadrilaterals of a discrete isothermic surface: the cross-ratiossatisfy q · q − , − = q − · q − point ℓ (1) . Thus, we are in the situation of Fig. 5, described by the Desargues theorem.Here, we apply the Menelaus theorem twice, to the triangle △ ( f − , f , f ) intersected bythe line ( f − , f , ), and to the triangle △ ( f − , f − , f ) intersected by the line ( f − , − f , − ): l ( f , f ) l ( f , f ) · l ( f − , f − , ) l ( f − , , f ) = − l ( f − , ℓ (1) ) l ( ℓ (1) , f ) = l ( f − , f , − ) l ( f , − , f ) · l ( f − , f − , − ) l ( f − , − , f − ) . This yields formula (42), again. (cid:3)
For multidimensional discrete isothermic nets Theorem 23 yields a similar characteri-zation.
Theorem 25 (Cross-ratios of three adjacent quadrilaterals)
A circular net f : Z m → R N is discrete isothermic, if and only if the cross-ratios of its elementary quadri-laterals satisfy the following condition: q ( f, f i , f ij , f j ) · q ( f, f j , f jk , f k ) · q ( f, f k , f ki , f i ) = 1 (43) for any triple of different indices i, j, k . Proof.
Again, perform a M¨obius transformation sending f to ∞ . Under such a transfor-mation, the three adjacent circles through f turn into three straight lines ( f i f j ), ( f j f k ) and( f k f i ), containing the (white) points f ij , f jk and f ki , respectively. Concircularity of thesewhite points with f means simply that they are collinear. The necessary and sufficientcondition for this is given by the Menelaus theorem: l ( f j , f ij ) l ( f ij , f i ) · l ( f k , f jk ) l ( f jk , f j ) · l ( f i , f ki ) l ( f ki , f k ) = − . (44)22ince the M¨obius-invariant meaning of the ratios of directed lengths is given by the corre-sponding cross-ratios, q ( f, f i , f ij , f j ) = − l ( f i , f ij ) l ( f ij , f j ) , eq. (44) is equivalent to eq. (43). (cid:3) The conclusions of Theorems 24, 25 can be summarized with the help of the followingnotion:
Definition 26 (Edge labelling)
A system of real-valued functions α i defined on theedges of Z m parallel to the i -th coordinate axis ( i = 1 , . . . , m ) is called an edge labelling ,if they take equal values on each pair of opposite edges of any elementary quadrilateral. Thus, both edges ( u, u + e i ) and ( u + e j , u + e i + e j ) of an elementary square of Z m parallelto the coordinate plane ( ij ) carry the label α i = α i ( u ) = α i ( u + e j ), and, similarly, bothother edges ( u, u + e j ) and ( u + e i , u + e i + e j ) carry the label α j = α j ( u ) = α j ( u + e i ),see Fig. 10. In this notation, there holds τ j α i = α i for i = j , so that each function α i ( u )depends on u i only. α i α j α j α i f f i f ij f j Figure 10: Labelling of edges of a discrete isothermic netThe following theorem is an immediate consequence of Theorems 24, 25.
Theorem 27 (Factorized cross-ratios)
A circular net f : Z m → R N is discrete isother-mic, if and only if the cross-ratios of its elementary quadrilaterals satisfy q ( f, f i , f ij , f j ) = α i α j , (45) where α i ( i = 1 , . . . , m ) constitute a real-valued labelling of the edges of Z m . Theorem 27 says that our definition of discrete isothermic nets coincides with theoriginal definition from [BP]. In the next subsection we will give a more concrete way ofdetermining the labelling α i for a given discrete isothermic net.23 .3 Metric of a discrete isothermic net Now we turn to a characterization of discrete Koenigs nets given in Corollary 18. Beingapplied to circular nets, it says that such a net f is Koenigs, if and only if there existsa function s : Z m → R ∗ such that for any circular quadrilateral ( f, f i , f ij , f j ) with theintersection point of diagonals M there holds: l ( M, f ij ) l ( M, f ) = s ij s , l ( M, f j ) l ( M, f i ) = s j s i . (46)(Note that the notation s comes to replace ν which we reserve for general Koenigs nets.)The function s for circular nets turns out to admit an additional property. Theorem 28 (Discrete metric for discrete isothermic nets)
For a discrete isother-mic net f , relations (46) define a function s : Z m → R uniquely, up to a black-whitere-scaling which can be fixed by prescribing s arbitrarily at one black and at one whitepoint. There exists a labelling α of edges of Z m such that | f i − f | = α i ss i ( i = 1 , . . . , m ) . (47) A black-white re-scaling of the function s ( s λs on black vertices, s µs on whitevertices) results in the re-scaling α ( λµ ) − α of the labelling α . Proof.
