Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization
Abstract
We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices). We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces. The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems. The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory. We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality. We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in \(\mathbb {R}^{3}\), as hyperelliptic curves, and as \(\mathbb {CP}^{1}\) modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization. Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller.
1 Introduction
The history of nonlinear theories of discrete conformal maps goes back to Thurston, who introduced patterns of circles as elementary geometric way to visualize hyperbolic polyhedra [45, Chapter 13]. His conjecture that circle packings could be used to approximate Riemann mappings was proved by Rodin and Sullivan [35]. This initiated a period of intensive research on circle packings and circle patterns, which lead to a fullfledged theory of discrete analytic functions and discrete conformal maps [44].
A related but different nonlinear theory of discrete conformal maps is based on a straightforward definition of discrete conformal equivalence for triangulated surfaces: Two triangulations are discretely conformally equivalent if the edge lengths are related by scale factors assigned to the vertices. This also leads to a surprisingly rich theory [5, 17, 18, 28]. In this article, we investigate different aspects of this theory (Fig. 1).
We extend the notion of discrete conformal equivalence from triangulated surfaces to polyhedral surfaces with faces that are inscribed in circles. The basic definitions and their immediate consequences are discussed in Sect. 2.
In Sect. 3, we generalize a variational principle for discretely conformally equivalent triangulations [5] to the polyhedral setting. This variational principle is the main tool for all our numerical calculations. It is also the basis for our uniqueness proof for discrete conformal mapping problems (Theorem 3.9).
Section 4 is concerned with the special case of quadrilateral meshes. We discuss the emergence of orthogonal circle patterns, a peculiar necessary condition for the existence of solutions for boundary angle problems, and we extend the method of constructing discrete Riemann maps from triangulations to quadrangulations.
In Sect. 5, we briefly discuss discrete conformal maps from multiply connected domains to circle domains, and special cases in which we can map to slit domains.
Section 6 deals with conformal mappings onto the sphere. We generalize the method for triangulations to quadrangulations, and we explain how the spherical version of the variational principle can in some cases be used for numerical calculations although the corresponding functional is not convex.
Section 7 is concerned with the uniformization of tori, i.e., the representation of Riemann surfaces as a quotient space of the complex plane modulo a period lattice. We consider Riemann surfaces represented as immersed surfaces in \(\mathbb R^{3}\), and as elliptic curves. We conduct numerical experiments to test the conjectured convergence of discrete conformal maps. We consider the difference between the true modulus of an elliptic curve (which can be calculated using hypergeometric functions) and the modulus determined by discrete uniformization, and we estimate the asymptotic dependence of this error on the number of vertices.
In Sect. 8, we consider the Fuchsian uniformization of Riemann surfaces represented in different forms. We consider immersed surfaces in \(\mathbb R^{3}\) (and \(S^{3}\)), hyperelliptic curves, and Riemann surfaces represented as a quotient of \(\hat{\mathbb C}\) modulo a classical Schottky group. That is, we convert from Schottky uniformization to Fuchsian uniformization. The section ends with two extended examples demonstrating, among other things, a remarkable geometric characterization of hyperelliptic surfaces due to Schmutz Schaller.
2 Discrete Conformal Equivalence of Cyclic Polyhedral Surfaces
2.1 Cyclic Polyhedral Surfaces
A euclidean polyhedral surface is a surface obtained from gluing euclidean polygons along their edges. (A surface is a connected twodimensional manifold, possibly with boundary.) In other words, a euclidean polyhedral surface is a surface equipped with, first, an intrinsic metric that is flat except at isolated points where it has conelike singularities, and, second, the structure of a CW complex with geodesic edges. The set of vertices contains all conelike singularities. If the surface has a boundary, the boundary is polygonal and the set of vertices contains all corners of the boundary.
Hyperbolic polyhedral surfaces and spherical polyhedral surfaces are defined analogously. They are glued from polygons in the hyperbolic and elliptic planes, respectively. Their metric is locally hyperbolic or spherical, except at conelike singularities.
We will only be concerned with polyhedral surfaces whose faces are all cyclic, i.e., inscribed in circles. We call them cyclic polyhedral surfaces. More precisely, we require the polygons to be cyclic before they are glued together. It is not required that the circumcircles persist after gluing; they may be disturbed by conelike singularities. A polygon in the hyperbolic plane is considered cyclic if it is inscribed in a curve of constant curvature. This may be a circle (the locus of points at constant distance from its center), a horocycle, or a curve at constant distance from a geodesic.
A triangulated surface, or triangulation for short, is a polyhedral surface all of whose faces are triangles. All triangulations are cyclic.
2.2 Notation
We will denote the sets of vertices, edges, and faces of a CW complex \(\Sigma \) by \(V_{\Sigma }\), \(E_{\Sigma }\), and \(F_{\Sigma }\), and we will often omit the subscript when there is no danger of confusion. For notational convenience, we require all CW complexes to be strongly regular. This means that we require that faces are not glued to themselves along edges or at vertices, that two faces are not glued together along more than one edge or one vertex, and that edges have distinct endpoints and two edges have at most one endpoint in common. This allows us to label edges and faces by their vertices. We will write \( ij \in E\) for the edge with vertices \(i,j\in V\) and \( ijkl \in F\) for the face with vertices \(i,j,k,l\in V\). We will always list the vertices of a face in the correct cyclic order, so that for example the face \( ijkl \) has edges \( ij \), \( jk \), \( kl \), and \( li \). The only reason for restricting our discussion to strongly regular CW complexes is to be able to use this simple notation. Everything we discuss applies also to general CW complexes.
2.3 Discrete Metrics
Proposition and Definition 2.1
If \(\Sigma \) is a surface with the structure of a CW complex and a function \(\ell :E_{\Sigma }\rightarrow \mathbb R_{>0}\) satisfies the polygon inequalities (1), then there is a unique euclidean cyclic polyhedral surface and also a unique hyperbolic cyclic polyhedral surface with CW complex \(\Sigma \) and discrete metric \(\ell \). If \(\ell \) also satisfies the inequalities (2), then there is a unique spherical cyclic polyhedral surface with CW complex \(\Sigma \) and discrete metric \(\ell \).
We will denote the euclidean, hyperbolic, and spherical polyhedral surface with CW complex \(\Sigma \) and discrete metric \(\ell \) by \((\Sigma ,\ell )_{ euc }\), \((\Sigma ,\ell )_{ hyp }\), and \((\Sigma ,\ell )_{ sph }\), respectively.
2.4 Discrete Conformal Equivalence
We extend the definition of discrete conformal equivalence from triangulations [5, 28] to cyclic polyhedral surfaces in a straightforward way (Definition 2.2). While some aspects of the theory carry over to the more general setting (e.g., Möbius invariance, Proposition 2.5), others do not, like the characterization of discretely conformally equivalent triangulations in terms of length crossratios (Sect. 2.5). We will discuss similar characterizations for polyhedral surfaces with 2colorable vertices and the particular case of quadrilateral faces in Sects. 2.7 and 2.8.
We define discrete conformal equivalence only for polyhedral surfaces that are combinatorially equivalent (see Remark 2.4). Thus, we may assume that the surfaces share the same CW complex \(\Sigma \) equipped with different metrics \(\ell \), \(\tilde{\ell }\).
