Linear variational principle for Riemann mappingsand discrete conformalityNadav 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 domainsin the plane to a triangle. We show that in this case the Riemannmapping has a linear variational principle: It is the minimizer ofthe Dirichlet energy over an appropriate affine space. By discretiz-ing the variational principle in a natural way we obtain discreteconformal maps which can be computed by solving a sparse linearsystem. We show that these discrete conformal maps convergeto the Riemann mapping in H1, even for non-Delaunay triangula-tions. Additionally, for Delaunay triangulations the discrete con-formal maps converge uniformly and are known to be bijective.As a consequence we show that the Riemann mapping betweentwo bounded Lipschitz domains can be uniformly approximatedby composing the discrete Riemann mappings between eachLipschitz domain and the triangle.

conformal maps | discrete differential geometry | finite elements

B iconformal mappings between surfaces are homeomor-phisms which preserve angles and orientation. The existence

of biconformal mappings between simply connected open sub-sets of the complex plane is guaranteed by the Riemann mappingtheorem. When both surfaces are Lipschitz domains (see ref.1, definition 12.9 for a definition of Lipschitz domains), thebiconformal mapping extends to a homeomorphism betweenthe closures of the domains. The uniformization theorem andKeobe’s theorem generalize the Riemann mapping theorem andestablish the existence of biconformal mappings between otherclasses of Riemann surfaces. Biconformal mappings have foundmany applications in engineering (2), morphology (3, 4), medi-cal imaging (5–7), and computer graphics and vision (8–10) andnow play an important role in the new frontier of geometric deeplearning (11–13).

Motivated in part by these applications, one of the centralthemes in the emerging field of discrete differential geometry(DDG) (14) aims at developing discrete analogues of conformalmappings. Often the discrete structure at question is a trian-gulation M= (V, E ,F) of a planar-bounded Lipschitz domainΩ, and the question asked is how to place its vertices or alter-natively set its edge lengths to define a discrete analogue of aconformal map into the plane. One important benchmark fordiscrete conformal mappings is convergence. Namely, Does thediscrete conformal map converge to the biconformal mappingunder refinement of the triangulationM?

In this paper we construct a linear variational principle for theRiemann mapping between a planar-bounded Lipschitz domainΩ and a triangle domain T . We use this principle to devisean algorithm, based on simple piecewise-linear finite elements,for defining discrete conformal mapping between a simply con-nected polygonal domain Ω with arbitrary triangulationM anda general triangle domain T . This class in particular includesthe recent Orbifold–Tutte algorithm (15) for the case whereMis a Delaunay triangulation and T is a triangle orbifold (i.e.,equilateral or right-angle isosceles).

The algorithm for computing discrete conformal maps is lin-ear in the sense that it consists of solving a single sparse linearsystem. We prove that these discrete mappings converge in the

H 1 norm to the Riemann mapping Φ : Ω→T under refine-ment of the triangulation M. Furthermore, in the case of theorbifold-Tutte algorithm, where the initial triangulation M isDelaunay and the triangle T is an Euclidean orbifold, the con-vergence is also uniform over the closure Ω. For two simplyconnected polygonal domains Ω, Ω′ with Delaunay triangula-tions we prove that the composition of the orbifold-Tutte map-pings converges uniformly to the Riemann mapping Φ : Ω→Ω′.Finally in SI Appendix we show that these convergence resultsobtained for polygonal domains can be extended to general Lip-schitz domains. In this scenario the convergence is uniform andH 1 on all compact subsets of Ω.

The linear variational principle is derived from a tight linearrelaxation of Plateau’s problem in the 2D case. Plateau’s problemseeks for a surface with minimal area spanning a prescribed curveΓ⊂Rd , d = 2, 3. It is well known that Plateau’s problem can besolved by minimizing the Dirichlet energy of a parameterizationX :B→Rd , where B is the open unit disc, among all admissiblemappings X ∈C(B , Γ) with a (weakly) homeomorphic bound-ary map X |∂B : ∂B→Γ fixing three points on the boundary(e.g., ref. 16). Formulated this way, Plateau’s problem is a varia-tional problem with a convex quadratic energy (Dirichlet) anda nonlinear admissible set of functions, C(B , Γ). Therefore, itcorresponds to a nonlinear partial differential equation in gen-eral. When Γ⊂R2, the unique minimizer of Plateau’s variationalproblem is the Riemann mapping. We consider a particularinstance of Plateau’s variational problem: Instead of B we con-sider Ω as the base domain, and we make a particular choice ofΓ = ∂T ⊂R2, fixing the preimages of the corners of T . Still, evenin this simplified setting, the respective set of admissible map-pings, C(Ω, ∂T ), for Plateau’s variational problem is nonlinear (itis convex, however). We introduce a relaxation of this variationalproblem by replacing the nonlinear admissible set C(Ω, ∂T ) with

Significance

Computing conformal (angle-preserving) mappings betweendomains is a central task in discrete differential geometry,which has found many applications in morphology, medicalimaging, computer graphics and vision, and related fields. Inthis paper we show that by choosing a suitable target domain,computing conformal mappings becomes a linear problem. Asa result conformal mappings can be computed robustly andefficiently.