For a circular quadrilateral ( f, f i , f ij , f j ) with the intersection point of diagonals M , one has two pairs of similar triangles, △ ( f, f i , M ) ∼ △ ( f j , f ij , M ) , △ ( f, f j , M ) ∼ △ ( f i , f ij , M ) . Hence, there holds: | M f ij || M f i | = | M f j || M f | = | f ij − f j || f i − f | , | M f ij || M f j | = | M f i || M f | = | f ij − f i || f j − f | . (48)There follows: | M f ij || M f | · |
M f j || M f i | = | f ij − f j | | f i − f | , | M f ij || M f | · |
M f i || M f j | = | f ij − f i | | f j − f | . (49)This can be written as l ( M, f ij ) l ( M, f ) · l ( M, f j ) l ( M, f i ) = | f ij − f j | | f i − f | , l ( M, f ij ) l ( M, f ) · l ( M, f i ) l ( M, f j ) = | f ij − f i | | f j − f | . (50)Indeed, contemplating Fig. 11, it is not difficult to realize that the fractions on the left-handside of each one of the two equations in (50) are either both negative (for an embeddedquadrilateral), or both positive (for a non-embedded quadrilateral), so that the replacementof the quotients of lengths in eq. (49) by quotients of directed lengths in eq. (50) is legitime.Substitute the defining relations (46) of the function s into eq. (50): s j s ij ss i = | f ij − f j | | f i − f | , s i s ij ss j = | f ij − f i | | f j − f | . (51)24Sfrag replacements ff f i f i f j f j f ij f ij MM Figure 11: Circular quadrilaterals, an embedded and a non-embedded ones.But this is equivalent to the claim that the functions α i = | f i − f | ss i (52)possess the labelling property, τ j α i = α i . (cid:3) The notations α i for edge labellings in Theorems 27 and 28 coincide not without areason. Theorem 29 (Origin of the edge labelling for factorized cross-ratios)
If the edgelabelling α i for a discrete isothermic net f : Z m → R N is introduced according to eq. (47),then the cross-ratios of its elementary quadrilaterals are factorized as in eq. (45). Proof.
For a circular quadrilateral ( f, f i , f ij , f j ) one has: q ( f, f i , f ij , f j ) = ǫ | f i − f | · | f ij − f j || f j − f | · | f ij − f i | , where ǫ < ǫ > q ( f, f i , f ij , f j ) = ǫ | f i − f | | f j − f | · | f ij − f j || f i − f | · | f j − f || f ij − f i | . Upon using eqs. (47) and (48), the latter equation can be re-written as q ( f, f i , f ij , f j ) = ǫ α i s i α j s j · | M f j || M f i | = α i s i α j s j · l ( M, f j ) l ( M, f i ) , and finally, due to eq. (51), we arrive at q ( f, f i , f ij , f j ) = α i s i α j s j · s j s i = α i α j , (cid:3) Theorem 28 as it stands cannot be reversed: existence of a function s satisfying (47)does not yield the Koenigs property. Indeed, from eqs. (47) and (50) one finds: l ( M, f ij ) l ( M, f ) · l ( M, f j ) l ( M, f i ) = s j s ij ss i , l ( M, f ij ) l ( M, f ) · l ( M, f i ) l ( M, f j ) = s i s ij ss j , (53)which is equivalent to l ( M, f ij ) l ( M, f ) = ± s ij s , l ( M, f i ) l ( M, f j ) = ± s i s j (54)(with the same sign ± in both equations). The latter equation is somewhat weaker than eq.(51), which is necessary and sufficient for the net f to be Koenigs. However, assuming someadditional information about f , it is possible to force the plus signs in the latter formula.For instance, if it is known that all elementary quadrilaterals of a two-dimensional circularnet f are embedded, then property (47) is sufficient to assure that f is Koenigs. Indeed,in this case α /α <
0, so that eq. (47) yields s /s < s /s <
0, and then the plussign has to be chosen in eq. (54).