Definition 2.2
 Two euclidean cyclic polyhedral surfaces \((\Sigma ,\ell )_{ euc }\) and \((\Sigma ,\tilde{\ell })_{ euc }\) are discretely conformally equivalent if there exists a function \(u:V_{\Sigma }\rightarrow \mathbb R\) such that$$\begin{aligned} \tilde{\ell }_ ij =e^{\frac{1}{2}(u_{i}+u_{j})}\ell _ ij . \end{aligned}$$(3)
 Two hyperbolic cyclic polyhedral surfaces \((\Sigma ,\ell )_{ hyp }\) and \((\Sigma ,\tilde{\ell })_{ hyp }\) are discretely conformally equivalent if there exists a function \(u:V_{\Sigma }\rightarrow \mathbb R\) such that$$\begin{aligned} \sinh \Big (\frac{\tilde{\ell }_ ij }{2}\Big ) = e^{\frac{1}{2}(u_{i}+u_{j})}\, \sinh \Big (\frac{\ell _ ij }{2}\Big ). \end{aligned}$$(4)
 Two spherical cyclic polyhedral surfaces \((\Sigma ,\ell )_{ sph }\) and \((\Sigma ,\tilde{\ell })_{ sph }\) are discretely conformally equivalent if there exists a function \(u:V_{\Sigma }\rightarrow \mathbb R\) such that$$\begin{aligned} \sin \Big (\frac{\tilde{\ell }_ ij }{2}\Big ) = e^{\frac{1}{2}(u_{i}+u_{j})}\, \sin \Big (\frac{\ell _ ij }{2}\Big ). \end{aligned}$$(5)
 A euclidean cyclic polyhedral surface \((\Sigma ,\ell )_{ euc }\) and a hyperbolic cyclic polyhedral surface \((\Sigma ,\tilde{\ell })_{ hyp }\) are discretely conformally equivalent if$$\begin{aligned} \sinh \Big (\frac{\tilde{\ell }_ ij }{2}\Big ) = e^{\frac{1}{2}(u_{i}+u_{j})}\ell _ ij . \end{aligned}$$(6)
 A euclidean cyclic polyhedral surface \((\Sigma ,\ell )_{ euc }\) and a spherical cyclic polyhedral surface \((\Sigma ,\tilde{\ell })_{ sph }\) are discretely conformally equivalent if$$\begin{aligned} \sin \Big (\frac{\tilde{\ell }_ ij }{2}\Big ) = e^{\frac{1}{2}(u_{i}+u_{j})}\ell _ ij . \end{aligned}$$(7)
 A hyperbolic cyclic polyhedral surface \((\Sigma ,\ell )_{ hyp }\) and a spherical cyclic polyhedral surface \((\Sigma ,\tilde{\ell })_{ sph }\) are discretely conformally equivalent if$$\begin{aligned} \sin \Big (\frac{\tilde{\ell }_ ij }{2}\Big ) = e^{\frac{1}{2}(u_{i}+u_{j})}\sinh \Big (\frac{\ell _ ij }{2}\Big ). \end{aligned}$$(8)
Remark 2.3
Note that relation (5) for spherical edge lengths is equivalent to relation (3) for the euclidean lengths of the chords in the ambient \(\mathbb R^{3}\) of the sphere (see Fig. 2, left). Likewise, relation (4) for hyperbolic edge lengths is equivalent to (3) for the euclidean lengths of the chords in the ambient \(\mathbb R^{2,1}\) of the hyperboloid model of the hyperbolic plane (see Fig. 2, right).
Remark 2.4
For triangulations, the definition of discrete conformal equivalence has been extended to meshes that are not combinatorially equivalent [5, Definition 5.1.4] [17, 18]. It is not clear whether or how the following definitions for cyclic polyhedral surfaces can be extended to combinatorially inequivalent CW complexes.
The discrete conformal class of a cyclic polyhedral surface embedded in ndimensional euclidean space is invariant under Möbius transformations of the ambient space:
Proposition 2.5
(Möbius invariance) Suppose P and \(\tilde{P}\) are two combinatorially equivalent euclidean cyclic polyhedral surfaces embedded in \(\mathbb R^{n}\) (with straight edges and faces), and suppose there is a Möbius transformation of \(\mathbb R^{n}\cup \{\infty \}\) that maps the vertices of P to the corresponding vertices of \(\tilde{P}\). Then P and \(\tilde{P}\) are discretely conformally equivalent.
Note that only vertices are related by the Möbius transformation, not edges and faces, which remain straight. The simple proof for the case of triangulations [5] carries over without change.
2.5 Triangulations: Characterization by Length CrossRatios
For euclidean triangulations, there is an alternative characterization of conformal equivalence in terms of length crossratios [5]. We review the basic facts in this section.
Proposition 2.6
Two euclidean triangulations \((\Sigma , \ell )_{ euc }\) and \((\Sigma , \tilde{\ell })_{ euc }\) are discretely conformally equivalent if and only if for each interior edge \({ ij}\in E^{ int }_{\Sigma }\), the induced length crossratios agree.
2.6 Triangulations: Reconstructing Lengths from Length CrossRatios
2.7 Bipartite Graphs: Characterization by Length MultiRatios
A different characterization of discrete conformal equivalence in terms of length multiratios holds if the 1skeleton of the polyhedral surface is bipartite, i.e., if the vertices can be colored with two colors so that no two neighboring vertices share the same color.
Proposition 2.8
Proof
(i) This is obvious, because all scale factors \(e^{u}\) cancel. (ii) It is easy to see that Eq. (3) can be solved for the scale factors \(e^{u/2}\) if the length multiratios are equal. Note that the scale factors are not uniquely determined: they can be multiplied by \(\lambda \) and \(1/\lambda \) on the two vertex color classes, respectively. To find a particular solution, one can fix the value of \(e^{u/2}\) at one vertex, and find the other values by alternatingly dividing and multiplying by \(\tilde{\ell }/\ell \) along paths. The equality of length multiratios implies that the obtained values do not depend on the path. \(\square \)
Remark 2.9
If a polyhedral surface is simply connected, then its 1skeleton is bipartite if and only if all faces are even polygons. If a polyhedral surface is not simply connected, then having even faces is only a necessary condition for being bipartite.
A polyhedral surface with bipartite 1skeleton has even faces. If a polyhedral surface has even faces and is simply connected, then its 1skeleton is bipartite, and the face boundaries generate all cycles. Thus, Proposition 2.8 implies the following corollary.
Corollary 2.10
Two simply connected combinatorially equivalent euclidean cyclic polyhedral surfaces with even faces and with discrete metrics \(\ell \) and \(\tilde{\ell }\) are discretely conformally equivalent if and only if the multiratio condition (14) holds for every face boundary cycle.
Remark 2.11
Analogous statements hold for spherical and hyperbolic cyclic polyhedral surfaces. In the multiratio condition, one has to replace noneuclidean lengths \(\ell \) with \(\sin \frac{\ell }{2}\) or \(\sinh \frac{\ell }{2}\), respectively (compare Remark 2.3).
2.8 Quadrangulations: The CrossRatio System on QuadGraphs
The case of cyclic quadrilateral faces is somewhat special (and we will return to it in Sect. 4), because equal length crossratio implies equal complex crossratio:
Proposition 2.12
Proof
This follows immediately from Proposition 2.8: The length multiratio of a quadrilateral is the modulus of the complex crossratio. If the (embedded) quadrilaterals are cyclic, then their complex crossratios are real and negative, so their arguments are also equal. \(\square \)
By Proposition 2.12, two discretely conformally equivalent planar quadrangulations correspond to two solutions of the crossratio system on the same quadgraph with the same crossratios Q. The following proposition says that in the simply connected case, one can find complex factors w on the vertices whose absolute values \(w=e^{u/2}\) govern the length change of edges according to (3), and whose arguments govern the rotation of edges. Note that (3) is obtained from (16) by taking absolute values.
Proposition 2.13
Proof
As in the proof of Proposition 2.8, it is easy to see that the system of equations (16) is solvable for w if and only if the complex multiratios for even cycles are equal. Because \(\Sigma \) is simply connected, this is the case if and only if the complex crossratios of corresponding faces are equal. \(\square \)
Remark 2.14
 (i)It takes the same value on opposite edges,$$\begin{aligned} a_{ ij}=a_{ kl}, \quad a_{ jk}=a_{ li}. \end{aligned}$$(17)
 (ii)$$\begin{aligned} Q_{ ijkl} = \frac{a_{ ij}}{a_{ jk}}\ . \end{aligned}$$(18)
In Adler et al. classification of integrable equations on quadgraphs [2], the integrable crossratio system is called \((\mathrm{Q1})_{\delta =0}\). It is also known as the discrete Schwarzian Korteweg–de Vries (dSKdV) equation, especially when it is considered on the regular square lattice [33] with constant crossratios.
If the crossratios Q have unit modulus, the crossratio system on quadgraphs is connected with circle patterns with prescribed intersection angles [6, 7].
Remark 2.15
3 Variational Principles for Discrete Conformal Maps
3.1 Discrete Conformal Mapping Problems
We will consider the following discrete conformal mapping problems. (The notation \((\Sigma ,\ell )_{g}\) was introduced in Definition 2.1.)
Problem 3.1

