Linear variational principle for Riemann mappings and discrete conformality Nadav Dyma,1,2, Raz Slutskyb,1, and Yaron Lipmana,1 aDepartment of Computer Science and Applied Mathematics, Faculty of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel; and bDepartment of Mathematics, Faculty of Computer Science and Applied Mathematics, Weizmann Institute of Science, Rehovot 7610001, Israel Edited by Robion Kirby, University of California, Berkeley, CA, and approved November 19, 2018 (received for review June 12, 2018) We consider Riemann mappings from bounded Lipschitz domains H 1 norm to the Riemann mapping Φ:Ω !T under refine- in the plane to a triangle. We show that in this case the Riemann ment of the triangulation M. Furthermore, in the case of the mapping has a linear variational principle: It is the minimizer of orbifold-Tutte algorithm, where the initial triangulation M is the Dirichlet energy over an appropriate affine space. By discretiz- Delaunay and the triangle T is an Euclidean orbifold, the con- ing the variational principle in a natural way we obtain discrete vergence is also uniform over the closure Ω¯. For two simply 0 conformal maps which can be computed by solving a sparse linear connected polygonal domains Ω, Ω with Delaunay triangula- system. We show that these discrete conformal maps converge tions we prove that the composition of the orbifold-Tutte map- 0 to the Riemann mapping in H1, even for non-Delaunay triangula- pings converges uniformly to the Riemann mapping Φ:Ω ! Ω . tions. Additionally, for Delaunay triangulations the discrete con- Finally in SI Appendix we show that these convergence results formal maps converge uniformly and are known to be bijective. obtained for polygonal domains can be extended to general Lip- As a consequence we show that the Riemann mapping between schitz domains. In this scenario the convergence is uniform and 1 two bounded Lipschitz domains can be uniformly approximated H on all compact subsets of Ω. by composing the discrete Riemann mappings between each The linear variational principle is derived from a tight linear Lipschitz domain and the triangle. relaxation of Plateau’s problem in the 2D case. Plateau’s problem seeks for a surface with minimal area spanning a prescribed curve d APPLIED conformal maps j discrete differential geometry j finite elements Γ ⊂ R , d = 2, 3. It is well known that Plateau’s problem can be MATHEMATICS solved by minimizing the Dirichlet energy of a parameterization d iconformal mappings between surfaces are homeomor- X : B ! R , where B is the open unit disc, among all admissible Bphisms which preserve angles and orientation. The existence mappings X 2 C(B¯ , Γ) with a (weakly) homeomorphic bound- of biconformal mappings between simply connected open sub- ary map X j@B : @B ! Γ fixing three points on the boundary sets of the complex plane is guaranteed by the Riemann mapping (e.g., ref. 16). Formulated this way, Plateau’s problem is a varia- theorem. When both surfaces are Lipschitz domains (see ref. tional problem with a convex quadratic energy (Dirichlet) and 1, definition 12.9 for a definition of Lipschitz domains), the a nonlinear admissible set of functions, C(B, Γ). Therefore, it biconformal mapping extends to a homeomorphism between corresponds to a nonlinear partial differential equation in gen- 2 the closures of the domains. The uniformization theorem and eral. When Γ ⊂ R , the unique minimizer of Plateau’s variational Keobe’s theorem generalize the Riemann mapping theorem and problem is the Riemann mapping. We consider a particular establish the existence of biconformal mappings between other instance of Plateau’s variational problem: Instead of B we con- classes of Riemann surfaces. Biconformal mappings have found sider Ω as the base domain, and we make a particular choice of 2 many applications in engineering (2), morphology (3, 4), medi- Γ = @T ⊂ R , fixing the preimages of the corners of T . Still, even cal imaging (5–7), and computer graphics and vision (8–10) and in this simplified setting, the respective set of admissible map- now play an important role in the new frontier of geometric deep pings, C(Ω, @T ), for Plateau’s variational problem is nonlinear (it learning (11–13). is convex, however). We introduce a relaxation of this variational Motivated in part by these applications, one of the central problem by replacing the nonlinear admissible set C(Ω, @T ) with themes in the emerging field of discrete differential geometry (DDG) (14) aims at developing discrete analogues of conformal mappings. Often the discrete structure at question is a trian- Significance gulation M = (V, E, F) of a planar-bounded Lipschitz domain Ω, and the question asked is how to place its vertices or alter- Computing conformal (angle-preserving) mappings between natively set its edge lengths to define a discrete analogue of a domains is a