Specializing the notion of duality from general Koenigs nets to circular ones, the firstessential observation is: the dual net for a discrete isothermic net is discrete isothermic, aswell. Indeed, any quadrilateral with sides parallel to the corresponding sides of a circularquadrilateral is, obviously, also circular. A more detailed description of duality for discreteisothermic nets is contained in the following theorem.
Theorem 30 (Dual discrete isothermic net)
Let f : Z m → R N be a discrete isother-mic net, with the factorized cross-ratios q ( f, f i , f ij , f j ) = α i α j (55) and with the discrete metric s : Z m → R ∗ . Then the R N -valued discrete one-form δf ∗ defined by δ i f ∗ = α i δ i f | δ i f | = δ i fss i , i = 1 , . . . , m, (56) is closed. Its integration defines (up to a translation) a net f ∗ : Z → R N , called dual tothe net f , or Christoffel transform of the net f . The net f ∗ is discrete isothermic, withthe cross-ratios q ( f ∗ , f ∗ i , f ∗ ij , f ∗ j ) = α i α j (57) and with the discrete metric s ∗ = s − : Z m → R ∗ . Conversely, if for a given net f : Z m → R N there exists an edge labelling α i such that the discrete one-form δ i f ∗ = α i δ i f | δ i f | (58) is closed, then f is a discrete isothermic net, with cross-ratios as in eq. (55). roof. The first part of the theorem is a consequence of the general construction of dualKoenigs nets. To prove to converse part, observe that closeness of the one-form (58) impliesthat the quadrilateral ( f, f i , f ij , f j ) is planar. Identifying its plane with C , we see that thecloseness condition is equivalent to (the complex conjugate of) α i f i − f − α i f ij − f j = α j f j − f − α j f ij − f i . Upon clearing denominators the latter equation turns into the cross-ratio equation (55)(in the generic situation, when f ij − f i − f j + f = 0). Thus, the closeness of the form (58)actually characterizes discrete isothermic nets. (cid:3) Corollary 31
The non-corresponding diagonals of any elementary quadrilateral of a dis-crete isothermic net f and of its dual are related by f ∗ i − f ∗ j = ( α i − α j ) f ij − f | f ij − f | , f ∗ ij − f ∗ = ( α i − α j ) f i − f j | f i − f j | . (59) Proof.
We put eq. (45) into several equivalent forms; these computations hold not onlyin the Clifford algebra C ℓ ( R N ), but in an arbitrary associative algebra with unit A . Beingwritten as α i ( f ij − f i )( f i − f ) − = α j ( f ij − f j )( f j − f ) − , (60)this equation displays the symmetry with respect to the diagonal flips of an elementaryquadrilateral, expressed as f i ↔ f j and f ↔ f ij , respectively (both have to be accompaniedby the change α i ↔ α j ). Writing eq. (60) as α i ( f ij − f )( f i − f ) − − α i = α j ( f ij − f )( f j − f ) − − α j , and dividing from the left by f ij − f , we arrive at the so-called three-leg form of thecross-ratio equation:( α i − α j )( f ij − f ) − = α i ( f i − f ) − − α j ( f j − f ) − . (61)According to eq. (56), the right-hand side of eq. (61) is equal to − ( f ∗ i − f ∗ ) + ( f ∗ j − f ∗ ) = f ∗ j − f ∗ i . This proves the first equation in (59). The second one is analogous. (cid:3) The discrete metric of a discrete isothermic net f can be used to produce its Moutardrepresentative, or, better, a Moutard representative of its lift ˆ f into the light cone of R N +1 , . This leads to a new characterization of discrete isothermic nets which is manifestlyM¨obius invariant, since it is given entirely within the formalism of the projective model ofM¨obius geometry. The following statement is a discrete analog of Theorem 6. Theorem 32 (Discrete isothermic nets = discrete Moutard nets in light cone) If f : Z m → R N is a discrete isothermic net, then its lift y = s − ˆ f : Z m → L N +1 , to thelight cone of R N +1 , satisfies the discrete Moutard equation (32). onversely, given a discrete Moutard net y : Z m → L N +1 , in the light cone, let thefunctions s : Z m → R and f : Z m → R N be defined by y = s − ( f + e + | f | e ∞ ) (62) (so that s − is the e -component, and s − f is the R N -part of y in the basis e , . . . , e N , e , e ∞ ).Then f is a discrete isothermic net. Proof.
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