A euclidean, spherical, or hyperbolic cyclic polyhedral surface \((\Sigma ,\ell )_{g}\), where \(g\in \{ euc , hyp , sph \}\),

a desired total angle \(\varTheta _{i}>0\) for each vertex \(i\in V_{\Sigma }\),

a choice of geometry \(\tilde{g}\in \{ euc , hyp , sph \}\),
For interior vertices, \(\varTheta \) prescribes a desired cone angle. For boundary vertices, \(\varTheta \) prescribes a desired interior angle of the polygonal boundary. If \(\varTheta _{i}=2\pi \) for all interior vertices i, then Problem 3.1 asks for a flat metric in the discrete conformal class, with prescribed boundary angles if the surface has a boundary.
More generally, we will consider the following problem, where the logarithmic scale factors u (see Definition 2.2) are fixed at some vertices and desired angle sums \(\varTheta \) are prescribed at the other vertices. The problems to find discrete Riemann maps (Sect. 4.2) and maps onto the sphere (Sect. 6.1) can be reduced to this mapping problem with some fixed scale factors.
Problem 3.2

A euclidean, spherical, or hyperbolic cyclic polyhedral surface \((\Sigma ,\ell )_{g}\), where \(g\in \{ euc , hyp , sph \}\),

a partition \(V_{\Sigma }=V_0\dot{\cup }V_1\)

a prescribed angle \(\varTheta _{i} > 0\) for each vertex \(i\in V_1\),

a prescribed logarithmic scale factor \(u_i\in \mathbb R\) for each vertex \(i\in V_0\),

a choice of geometry \(\tilde{g}\in \{ euc , hyp , sph \}\),
3.2 Analytic Formulation of the Mapping Problems
Lemma 3.3

the polyhedral surface \((\Sigma ,\ell )_{g}\),

the partition \(V_{0}\dot{\cup }V_{1}\),

\(\varTheta _{i}\) for \(i\in V_{1}\),

\(u_{i}\) for \(i\in V_{0}\),

the geometry \(\tilde{g}\in \{ euc , hyp , sph \}\)
Problem 3.4

an abstract triangulation \(\varDelta \),

a partition \(V_{\varDelta }=V_0\dot{\cup }V_1\),

\(u_{i}\in \mathbb R\) for \(i\in {V_{0}}\)

\(\varTheta _{i}\in \mathbb R_{>0}\) for \(i\in {V_{1}}\),

a partition \(E_{\varDelta }=E_{0}\dot{\cup }E_{1}\),

\(\lambda _{ ij}\) for \({ ij}\in {E_{0}}\),

\(\tilde{g}\in \{ euc , hyp , sph \}\),
Proof
(of Lemma 3.3) Note that (27) says that the angle sums at vertices in \(V_{1}\) have the prescribed values, and (28) says that neighboring triangles of \((\varDelta ,\tilde{\ell })_{\tilde{g}}\) belonging to the same face of \(\Sigma \) share the same circumcircle. So deleting the edges in \(E_{\varDelta }\setminus E_{\Sigma }\), one obtains a cyclic polyhedral surface \((\Sigma ,\tilde{\ell }_{E_{\Sigma }})_{\tilde{g}}\). \(\square \)
3.3 Variational Principles
Definition 3.5
We will often omit the subscripts and write simply \(E^{ euc },E^{ hyp },E^{ sph }\) when this is unlikely to cause confusion.
Definition 3.6

The feasible region of \(E^{ euc }\) and \(E^{ hyp }\) is the set of all \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) such that \(\tilde{\ell }\in \mathbb R_{>0}^{E}\) defined by (23) and (24) satisfies the triangle inequalities (25)

