Advances in Discrete Differential Geometry pp 133149  Cite as
Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices
Abstract
Two triangle meshes are conformally equivalent if their edge lengths are related by scale factors associated to the vertices. Such a pair can be considered as preimage and image of a discrete conformal map. In this article we study the approximation of a given smooth conformal map f by such discrete conformal maps \(f^\varepsilon \) defined on triangular lattices. In particular, let T be an infinite triangulation of the plane with congruent strictly acute triangles. We scale this triangular lattice by \(\varepsilon >0\) and approximate a compact subset of the domain of f with a portion of it. For \(\varepsilon \) small enough we prove that there exists a conformally equivalent triangle mesh whose scale factors are given by \(\log f'\) on the boundary. Furthermore we show that the corresponding discrete conformal (piecewise linear) maps \(f^\varepsilon \) converge to f uniformly in \(C^1\) with error of order \(\varepsilon \).
Keywords
Dirichlet Problem Edge Length Circle Pattern Triangular Lattice Boundary Vertex1 Introduction
Holomorphic functions build the basis and heart of the rich theory of complex analysis. Holomorphic functions with nowhere vanishing derivative, also called conformal maps, have the property to preserve angles. Thus they may be characterized by the fact that they are infinitesimal scalerotations.
In the discrete theory, the idea of characterizing conformal maps as local scalerotations may be translated into different concepts. Here we consider the discretization coming from a metric viewpoint: Infinitesimally, lengths are scaled by a factor, i.e. by \(f'(z)\) for a conformal function f on \(D\subset \mathbb C\). More generally, on a smooth manifold two Riemannian metrics g and \(\tilde{g}\) are conformally equivalent if \(\tilde{g}=\text {e}^{2u}g\) for some smooth function u.
The smooth complex domain (or manifold) is replaced in this discrete setting by a triangulation of a connected subset of the plane \(\mathbb C\) (or a triangulated piecewise Euclidean manifold).
1.1 Convergence for Discrete Conformal PLMaps on Triangular Lattices
On a subcomplex of T we now define a discrete conformal mapping. The main idea is to change the lengths of the edges of the triangulation according to scale factors at the vertices. The new triangles are then ‘glued together’ to result in a piecewise linear map, see Fig. 2 for an illustration. More precisely, we have
Definition 1.1
In fact, the definition of a discrete conformal PLmap relies on the notion of discrete conformal triangle meshes. These have been studied by Luo, Gu, Sun, Wu, Guo [8, 9, 14], Bobenko, Pinkall, and Springborn [1] and others.
As possible application, discrete conformal PLmaps can be used for discrete uniformization. The simplest case is a discrete Riemann mapping theorem, i.e. the problem of finding a discrete conformal mapping of a simply connected domain onto the unit disc. Similarly, we may consider a related Dirichlet problem. Given some function \(u_\partial \) on the boundary of a subcomplex \(T_S\), find a discrete conformal PLmap whose associated scale factors agree on the boundary with \(u_\partial \). For such a Dirichlet problem (with assumptions on \(u_\partial \) and \(T_S\)) we will prove existence as part of our convergence theorem.
In this article we present a first answer to the following problem: Given a smooth conformal map, find a sequence of discrete conformal PLmaps which approximate the given map. We study this problem on triangular lattices T with acute angles and always assume for simplicity that the origin is a vertex. Denote by \(\varepsilon T\) the lattice T scaled by \(\varepsilon >0\). Using the values of \(\log f'\), we obtain a discrete conformal PLmap \(f^\varepsilon \) on a subcomplex of \(\varepsilon T\) from a boundary value problem for the associated scale factors. More precisely, we prove the following approximation result.
Theorem 1.2
Let \(f:D\rightarrow \mathbb C\) be a conformal map (i.e. holomorphic with \(f'\not =0\)). Let \(K\subset D\) be a compact set which is the closure of its simply connected interior int(K) and assume that \(0\in int(K)\). Let T be a triangular lattice with strictly acute angles. For each \(\varepsilon >0\) let \(T^\varepsilon _K\) be a subcomplex of \(\varepsilon T\) whose support is contained in K and is homeomorphic to a closed disc. We further assume that 0 is an interior vertex of \(T^\varepsilon _K\). Let \(e_0=[0,{\hat{v}}_0]\in E^\varepsilon _K\) be one of its incident edges.