central task in discrete differential geometry, conformal map into the plane. One important benchmark for which has found many applications in morphology, medical discrete conformal mappings is convergence. Namely, Does the imaging, computer graphics and vision, and related fields. In discrete conformal map converge to the biconformal mapping this paper we show that by choosing a suitable target domain, under refinement of the triangulation M? computing conformal mappings becomes a linear problem. As In this paper we construct a linear variational principle for the a result conformal mappings can be computed robustly and Riemann mapping between a planar-bounded Lipschitz domain efficiently. Ω and a triangle domain T . We use this principle to devise an algorithm, based on simple piecewise-linear finite elements, Author contributions: N.D., R.S., and Y.L. designed research, performed research, and for defining discrete conformal mapping between a simply con- wrote the paper.y nected polygonal domain Ω with arbitrary triangulation M and The authors declare no conflict of interest.y a general triangle domain T . This class in particular includes This article is a PNAS Direct Submission.y the recent Orbifold–Tutte algorithm (15) for the case where M Published under the PNAS license.y is a Delaunay triangulation and T is a triangle orbifold (i.e., 1 N.D, R.S., and Y.L. contributed equally to this work.y equilateral or right-angle isosceles). 2 To whom correspondence should be addressed. Email: [email protected] The algorithm for computing discrete conformal maps is lin- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. ear in the sense that it consists of solving a single sparse linear 1073/pnas.1809731116/-/DCSupplemental.y system. We prove that these discrete mappings converge in the www.pnas.org/cgi/doi/10.1073/pnas.1809731116 PNAS Latest Articles j 1 of 6 Downloaded by guest on October 2, 2021 a linear superset of admissible mappings C∗(Ω, @T ) ⊃ C(Ω, @T ). Schwarz–Christoffel formula (17, 18), the zipper algorithm (19), Surprisingly, this relaxed variational problem is tight; that is, it polynomial methods (20), and others (21, 22). has a unique solution and this solution is the Riemann map- Probably the first discrete conformal mapping is the circle ping Φ:Ω !T . Since this variational problem corresponds to packing introduced in ref. 23. Circle packing defines a discrete a linear partial differential equation, we can employ more or conformal (more generally, analytic) mapping of a triangulation less standard finite-element theory to prove convergence of the by packing circles with different radii centered at vertices in the algorithm. plane. These radii can be seen as setting edge lengths in M. Con- The power of this approach is illustrated in Fig. 1, where vergence of circle packing to the Riemann mapping was proved we consider the problem of computing the Riemann mapping in refs. 24 and 25. An efficient algorithm for circle packing was from a “Koch polygon” to a triangle. This Koch polygon is the developed in ref. 26. A variational principle for circle packing polygon obtained by the first six iterations of the iterative pro- was found in ref. 27. Discrete Ricci flow was developed in ref. 28 cess used to create the fractal Koch snowflake. Mapping the and was shown to converge to a circle packing. Circle patterns Koch polygon is challenging due to the fragmented nature of (29) generalize circle packings and allow nontrivial intersection the boundary and requires a high-resolution map which would of circles; a variational principle for circle patterns was discov- be difficult to achieve using nonlinear methods. We obtain such ered in ref. 30. In ref. 31 discrete conformality is defined by a high-resolution map by computing our discrete conformal map averaging conformal scales at vertices; in ref. 32 an explicit varia- from a triangulation of the polygon with approximately 6 mil- tional principle and an efficient algorithm are developed for this lion vertices. Solving for the conformal map approximation in equivalence discrete conformality relation. Note that while cir- this case, using Matlab’s linear solver, takes approximately 2 min cle packing has a convex variational principle, it is not linear. on an Intel Xeon processor. Additionally circle packing was shown to converge uniformly on The quality of the discrete conformal mapping is visualized in compact subsets of Ω while our algorithm converges uniformly the standard fashion: A scalar function is defined on the trian- on all of Ω¯ and also converges in H 1. gle, which represents a black-and-white coloring of a grid. The A natural tool, which we also use in this paper, to han- function is then pulled back by the computed discrete Riemann dle discrete conformality of triangulations is the finite-elements mapping. Note that the 90◦ angles are preserved by the map.