The feasible region of \(E^{ sph }\) is the set of all \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) such that \(\tilde{\lambda }\) defined by (23) is negative, and \(\tilde{\ell }\), which is then welldefined by (24), satisfies the triangle inequalities (25) and the inequalities (26).
Theorem 3.7
(Variational principles) Every solution \((\Sigma ,\tilde{\ell })_{\tilde{g}}\) of Problem 3.2 corresponds via (23) and (24) to a critical point \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) of the function \(E^{\tilde{g}}_{\varDelta ,\varTheta }\) under the constraints that \(\lambda _{ ij}\) and \(u_{i}\) are fixed for \({ ij}\in E_{0}\) and \(i\in V_{0}\), respectively. (The triangulation \(\varDelta \), and \(E_{0}=E_{\Sigma }\) and \(E_{1}=E_{\varDelta }\setminus E_{\Sigma }\) are as in Lemma 3.3, and the given function \(\varTheta \) is extended from \(V_{1}\) to V by arbitrary values on \(V_{0}\).)
Conversely, if \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) is a critical point of the function \(E^{\tilde{g}}_{\varDelta ,\varTheta }\) under the same constraints, and if \((\lambda , u)\) is contained in the feasible region of \(E^{\tilde{g}}_{\varDelta ,\varTheta }\), then \((\Sigma ,\tilde{\ell })_{\tilde{g}}\) defined by (23) and (24) is a solution of Problem 3.2.
Proof
This follows from the analytic formulation of Problem 3.2 (see Sect. 3.2) and Proposition 3.8. \(\square \)
Proposition 3.8
Proof
Equations (30) and (31) follow from the definition of \(E^{\tilde{g}}\) and Proposition 3.14 on the partial derivatives of \(f^{g}\). \(\square \)
Theorem 3.9
(Uniqueness for mapping problems) If Problem 3.2 with target geometry \(\tilde{g}\in \{ euc , hyp \}\) has a solution, then the solution is unique—except if \(\tilde{g}= euc \) and \(V_{0}=\emptyset \) (the case of Problem 3.1). In this case, the solution is unique up to scale.
The critical point \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) that corresponds, via (23) and (24), to a solution \((\Sigma ,\tilde{\ell })_{\tilde{g}}\) of Problem 3.2 with \(\tilde{g}\in \{ euc , hyp \}\) is a minimizer of \(E^{\tilde{g}}_{\varDelta ,\varTheta }\) under the constraints described in Theorem 3.7. The minimizer is unique except in the following cases. If \(\tilde{g}= euc \) and \(V_{0}=\emptyset \), then \(E^{\tilde{g}}_{\varDelta ,\varTheta }\) is constant along all lines in the “scaling direction” \((0,1_{V_{\varDelta }})\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\). If the 1skeleton of \(\Sigma \) is bipartite and \(V_{0}=\emptyset \), then \(E^{\tilde{g}}_{\varDelta ,\varTheta }\) is constant in the direction that is \(\pm 1\) on the two color classes of \(V_{\varDelta }\), respectively, and takes appropriate values on \(E_{\varDelta }\setminus E_{\Sigma }\) so that \(\tilde{\lambda }_{ ij}\) defined by (23) remains constant for all \({ ij}\in E_{\varDelta }\). (In both exceptional cases, one can obtain a unique minimizer by adding the constraint of fixing \(u_{i}\) for some \(i~\in V_{\varDelta }\).)
Proof
The theorem follows from Theorem 3.7 and the following observations.
 (1)
If the point \((\lambda , u)\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) corresponds to a solution of Problem 3.2, it is contained in the feasible region of \(E^{\tilde{g}}_{\varDelta ,\varTheta }\).
 (2)
 (3)For \((\lambda , u)\) in the feasible region, the second derivative \(D^{2}E^{ hyp }(\lambda ,u)\) is a positive definite quadratic form of \(d\tilde{\lambda }\), i.e., \(D^{2}E^{ hyp }(\lambda ,u)(\dot{\lambda },\dot{u})\ge 0\) for all \((\dot{\lambda },\dot{u})\in \mathbb R^{E_{\varDelta }}\times \mathbb R^{V_{\varDelta }}\) and \(D^{2}E^{ hyp }(\lambda ,u)(\dot{\lambda },\dot{u})=0\) if and only if$$\begin{aligned} \dot{\lambda }_{ ij}+\dot{u}_{i}+\dot{u}_{j}=0 \quad \text {for all }{} { ij}\in E_{\varDelta }. \end{aligned}$$
 (4)Similarly, for \((\lambda , u)\) in the feasible region, the second derivative \(D^{2}E^{ euc }(\lambda ,u)\) is a positive semidefinite quadratic form with \(D^{2}E^{ euc }(\lambda ,u)(\dot{\lambda },\dot{u})=0\) if and only if$$\begin{aligned} \dot{\lambda }_{ ij}+\dot{u}_{i}+\dot{u}_{j}=c \quad \text {for all }{} { ij}\in E_{\varDelta }, \text { for some } c\in \mathbb R. \end{aligned}$$
In the following proposition, we collect explicit formulas for the second derivatives of the functions \(E^{\tilde{g}}\). They are useful for the numerical minimization of \(E^{ euc }\) and \(E^{ hyp }\), and even for finding critical points of \(E^{ sph }\), as explained in Sect. 6.2.
Proposition 3.10
Proposition 3.10 follows from (29) and Proposition 3.15 about the second derivatives of \(f^{g}\).
3.4 The Triangle Functions
Definition 3.11
Definition 3.12