 The associated scale factors \(u^\varepsilon :V^\varepsilon _K\rightarrow \mathbb {R}\) satisfy$$\begin{aligned} u^\varepsilon (v)=\log f'(v)\qquad \text {for all boundary vertices } v \text { of } V^\varepsilon _K. \end{aligned}$$(2)

The discrete conformal PLmap is normalized according to \(f^\varepsilon (0)=f(0)\) and \(\arg (f^\varepsilon ({\hat{v}}_0)f^\varepsilon (0))= \arg ({\hat{v}}_0)+ \arg (f'(\frac{{\hat{v}}_0}{2})) \pmod {2\pi }\).
 (i)The scale factors \(u^\varepsilon \) approximate \(\log f'\) uniformly with error of order \(\varepsilon ^2\):$$\begin{aligned} \left u^\varepsilon (v)\log f'(v)\right \leqslant C_1\varepsilon ^2. \end{aligned}$$(3)
 (ii)The discrete conformal PLmappings \(f^\varepsilon \) converge to f for \(\varepsilon \rightarrow 0\) uniformly with error of order \(\varepsilon \):$$\begin{aligned} \left f^\varepsilon (x)f(x)\right \leqslant C_2\varepsilon . \end{aligned}$$
 (iii)The derivatives of \(f^\varepsilon \) (in the interior of the triangles) converge to \(f'\) uniformly for \(\varepsilon \rightarrow 0\) with error of order \(\varepsilon \):for all points x in the interior of a triangle \(\varDelta \) of \(T^\varepsilon _K\). Here \(\partial _z\) and \(\partial _{\bar{z}}\) denote the Wirtinger derivatives applied to the linear maps \(f^\varepsilon _\varDelta \).$$\begin{aligned} \left \partial _z f^\varepsilon (x)f'(x)\right \leqslant C_3\varepsilon \qquad \text {and} \qquad \left \partial _{\bar{z}} f^\varepsilon (x)\right \leqslant C_3\varepsilon \end{aligned}$$
The proof of Theorem 1.2 is given in Sect. 4. The arguments are based on estimates derived in Sect. 3.
The problem of actually computing the scale factors u for given boundary values \(u_\partial \) such that u gives rise to a discrete conformal PLmap (in case it exists) can be solved using a variational principle, see [1, 20]. Our proof relies on investigations using the corresponding convex functional, see Theorem 2.2 in Sect. 2.
Remark 1.3
The convergence result of Theorem 1.2 also remains true if linear interpolation is replaced with the piecewise projective interpolation schemes described in [1, 3], i.e., circumcircle preserving, angle bisector preserving and, generally, exponenttcenter preserving for all \(t\in \mathbb {R}\). The proof is the same with only small adaptations. This is due to the fact that the image of the vertices is the same for all these interpolation schemes and these image points converge uniformly to the corresponding image points under f with error of order \(\varepsilon \). The estimates for the derivatives similarly follow from Theorem 1.2(i).
1.2 Other Convergence Results for Discrete Conformal Maps
Smooth conformal maps can be characterized in various ways. This leads to different notions of discrete conformality. Convergence issues have already been studied for some of these discrete analogs. We only give a very short overview and cite some results of a growing literature.
In particular, linear definitions can be derived as discrete versions of the CauchyRiemann equations and have a long and still developing history. Connections of such discrete mappings to smooth conformal functions have been studied for example in [2, 6, 7, 13, 16, 19, 22].
The idea of characterizing conformal maps as local scalerotations has lead to the consideration of circle packings, more precisely to investigations on circle packings with the same (given) combinatorics of the tangency graph. Thurston [21] first conjectured the convergence of circle packings to the Riemann map, which was then proven by [10, 11, 17].
The theory of circle patterns generalizes the case of circle packings. Also, there is a link to integrable structures via isoradial circle patterns. The approximation of conformal maps using circle patterns has been studied in [4, 5, 12, 15, 18].