if \(\lambda \) is in the feasible region of \(f^{g}\), then the angles \(\alpha \), \(\beta \) are defined as the angles (shown in Fig. 4) in a euclidean, hyperbolic, or spherical triangle (depending on \(g\)) with sides \(\ell _{1}, \ell _{2}, \ell _{3}\) determined by (22). That is, \(\alpha \) and \(\beta \) are defined by (21) and (20).
 Otherwise, if \(g= sph \), and if either at least two \(\lambda \)s are nonnegative or \(\lambda <0\) and inequality (33) is violated, let$$\begin{aligned} \alpha _{k}=\alpha _{i}=\alpha _{j}=\pi , \quad \beta _{k}=\beta _{i}=\beta _{j}=0. \end{aligned}$$
 Otherwise, if the triangle inequality (32) is violated, or if \(g= sph \) and \(\lambda _{k}\ge 0\), let$$\begin{aligned} \alpha _{k}=\beta _{k}=\pi , \quad \alpha _{i}=\alpha _{j}=\beta _{i}=\beta _{j}=0. \end{aligned}$$
Figure 5 shows a graph of Milnor’s Lobachevsky function. It is continuous, \(\pi \)periodic, odd, has zeros at the integer multiples of \(\pi /2\), and is real analytic except at integer multiples of \(\pi \), where the derivative tends to \(+\infty \).
Remark 3.13
Proposition 3.14
Proof
Note that the angles \(\alpha \), \(\beta \) are continuous functions of \(\lambda \) on \(\mathbb R^{3}\). Hence \(f^{g}\) defined by (34) is also continuous. We will show that \(f^{g}\) is continuously differentiable with derivative (37) on an open dense subset of the domain, namely, the union of (a) the feasible region and (b) the interior of its complement. Since \(f^{g}\) is continuous and \( df^{g}\) extends continuously to \(\mathbb R^{3}\), the claim follows.
Proposition 3.15
A proof of (39) is contained in [5] (Proposition 4.2.3), see Remark 3.17 below. Equations (40) and (41) can be derived by lengthy calculations.
Proposition 3.16
(i) The function \(f^{ euc }\) is convex. On its feasible set, the second derivative \(D^{2}f^{ euc }\) is positive semidefinite with onedimensional kernel spanned by the “scaling direction” (1, 1, 1).
(ii) The function \(f^{ hyp }\) is convex. On its feasible set, the second derivative \(D^{2}f^{ hyp }\) is positive definite, so the functions is locally strictly convex.
Part (i) is proved in [5] (Propositions 4.2.4, 4.2.5, note the following remark) directly from (39). We do not know a similarly straightforward proof of part (ii). The proof in [5] (Sect. 6.2) is based on a connection with 3dimensional hyperbolic geometry: \(f^{ hyp }\) is the Legendre dual of the volume of an ideal hyperbolic prism considered as a function of the dihedral angles. This volume function is strictly concave, as shown by Leibon [26]. His argument uses the decomposition of an ideal prism into three ideal tetrahedra.
Remark 3.17
4 Conformal Maps of Cyclic Quadrangulations
Having introduced the mapping problems and variational principles, we return to conformal maps of cyclic quadrangulations. Some basic facts were already discussed in Sect. 2.8. Here, in Sect. 4.1, we consider a simple experiment that demonstrates the somewhat unexpected appearance of orthogonal circle patterns, and also a necessary condition for the boundary angles. In Sect. 4.2, we discuss a discrete version of the Riemann mapping problem for quadrangulations.
4.1 Emerging Circle Patterns and a Necessary Condition
On first sight, the \(6\times 6\) example shown in the top row behaves rather like one would expect from a conformal map. The horizontal and vertical “coordinate lines” of the domain are mapped to polygonal curves that look more or less like they could be discretizations of reasonable smooth curves. In the \(6\times 5\) example shown in the bottom row, the images of the vertical lines zigzag noticeably.
A closer look at the \(6\times 6\) example reveals a remarkable phenomenon. Let us bicolor the vertices black and white so that neighboring vertices have different colors, with the corners colored white. Then, in the image quadrangulation, the edges incident with a black vertex meet at right angles, and the edges incident with a white vertex have the same length. One can therefore draw a circle around each white vertex through the neighboring black vertices as shown in Fig. 6 (top right). At the black vertices, these circles touch and intersect orthogonally. Such circle patterns were studied by Schramm [38] as discrete analogs of conformal maps.
Given such a circle pattern with orthogonally intersecting circles, the quadrangulation formed by drawing edges between circle centers and intersection points consists of quadrilaterals that are rightangled kites. Such kites have complex crossratio \(1\). Hence, the quadrangulation coming from an orthogonal circle pattern is discretely conformally equivalent (in our sense) to a combinatorially equivalent quadrangulation consisting of squares.
The conformal map shown in the top row of Fig. 6 “finds” the orthogonal circle pattern because that circle pattern exists and the conformal map is unique (by Theorem 3.9). For the \(6\times 5\) example shown in the bottom row, a corresponding orthogonal circle pattern does not exist. No matter which coloring is chosen, there are two black vertices at which the total angle changes (from \(90^{\circ }\) to \(50^{\circ }\) and \(130^{\circ }\), respectively). The neighbors of a vertex do not lie on a circle. Figure 6 (bottom right) shows two circles drawn through three out of four neighbors.
If we map an \(m\times n\) square grid to a parallelogram like in Fig. 6, an orthogonal circle pattern will appear if m an n are even. No such pattern will appear if one of the numbers is even and the other is odd. What happens if both m and n are odd? In this case, the conformal map does not exist. The corners with increasing angle and the corners with decreasing angle would have different colors. This violates the necessary condition expressed in the following theorem.
Theorem 4.1
Proof
4.2 Riemann Maps with Cyclic Quadrilaterals
The basic idea is the same as for triangulations: First, map the polyhedral surface to the half plane with one boundary vertex at infinity. Then apply a Möbius transformation. This leads to a combinatorial restriction: No face may have more than one edge on the boundary. (The face would degenerate when the boundary is mapped to a straight line.) For triangulations, this means that no triangle may be connected to the surface by only one edge. If this condition is violated, cutting off such “ears” often leads to an admissible triangulation. For quadrangulations, this fix does not work in typical situations. Instead, if a quadrilateral contains two consecutive edges on the boundary, cut off a triangle. The resulting polyhedral surface will consist mostly of quadrilaterals with some triangles on the boundary, as in the example shown in Figs. 7, 8.
 (1)
Choose a vertex k on the boundary of \(\Sigma \) such that all incident faces are quadrilaterals.
 (2)
Apply a discrete conformal change of metric (3) such that all edges incident with k have the same length. One may choose \(u=0\) for all vertices except the neighbors of k. It does not matter if polygon inequalities are violated after this step.
 (3)
Let \((\Sigma ',\ell ')_{ euc }\) be the cyclic polyhedral complex obtained by removing vertex k and all incident quadrilaterals.
 (4)
Solve Problem 3.2 for \((\Sigma ',\ell ')_{ euc }\) with prescribed total angles \(\varTheta _i=2\pi \) for interior vertices of \(\Sigma '\), \(\varTheta _i=\pi \) for boundary vertices of \(\Sigma '\) that were not neighbors of k in \(\Sigma \), and fixed logarithmic scale factors \(u_i=0\) for those that were neighbors of k. The result is a planar polyhedral surface as shown in Fig. 7, bottom. The boundary consists of one straight line segment containing all boundary edges of \(\Sigma '\) that were also boundary edges of \(\Sigma \), and two or more straight line segments, each consisting of two edges that were incident with a removed quadrilateral.
 (5)
Apply a Möbius transformation (e.g., \(z\mapsto 1/z\)) to the vertices that maps the boundary vertices of \(\Sigma \) to a circle and the other vertices to the inside of this circle. Reinsert k at the image point of \(\infty \) under this Möbius transformation. Each face \({ ijmk} \in \Sigma \) incident with k is cyclic because the three vertices i, j, and m are contained in a line before transformation.
 (6)
Optionally apply a 2dimensional version of the Möbius normalization described in Sect. 6.3.
Proposition 4.2
The result of this procedure is a planar cyclic polyhedral surface that is discretely conformally equivalent to \((\Sigma , \ell )_{ euc }\) and has its boundary polygon inscribed in a circle.
Proof
That the boundary polygon is inscribed in a circle is obvious from the construction. Using the Möbius invariance of discrete conformal equivalence (Proposition 2.5), it is not difficult to see that the surfaces without quadrilaterals incident with k are discretely conformally equivalent. To show that the whole surfaces are equivalent, it suffices to show that corresponding quadrilaterals incident with k have the same complex crossratio.
After step (2), the length crossratio of a quadrilateral incident with k is equal to the simple length ratio of the two edges that are not incident with k.
After step (4), the length crossratio of these edges is unchanged due to the fixed logarithmic scale factors \(u=0\) on the neighbors of k. Also, these edges are now collinear because of the prescribed angle \(\varTheta =\pi \) between them.
After applying the Möbius transformation in step (5), the image of the point at infinity and the other three vertices of our quadrilateral incident with k form again a cyclic quadrilateral with the same complex crossratio as in the beginning. \(\square \)
5 Multiply Connected Domains
5.1 Circle Domains
 (1)
Fill holes by gluing faces to all but one boundary component, so that the resulting surface is homeomorphic to a disk.
 (2)
Construct the discrete Riemann map.
 (3)
Remove the faces that were added in step (1).
5.2 Special Slit Domains
Sometimes, the symmetry of the problem determines the right positions of the endvertices, so that discrete conformal maps to slit surfaces can be computed. The first two rows of Fig. 10 show examples. The bottom row visualizes a discrete conformal map where circular holes are mapped to slits. Here, we use the following trick: We start with the slit surface and map it to a surface with circular holes as described in Sect. 5.1.
6 Uniformization of Spheres
This section is concerned with discrete conformal maps of polyhedral surfaces of genus 0 onto the round sphere. For triangulations, this is described in [5] (Sect. 3.2). In Sect. 6.1, we adapt this method to quadrangulations. This is similar to the discrete Riemann mapping with quadrilaterals described in Sect. 4.2. Effectively, this method reduces the problem to minimizing the convex euclidean functional \(E^{ euc }\). The spherical version of the variational principle (Theorem 3.7) involves the nonconvex function \(E^{ sph }\). It is not as practical for calculations, because one has to find a saddle point instead of a minimum. Nevertheless, the spherical functional can often be used to calculate maps to the sphere. This is explained in Sect. 6.2.
6.1 Uniformizing Quadrangulations of the Sphere
Suppose \((\Sigma ,\ell )_{ euc }\) is a cyclic polyhedral surface with quadrilateral faces that is homeomorphic to the sphere.
 (1)
Choose a vertex \(k\in V_{\Sigma }\).
 (2)
Apply a discrete conformal change of metric (3) such that all edges incident with k have the same length. One may choose \(u=0\) for all vertices except the neighbors of k. It does not matter if polygon inequalities are violated after this step.
 (3)
Let \((\Sigma ',\ell ')_{ euc }\) be the complex obtained by removing vertex k and all incident quadrilaterals.
 (4)
Solve Problem 3.2 for \((\Sigma ',\ell ')_{ euc }\) with prescribed total angles \(\varTheta _i=2\pi \) for interior vertices of \(\Sigma '\), \(\varTheta _i=\pi \) for boundary vertices of \(\Sigma '\) that were not neighbors of k in \(\Sigma \), and fixed scale factors \(u_i=0\) for vertices that were neighbors of k in \(\Sigma \). The result is a planar polyhedral surface with cyclic quadrilaterals. Consecutive boundary edges that belonged to a face incident with vertex k in \(\Sigma \) are contained in a straight line.
 (5)
Map the vertices to the unit sphere by stereographic projection and reinsert the vertex k at the image point of \(\infty \).
 (6)
Optionally apply Möbius normalization, see Sect. 6.3.
Proposition 6.1
The result is a cyclic polyhedral surface with vertices on the unit sphere that is discretely conformally equivalent to \((\Sigma , \ell )_{ euc }\).
6.2 Using the Spherical Functional
It is possible to use the spherical functional \(E^{ sph }\) to calculate maps to the sphere even though it is not convex. For simplicity, we consider only triangulations, so all \(\lambda \) variables are fixed and we may consider \(E^{ sph }\) as function of the logarithmic scale factors u only (see Sect. 3.3). A numerical method has to find a saddle point of \(E^{ sph }(u)\).
Note that the scaling direction \(1_{V_{\varDelta }}\in \mathbb R^{V_{\varDelta }}\) is a negative direction of the Hessian at a critical point: Suppose \((\varDelta ,\ell )_{ sph }\) is a spherical triangulation with the desired angle sum \(\varTheta _{i}\) at each vertex i. Then \(0\in \mathbb R^{V_{\varDelta }}\) is a critical point of \(E^{ sph }_{\varDelta ,\varTheta }(u)\). If we enlarge all edge lengths by a common factor \(e^{h}>1\), then all angles become larger, so every component (30) of the gradient of \(E^{ sph }\) becomes negative. Following the negative gradient would result in even larger lengths.
6.3 Möbius Normalization
7 Uniformization of Tori
Every Riemann surface R of genus one is conformally equivalent to a flat torus, i.e., to a quotient space \(\mathbb C/\Gamma \), where \(\Gamma =\mathbb Z\omega _{1}+\mathbb Z\omega _{2}\) is some twodimensional lattice in \(\mathbb C\). The biholomorphic map from R to \(\mathbb C/\Gamma \), or from the universal cover of R to \(\mathbb C\), is called a uniformizing map. For a polyhedral surface of genus one, constructing a discrete uniformizing map amounts to solving Problem 3.1 with prescribed total angle \(\varTheta =2\pi \) at all vertices. This provides us with a method to calculate approximate uniformizing maps for Riemann surfaces of genus one given in various forms. We consider examples of tori immersed in \(\mathbb R^{3}\) in Sect. 7.1 and elliptic curves in Sect. 7.2. (We will also consider tori in the form of Schottky uniformization in Sect. 8.2, as a toy example after treating the higher genus case.)
The belief that discrete conformal maps approximate conformal maps is not based on a proven theorem but on experimental evidence like the Wente torus example of Sect. 7.1 and the numerical experiments of Sect. 7.4.
7.1 Immersed Tori
To test the numerical accuracy of our discrete uniformizing maps, we consider the famous torus of constant mean curvature discovered by Wente [47]. Explicit doubly periodic conformal immersion formulas (i.e., formulas for the inverse of a uniformizing map) are known in terms of elliptic functions [1, 3, 46].
7.2 Elliptic Curves
Remark 7.1
Instead of constructing a doubly covered convex euclidean polyhedron with vertices on the unit sphere as described above, one could also construct a spherical triangulation of the doubly covered sphere that is invariant under the elliptic involution (exchanging sheets). These two approaches are in fact equivalent due to Remark 2.3.
Mapping a flat torus to an elliptic curve. We can also go the opposite way, mapping a flat torus to a double cover of \(S^{2}\). Start with a triangulated flat torus. The triangulation should be symmetric with respect to the elliptic involution, i.e., symmetric with respect to a half turn around one vertex (which is then also a half turn around three other vertices). The quotient space of the triangulated torus modulo the elliptic involution is then a triangulated sphere. Map it to the sphere by the procedure explained in [5] (Sect. 3.2), see also Sect. 6.1 of the present article.
7.3 Choosing Points on the Sphere
As initial guess we choose points uniformly distributed in \(S^{2}\). To achieve this we choose points with normally distributed coordinates and project them to \(S^2\) [31].
7.4 Numerical Experiments
Given the branch points of an elliptic curve, the modulus \(\tau \) can be calculated in terms of hypergeometric functions. In this section, we compare the theoretical value of \(\tau \) with the value \(\hat{\tau }\) that we obtain by the discrete uniformization method explained in Sect. 7.2.
The results from these two experiments suggest that the error depends on the number n of vertices asymptotically like \(n^{\alpha }\), where the exponent \(\alpha <0\) depends on the mesh.
7.5 Putting a Square Pattern on a Spherical Mesh
8 Uniformization of Surfaces of Higher Genus
As in the case of tori (Sect. 7), we can find uniformizing maps for cyclic polyhedral surfaces of genus \(g\ge 2\) by solving the hyperbolic version (\(\tilde{g}= hyp \)) of Problem 3.1 with prescribed total angle \(\varTheta =2\pi \) at all vertices. (We will only consider triangulations in the following.) This allows us to calculate approximate uniformizations for Riemann surfaces of genus \(g\ge 2\) given in various forms, by approximating them with polyhedral surfaces.