The approach taken in this article constructs discrete conformal maps from given boundary values. Our approximation results and some ideas of the proof are therefore similar to those in [4, 5, 18] for circle patterns which also rely on boundary value problems.
2 Some Characterizations of Associated Scale Factors of Discrete Conformal PLMaps
Proposition 2.1
Let \(T_S\) be a subcomplex of a triangular lattice T and \(u:V_S\rightarrow \mathbb {R}\) a function satisfying the following two conditions.
 (i)For every triangle \(\varDelta [v_1,v_2,v_3]\) of \(T_S\) the triangle inequalities for \(\tilde{l}\) defined by (4) hold, in particularfor all permutations (ijk) of (123).$$\begin{aligned} v_iv_j\text {e}^{(u(v_i)+u(v_j))/2}< v_iv_k\text {e}^{(u(v_i)+u(v_k))/2} +v_jv_k\text {e}^{(u(v_j)+u(v_k))/2} \end{aligned}$$(7)
 (ii)For every interior vertex \(v_0\) with neighbors \(v_1,v_2,\dots ,v_k,v_{k+1}=v_1\) in cyclic order we havewhere \(\lambda (v_a,v_b,v_c)= 2\log (v_bv_c/v_av_b)\) for a triangle \(\varDelta [v_a,v_b,v_c]\).$$\begin{aligned} \sum _{j=1}^k \theta (\lambda (v_0,v_j,v_{j+1})+ u(v_{j+1})u(v_0), \lambda (v_0,v_{j+1},v_j) +u(v_j)u(v_0))=2\pi , \end{aligned}$$(8)
Then there is a discrete conformal PLmap (unique up to postcomposition with Euclidean motions) such that its associated scale factors are the given function \(u:V_S\rightarrow \mathbb {R}\).
Conversely, given a discrete conformal PLmap on a subcomplex \(T_S\) of a triangular lattice T, its associated scale factors \(u:V_S\rightarrow \mathbb {R}\) satisfy conditions (i) and (ii).
In order to obtain discrete conformal PLmaps from a given smooth conformal map we will consider a Dirichlet problem for the associated scale factors. Therefore we will apply a theorem from [1] which characterizes the scale factors u for given boundary values using a variational principle for a functional E defined in [1, Sect. 4]. Note that we will not need the exact expression for E but only the formula for its partial derivatives. In fact, the vanishing of these derivatives is equivalent to the necessary condition (8) for the scale factors to correspond to a discrete conformal PLmap.
Theorem 2.2
([1]) Let \(T_S\) be a subcomplex of a triangular lattice and let \(u_\partial :V_\partial \rightarrow \mathbb {R}\) be a function on the boundary vertices \(V_\partial \) of \(T_S\). Then the solution \(\tilde{u}\) (if it exists) of Eq. (8) at all interior vertices with \({\tilde{u}}_{V_\partial }=u_\partial \) is the unique argmin of a locally strictly convex functional \(E(u)=E_{T_S}(u)\) which is defined for functions \(u:V\rightarrow \mathbb {R}\) satisfying the inequalities (7).
By Proposition 2.1 such a solution \(\tilde{u}\) are then scale factors associated to a discrete conformal PLmap.
Remark 2.3
The functional E can be extended to a convex continuously differentiable function on \(\mathbb {R}^V\), see [1] for details.
3 Taylor Expansions
We now examine the effect when we take \(u=\log f'\) as ‘scale factors’, i.e. for each triangle we multiply the length \(vw\) of an edge [v, w] by the geometric mean \(\sqrt{f'(v)f'(w)}\) of \(f'\) at the vertices. The proof of Theorem 1.2 is based on the idea that \(u=\log f'\) almost satisfies the conditions for being the associated scale factors of an discrete conformal PLmap, that is conditions (i) and (ii) of Proposition 2.1, and therefore is close to the exact solution \(u^\varepsilon \).
The following lemma summarizes the main properties of \(w^\pm \) which follow from the definition of \(w^\pm \) together with the preceding estimates.
Lemma 3.1
\(w^\pm \) satisfies the boundary condition \(w^\pm _{\partial V^\varepsilon _K} = \log f' \big _{\partial V^\varepsilon _K}\).