In Sect. 8.1, we briefly discuss how to construct fundamental polygons and group generators.
We explain how to calculate the Fuchsian uniformization for Riemann surfaces given in the form of a Schottky uniformization in Sect. 8.2. We discuss the uniformization of hyperelliptic curves in Sect. 8.3 and a geometric characterization of hyperelliptic Riemann surfaces in Sect. 8.4.
8.1 Fundamental Polygons and Group Generators
We simplify this fundamental polygon by connecting vertices that are identified with more than one partner by geodesic arcs, as shown in Fig. 24b. The resulting polygon has in general more than one vertex class.
Hyperbolic translations tend to accumulate numerical errors quite fast when building products. The situation could be ameliorated somewhat by using the \(PSL(2,\mathbb R)\)representation of hyperbolic isometries [16], but the fundamental problem remains. For this reason, it is desirable to perform the cutandglue algorithm in such a way that the number of matrix products required to maintain the gluing translations is small. We follow the following greedy approach. Repeatedly we have to choose labels x, y such that the order in the polygon is \(x\cdots y\cdots x'\cdots y'\), and then perform a cutandglue sequence to bring the labels next to each other, \(xyx'y'\). We always choose a pair x, y, for which this requires the minimal number of matrix multiplications.
This algorithm is not optimal with respect to the number of multiplications necessary to maintain the edgegluing translations. Especially if the original polygon is already “close” to one with opposite sides identified, the detour via a canonical polygon is inefficient.
In all examples, we use a heuristic method based on the following idea: Find a longest sequence of different letters in the edge labeling (ignoring primes), and then try to move a different letter into this sequence by cutting and gluing.
8.2 From Schottky to Fuchsian Uniformization
In this section, we consider Riemann surfaces presented as quotient spaces of classical Schottky groups.
Definition 8.1
Let \(C_1,C'_1\ldots ,C_g,C'_g\) be circles in \(\hat{\mathbb C}\) that bound disjoint disks. A classical Schottky group is a Kleinian group generated by Möbius transformations \(\sigma _1,\ldots ,\sigma _g\), where \(\sigma _j\) maps the outside of \(C_j\) onto the inside of \(C'_j\).
Each generator \(\sigma _{j}\) has fixed points \(A_{j}\), \(B_{j}\) inside \(C_{j}\) and \(C'_{j}\), respectively. The limiting set A of G is the union of orbits of the fixed points \(A_{j}\), \(B_{j}\). G acts freely and properly discontinuously on the domain of discontinuity \(\Omega =\hat{\mathbb C}\setminus A\). The quotient space \(R=\Omega /G\) is a Riemann surface of genus g. The domain outside all of the circles is a fundamental domain of G. The identified pairs of circles form handles.
We discretize the Riemann surface \(R=\Omega /G\) determined by a classical Schottky group G as follows. First, construct a triangulation of \(\Omega \) whose vertex set and combinatorics are invariant under the action of G. (Ignore the fact that a Möbius transformation maps straight edges to circular arcs as in Proposition 2.5 on the Möbius invariance of conformal classes.) For example, the triangulation may be the Delaunay triangulation of a Ginvariant point set. The following construction avoids Delaunay triangulations of infinite (but symmetric) point configurations:
If necessary, choose a Möbius normalization for which the fundamental domain is bounded in \(\mathbb C\). For each pair of circles \(C_{j}, C'_{j}\) we construct polygons \(p_{1j},\ldots ,p_{n_{j}j}\) inscribed in \(C_{j}\) and \(p'_{1j},\ldots ,p'_{n_{j}j}\) inscribed in \(C'\) such that \(\sigma _{j}(p_{kj})=p'_{kj}\). For example, we may choose a regular ngon inscribed in \(C_{j}\) and map the vertices by \(\sigma _{j}\) to \(C'_{j}\). Triangulate the compact region bounded by these polygons, adding vertices in the interior as wanted. (For example, use a constrained Delaunay triangulation.) The images of this triangulation under the action of G (again, considering only combinatorics and vertex positions) form a Ginvariant triangulation \({\hat{\varDelta }}\) of the universal cover of R, hence a triangulation \(\varDelta \) of R. More precisely, the triangulations \({\hat{\varDelta }}\) and \(\varDelta \) are only defined up to isotopy fixing the vertices. The edgelengths \(\hat{\ell }\) (distances of vertices) do not project from \({\hat{\varDelta }}\) to \(\varDelta \), but the length crossratios \(\widehat{\text {lcr}}\) calculated from these edge lengths do, because they are Möbius invariant. The projected length crossratios \({\text {lcr}}\) determine a discrete conformal class for \(\varDelta \) (see Sect. 2.5).
To obtain a Fuchsian uniformization of R, construct edge lengths \(\ell \) from the length crossratios \({\text {lcr}}\) as described in Sect. 2.6. Then solve Problem 3.1 (or rather the corresponding analytic version, Problem 3.4) for \((\varDelta ,\ell )_{ euc }\) with \(\tilde{g}= hyp \) and desired angle sums \(\varTheta =2\pi \) at all vertices.
Figure 26 shows an example of the Fuchsian uniformization of a genus three surface presented by its Schottky uniformization.
8.3 Hyperelliptic Curves
We construct a polyhedral approximation of a hyperelliptic curve in the same way as for elliptic curves (Sect. 7.2). We choose points \(p_{1},\ldots ,p_{n}\) in addition to the \(2g+2\) branch points and take the convex hull. We cut the resulting polyhedron open along edge paths joining pairs of branch points and glue a second copy along the cuts.
8.4 Geometric Characterization of Hyperelliptic Surfaces
 1.
R is conformally equivalent to some hyperelliptic curve.
 2.
R is conformally equivalent to a branched cover of the sphere with \(2g+2\) branch points.
 3.
There is a conformal involution \(\tau :R\rightarrow R\) with exactly \(2g+2\) fixed points.
The involution \(\tau \) is called the hyperelliptic involution of R. By the RiemannHurwitz formula, the quotient surface \(R/\tau \) is a sphere.
All Riemann surfaces of genus two are hyperelliptic, but for every genus greater than two, there are Riemann surfaces that are not hyperelliptic. The following geometric characterization of hyperelliptic Riemann surfaces is due to Schmutz Schaller [36, 37].
Theorem 8.2
 (i)
R is hyperelliptic.
 (ii)
R has a set of \(2g2\) simple closed geodesics which all intersect in one point and which intersect in no other point.
 (iii)
R has a set of 2g simple closed geodesics which all intersect in one point and which intersect in no other point.
 (iv)
R has a fundamental polygon that is a 4ggon with opposite sides identified and equal opposite angles.
The fundamental polygon of condition (iv) is symmetric with respect to a \(180^{\circ }\) rotation around its center, which corresponds to the hyperelliptic involution on R. The \(2g+2\) fixed points on R are the vertex of the polygon, its center, and the 2g edge midpoints. The axes of the 2g edgegluing translations all go through the center. They project to 2g simple closed geodesics on R which all intersect in one point and intersect in no other point.
8.5 Example: Deforming a Hyperelliptic Surface
For this example, we construct an elliptichyperelliptic triangulated surface with additional symmetry. A surface is called elliptichyperelliptic if it is conformally equivalent to a twosheeted branched cover of the torus.
Take two regular tetrahedra (the faces of which are subdivided several times to obtain a finer mesh), cut them across pairs of opposite edges and glue them together to obtain a twosheeted cover of a tetrahedron branched at the four vertices. Now choose two paths in one of the sheets that connect the centers of the tetrahedron’s faces in pairs. Cut the surface along these paths, take another copy of this cut surface and glue corresponding cuts together to form an elliptichyperelliptic surface of genus three that is a fourfold cover of a regular tetrahedron. The surface possesses six antiholomorphic involutions corresponding to the six reflectional symmetries of the tetrahedron, and three holomorphic involutions corresponding to the rotational symmetries of the tetrahedron of order two. Each of the holomorphic involutions has eight fixed points covering the midpoints of a pair of opposite edges. Thus, this elliptichyperelliptic surface is also hyperelliptic.
8.6 Example: Different Forms of the Same Genus2 Surface

an octagon with canonical edge pairing \(aba'b'cdc'd'\),

an octagon with opposite sides identified, \(abcda'b'c'd'\),

a 12gon that is adapted to the sixsquares surface.
All data presented in this section is available on the DGD Gallery webpage [39].
Hyperelliptic curve. We uniformize the hyperelliptic curve \(\mu ^2=\lambda ^61\) as described in Sect. 7.2. The results are shown in Fig. 32.
To understand the cuts on the hyperelliptic surface that lead to the 12gon in the bottom row, imagine taking the canonical system of loops in the top row, meeting at the north pole, and deform them until they also meet at the south pole. This introduces a second vertex class in the fundamental polygon.
Lawson’s surface. Figure 33 shows Fuchsian uniformizations of Lawson’s minimal surface in \(S^{3}\). The triangulated surface model was kindly provided by Konrad Polthier [34].
Sixsquares surface. Figure 34 (left) shows a surface glued from six squares, which is conformally equivalent to Lawson’s surface and the hyperelliptic curve. Edges with the same marking are glued together. We calculate a uniformization using the triangulation with vertices added in the centers of the squares as shown. An adapted fundamental domain for this squaretiled translational surface arranges all squares around a single vertex, see Fig. 34 (right). By comparison with Fig. 32 (bottom) we see that the vertices in the center of the squares correspond to the branch points of the hyperelliptic representation of the surface. The black, gray, and white vertices correspond to the north and south pole of the hyperelliptic representation.
Notes
Acknowledgments
This research was supported by DFG SFB/TRR 109 “Discretization in Geometry and Dynamics”.
References
 1.Abresch, U.: Constant mean curvature tori in terms of elliptic functions. J. Reine Angew. Math. 374, 169–192 (1987)MathSciNetzbMATHGoogle Scholar
 2.Adler, V.E., Bobenko, A.I., Suris, Y.B.: Classification of integrable equations on quadgraphs. The consistency approach. Commun. Math. Phys. 233(3), 513–543 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
 3.Bobenko, A.I.: All constant mean curvature tori in \(R^3,\;S^3,\;H^3\) in terms of thetafunctions. Math. Ann. 290(2), 209–245 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
 4.Bobenko, A.I., Günther, F.: Discrete complex analysis on planar quadgraphs. In this volumeGoogle Scholar
 5.Bobenko, A.I., Pinkall, U., Springborn, B.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topology 19(4), 2155–2215 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Bobenko, A.I., Suris, Y.B.: Integrable systems on quadgraphs. Int. Math. Res. Not. 11, 573–611 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Bobenko, A.I., Suris, Y.B.: Discrete Differential GeometryIntegrable Structure, Graduate Studies in Mathematics, vol. 98. American Mathematical Society, Providence (2008)CrossRefzbMATHGoogle Scholar
 8.Chelkak, D., Smirnov, S.: Discrete complex analysis on isoradial graphs. Adv. Math. 228(3), 1590–1630 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
 9.Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. Math. 189(3), 515–580 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 10.Courant, R., Friedrichs, K., Lewy, H.: Über partielle Differentialgleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928)MathSciNetCrossRefzbMATHGoogle Scholar
 11.Duffin, R.J.: Discrete potential theory. Duke Math. J. 20, 233–251 (1953)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Duffin, R.J.: Basic properties of discrete analytic functions. Duke Math. J. 23, 335–363 (1956)MathSciNetCrossRefzbMATHGoogle Scholar
 13.Duffin, R.J.: Distributed and lumped networks. J. Math. Mech. 8, 793–826 (1959)MathSciNetzbMATHGoogle Scholar
 14.Duffin, R.J.: Potential theory on a rhombic lattice. J. Comb. Theory 5, 258–272 (1968)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Erickson, J., HarPeled, S.: Optimally cutting a surface into a disk. In: Symposium on Computational Geometry, pp. 244–253 (2002)Google Scholar
 16.Floyd, W., Weber, B., Weeks, J.: The Achilles’ heel of \({\rm{O}}(3,1)\)? Experiment. Math. 11(1), 91–97 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 17.Gu, X., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for ployhedral surfaces. arXiv:1309.4175 (2013)
 18.Gu, X., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces II. arXiv:1401.4594 (2014)
 19.Heil, M.: Numerical tools for the study of finite gap solutions of integrable systems. Ph.D. thesis, TUBerlin (1995)Google Scholar
 20.Jost, J.: Compact Riemann Surfaces, 3rd edn. Universitext. SpringerVerlag, Berlin (2006)CrossRefzbMATHGoogle Scholar
 21.Keen, L.: Canonical polygons for finitely generated Fuchsian groups. Acta Math. 115, 1–16 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
 22.Kenyon, R.: Conformal invariance of domino tiling. Ann. Probab. 28(2), 759–795 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
 23.Kenyon, R.: The Laplacian and Dirac operators on critical planar graphs. Invent. Math. 150(2), 409–439 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 24.Kouřimská, H., Skuppin, L., Springborn, B.: A variational principle for cyclic polygons with prescribed edge lengths. In this volumeGoogle Scholar
 25.Lawson Jr., H.B.: Complete minimal surfaces in \(\mathbb{S}^3\). Ann. Math. 92(3), 335–374 (1970)MathSciNetCrossRefzbMATHGoogle Scholar
 26.Leibon, G.: Characterizing the Delaunay decompositions of compact hyperbolic surfaces. Geom. Topol. 6, 361–391 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
 27.LelongFerrand, J.: Représentation conforme et transformations à intégrale de Dirichlet bornée. GauthierVillars, Paris (1955)zbMATHGoogle Scholar
 28.Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 29.Mercat, C.: Discrete Riemann surfaces and the Ising model. Commun. Math. Phys. 218(1), 177–216 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
 30.Milnor, J.: Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.) 6(1), 9–24 (1982)Google Scholar
 31.Muller, M.E.: A note on a method for generating points uniformly on ndimensional spheres. Commun. ACM 2(4), 19–20 (1959)CrossRefzbMATHGoogle Scholar
 32.Nehari, Z.: Conformal Mapping. McGrawHill Book Co., Inc., New York (1952)zbMATHGoogle Scholar
 33.Nijhoff, F., Capel, H.: The discrete Kortewegde Vries equation. Acta Appl. Math. 39(1–3), 133–158 (1995). KdV ’95 (Amsterdam, 1995)Google Scholar
 34.Oberknapp, B., Polthier, K.: An algorithm for discrete constant mean curvature surfaces. In: Hege, H.C., Polthier, K. (eds.) Visualization and Mathematics, pp. 141–161. SpringerVerlag, Berlin (1997)CrossRefGoogle Scholar
 35.Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Differ. Geom. 26(2), 349–360 (1987)MathSciNetzbMATHGoogle Scholar
 36.Schmutz Schaller, P.: Geometry of Riemann surfaces based on closed geodesics. Bull. Amer. Math. Soc. (N.S.) 35(3), 193–214 (1998)Google Scholar
 37.Schmutz Schaller, P.: Teichmüller space and fundamental domains of Fuchsian groups. Enseign. Math. (2) 45(12), 169–187 (1999)Google Scholar
 38.Schramm, O.: Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
 39.Sechelmann, S., Bobenko, A.I., Springborn, B.: DGD Gallery, Lawson’s surface uniformization. https://gallery.discretization.de/models/lawsons_surface_uniformization (2015)
 40.Smirnov, S.: Critical percolation in the plane: conformal invariance, Cardy’s formula, scaling limits. C. R. Acad. Sci. Paris Sér. I Math. 333(3), 239–244 (2001)Google Scholar
 41.Smirnov, S.: Conformal invariance in random cluster models. I. Holomorphic fermions in the Ising model. Ann. Math. (2) 172(2), 1435–1467 (2010)Google Scholar
 42.Smirnov, S.: Discrete complex analysis and probability. In: Proceedings of the International Congress of Mathematicians 2010 (ICM 2010), vol. I: Plenary Lectures and Ceremonies, vols. IIIV: Invited Lectures, pp. 595–621. Hyderabad, India (2010)Google Scholar
 43.Springborn, B.: A unique representation of polyhedral types. Centering via Möbius transformations. Math. Z. 249(3), 513–517 (2005)MathSciNetzbMATHGoogle Scholar
 44.Stephenson, K.: Introduction to Circle Packing. The Theory of Discrete Analytic Functions. Cambridge University Press, Cambridge (2005)zbMATHGoogle Scholar
 45.Thurston, W.P.: The geometry and topology of threemanifolds. Electronic version 1.1, March 2002, http://www.msri.org/publications/books/gt3m/
 46.Walter, R.: Constant mean curvature tori with spherical curvature lines in nonEuclidean geometry. Manuscripta Math. 63(3), 343–363 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
 47.Wente, H.C.: Counterexample to a conjecture of H. Hopf. Pacific J. Math. 121(1), 193–243 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
Copyright information
Open Access This chapter is distributed under the terms of the Creative Commons AttributionNoncommercial 2.5 License (http://creativecommons.org/licenses/bync/2.5/) which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
The images or other third party material in this chapter are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material.