 (i)
\(q^+(v_0)>0\) and \(q^(v_0)<0\)
 (ii)
If \(v_1, v_2,\dots , v_6,v_7=v_1\) denote the chain of neighboring vertices of \(v_0\) in cyclic order and \(\lambda (v_a,v_b,v_c)= 2\log (v_bv_c/v_av_b)\) for any triangle \(\varDelta [v_a,v_b,v_c]\), we have
In analogy to the continuous case we interpret Eq. (8) as a nonlinear Laplace equation for u. In this spirit \(w^+\) may be taken as superharmonic function and \(w^\) as subharmonic function.
4 Existence of Discrete Conformal PLMaps and Estimates
Our aim is to show that for \(\varepsilon \) small enough there exists a function \(u^\varepsilon \) satisfying conditions (i) and (ii) of Proposition 2.1 and \(u^\varepsilon (v)=\log f'(v)\) for all boundary vertices \(v\in \partial V^\varepsilon _K\). This function then defines a discrete conformal PLmap \(f^\varepsilon \) (uniquely if we use the normalization of Theorem 1.2).
Theorem 4.1
Assume that all angles of the triangular lattice T are strictly smaller than \(\pi /2\). There is an \(\varepsilon _0>0\) (depending on f, K and the triangulation parameters) such that for all \(0<\varepsilon <\varepsilon _0\) the minimum of the functional E (see Theorem 2.2) with boundary conditions (2) is attained in \(W^\varepsilon \).
Corollary 4.2
For all \(0<\varepsilon <\varepsilon _0\) there exists a discrete conformal PLmap on \(T^\varepsilon _K\) whose associated scale factors satisfy the boundary conditions (2).
The proof of Theorem 4.1 follows from Lemma 4.4 below. It is based on Theorem 2.2 and on monotonicity estimates of the angle function \(\theta (x,y)\) defined in (6). It is only here where we need the assumption that all angles of the triangular lattice T are strictly smaller than \(\pi /2\).
Lemma 4.3
(Monotonicity lemma) Consider the star of a vertex \(v_0\) of a triangular lattice T and its neighboring vertices \(v_1,\dots , v_6,v_7=v_1\) in cyclic order. Denote \(\lambda _{0,k}:=2\log (v_{k+1}v_{k}/v_{0}v_{k})\). Assume that all triangles \(\varDelta (v_0,v_k,v_{k+1})\) are strictly acute angled, i.e. all angles \(<\pi /2\).
Proof
Lemma 4.4
There is an \(\varepsilon _0>0\) such that for all \(0<\varepsilon <\varepsilon _0\) the negative gradient \(\text {grad}(E)\) on the boundary of \(W^\varepsilon \) points into the interior of \(W^\varepsilon \).
Proof
For notational simplicity, set \(u_k=u(v_k)\), \(w_k^\pm =w^\pm (v_k)\) for vertices \(v_k\in V^\varepsilon _K\) and \(\lambda _{a,b,c}=2\log (v_bv_c/v_av_b)\).
We are now ready to deduce our convergence theorem.
Proof
(of Theorem 1.2) The existence part follows from Theorem 4.1. The uniqueness is obvious as the translational and rotational freedom of the image of \(f^\varepsilon \) is fixed using values of f.
We now deduce the remaining estimates.
Part (ii): Given the scale factors \(u^\varepsilon \) associated to the discrete conformal PLmap \(f^\varepsilon \) on \(T^\varepsilon _K\), we can in every image triangle determine the interior angles (using for example (5)). In particular, we begin by deducing from estimate (3) the change of these interior angles of the triangles.
Part (iii): As last step we consider the derivatives of \(f^\varepsilon \) restricted to a triangle.
Remark 4.5
Theorem 1.2 focuses on a particular way to approximate a given conformal map f by a sequence of discrete conformal PLmaps. Namely, we consider corresponding smooth and discrete Dirichlet boundary value problems and compare the solutions. There is of course a corresponding problem for Neumann boundary conditions, i.e. prescribing angle sums of the triangles at boundary vertices using \(\arg f'\). Also, there is a corresponding variational description for conformally equivalent triangle meshes or discrete conformal PLmaps in terms of angles, see [1]. But unfortunately, the presented methods for a convergence proof seem not to generalize in a straightforward manner to this case, as the order of the corresponding Taylor expansion is lower .
Notes
Acknowledgments
The author would like to thank the anonymous referees for the careful reading of the initial manuscript and various suggestions for improvement. This research was supported by the DFG Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics”.
References
 1.Bobenko, A.I., Pinkall, U., Springborn, B.: Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19, 2155–2215 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
 2.Bobenko, A.I., Skopenkov, M.: Discrete Riemann surfaces: linear discretization and its convergence. To appear in J. Reine Angew, Math (2014)Google Scholar
 3.Born, S., Bücking, U., Springborn, B.: Quasiconformal distortion of projective transformations, with an application to discrete conformal maps. arXiv:1505.01341 [math.CV]
 4.Bücking, U.: Approximation of conformal mappings by circle patterns and discrete minimal surfaces. Ph.D. thesis, Technische Universität Berlin (2007). http://opus.kobv.de/tuberlin/volltexte/2008/1764/
 5.Bücking, U.: Approximation of conformal mapping by circle patterns. Geom. Dedicata 137, 163–197 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
 6.Chelkak, D., Smirnov, S.: Universality in the 2D Ising model and conformal invariance of fermionic observables. Invent. math. 189, 515–580 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
 7.Courant, R., Friedrichs, K., Lewy, H.: Über die partiellen Differenzengleichungen der mathematischen Physik. Math. Ann. 100, 32–74 (1928). English translation: IBM Journal (1967), 215–234Google Scholar
 8.Gu, X., Guo, R., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces II. arXiv:1401.4594 [math.GT]
 9.Gu, X., Luo, F., Sun, J., Wu, T.: A discrete uniformization theorem for polyhedral surfaces. arXiv:1309.4175 [math.GT]
 10.He, Z.X., Schramm, O.: On the convergence of circle packings to the Riemann map. Invent. Math. 125, 285–305 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
 11.He, Z.X., Schramm, O.: The \(C^\infty \)convergence of hexagonal disk packings to the Riemann map. Acta Math. 180, 219–245 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
 12.Lan, S.Y., Dai, D.Q.: The \(C^\infty \)convergence of SG circle patterns to the Riemann mapping. J. Math. Anal. Appl. 332, 1351–1364 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
 13.LelongFerrand, J.: Représentation conforme et transformations à intégrale de Dirichlet bornée. GauthierVillars, Paris (1955)zbMATHGoogle Scholar
 14.Luo, F.: Combinatorial Yamabe flow on surfaces. Commun. Contemp. Math. 6(5), 765–780 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
 15.Matthes, D.: Convergence in discrete Cauchy problems and applications to circle patterns. Conform. Geom. Dyn. 9, 1–23 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
 16.Mercat, C.: Discrete Riemann Surfaces. In: Papadopoulos, A. (ed.) Handbook of Teichmüller theory, vol. I, pp. 541–575. Eur. Math. Soc., Zürich (Ed.) (2007)Google Scholar
 17.Rodin, B., Sullivan, D.: The convergence of circle packings to the Riemann mapping. J. Diff. Geom. 26, 349–360 (1987)MathSciNetzbMATHGoogle Scholar
 18.Schramm, O.: Circle patterns with the combinatorics of the square grid. Duke Math. J. 86, 347–389 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
 19.Skopenkov, M.: The boundary value problem for discrete analytic functions. Adv. Math. 240, 61–87 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
 20.Springborn, B., Schröder, P., Pinkall, U.: Conformal equivalence of triangle meshes. ACM Trans. Graph. 27(3) (2008)Google Scholar
 21.Thurston, B.: The finite Riemann mapping theorem. Invited address at the International Symposioum in Celebration of the proof of the Bieberbach Conjecture, Purdue University (1985)Google Scholar
 22.Werness, B.M.: Discrete analytic functions on nonuniform lattices without global geometric control (2014). PreprintGoogle Scholar
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