Author contributions: N.D., R.S., and Y.L. designed research, performed research, andwrote the paper.y

The authors declare no conflict of interest.y

This article is a PNAS Direct Submission.y

Published under the PNAS license.y1 N.D, R.S., and Y.L. contributed equally to this work.y2 To whom correspondence should be addressed. Email: nadavdym@gmail.com.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1809731116/-/DCSupplemental.y

Published online December 28, 2018.

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a linear superset of admissible mappings C∗(Ω, ∂T )⊃C(Ω, ∂T ).Surprisingly, this relaxed variational problem is tight; that is, ithas a unique solution and this solution is the Riemann map-ping Φ : Ω→T . Since this variational problem corresponds toa linear partial differential equation, we can employ more orless standard finite-element theory to prove convergence of thealgorithm.

The power of this approach is illustrated in Fig. 1, wherewe consider the problem of computing the Riemann mappingfrom a “Koch polygon” to a triangle. This Koch polygon is thepolygon obtained by the first six iterations of the iterative pro-cess used to create the fractal Koch snowflake. Mapping theKoch polygon is challenging due to the fragmented nature ofthe boundary and requires a high-resolution map which wouldbe difficult to achieve using nonlinear methods. We obtain sucha high-resolution map by computing our discrete conformal mapfrom a triangulation of the polygon with approximately 6 mil-lion vertices. Solving for the conformal map approximation inthis case, using Matlab’s linear solver, takes approximately 2 minon an Intel Xeon processor.

The quality of the discrete conformal mapping is visualized inthe standard fashion: A scalar function is defined on the trian-gle, which represents a black-and-white coloring of a grid. Thefunction is then pulled back by the computed discrete Riemannmapping. Note that the 90 angles are preserved by the map. Theright-hand side of Fig. 1 visualizes the high resolution of the mapnear the boundary.

Related WorkThe notion of a discrete conformal mapping of a triangulationM is a rather well-researched area. It is rich with constructionsand algorithms, each with its own definition of discrete confor-mality, often inspired by some property of smooth conformalmappings. Although we focus here on discrete conformal map-pings, we note that there are other numerical algorithms withconvergence guarantees to the Riemann mapping based on the

Schwarz–Christoffel formula (17, 18), the zipper algorithm (19),polynomial methods (20), and others (21, 22).

Probably the first discrete conformal mapping is the circlepacking introduced in ref. 23. Circle packing defines a discreteconformal (more generally, analytic) mapping of a triangulationby packing circles with different radii centered at vertices in theplane. These radii can be seen as setting edge lengths inM. Con-vergence of circle packing to the Riemann mapping was provedin refs. 24 and 25. An efficient algorithm for circle packing wasdeveloped in ref. 26. A variational principle for circle packingwas found in ref. 27. Discrete Ricci flow was developed in ref. 28and was shown to converge to a circle packing. Circle patterns(29) generalize circle packings and allow nontrivial intersectionof circles; a variational principle for circle patterns was discov-ered in ref. 30. In ref. 31 discrete conformality is defined byaveraging conformal scales at vertices; in ref. 32 an explicit varia-tional principle and an efficient algorithm are developed for thisequivalence discrete conformality relation. Note that while cir-cle packing has a convex variational principle, it is not linear.Additionally circle packing was shown to converge uniformly oncompact subsets of Ω while our algorithm converges uniformlyon all of Ω and also converges in H 1.

A natural tool, which we also use in this paper, to han-dle discrete conformality of triangulations is the finite-elementsmethod (FEM) (33). Since Riemann mappings consist of twoconjugate harmonic functions, researchers have constructed dis-crete conformal mappings by pairs of conjugate discrete har-monic functions defined via the Dirichlet integral (34–36). Thesealgorithms are linear but do not satisfy any prescribed bound-ary conditions and are not known to converge to the Rie-mann mapping. Convergence to the Riemann mapping, or moregenerally the solution of Plateau’s problem, can be obtainedby minimizing the Dirichlet energy (37, 38) or a conformalenergy (39), while imposing nonlinear boundary conditions. Solv-ing these nonconvex variational problems is a computationalchallenge.

Fig. 1. (Left) An approximation of the notorious Riemann map from a polygonal approximation of the Koch snowflake (computed with six recursions) toa triangle. The approximation consists of a mesh with roughly 6 million vertices and captures different resolutions of this map as shown in the zoom-in(Right).

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A Linear Variational Principle for Riemann MappingsWe consider a bounded, simply connected Lipschitz domainΩ⊂R2 with an oriented boundary ∂Ω and a target triangledomain T ⊂R2 with corners c1, c2, c3 ∈R2 positively orientedwith respect to (w.r.t.) ∂Ω. A 2D version of Plateau’s variationalproblem is

minX

ED(X ) [1a]

s.t. X ∈C(Ω, ∂T ), [1b]

where the Dirichlet energy of a map X (u, v) = (x (u, v),y(u, v))T : Ω→R2 is defined as

ED(X ) =1

2

∫Ω

|Xu |2 + |Xv |2

and |·| denotes the standard Euclidean norm of a 2-vector in R2,and the partial derivatives are interpreted in the distributionalsense. The set of admissible mappings C(Ω, ∂T ) is defined asfollows: We denote by H 1(Ω,R2) the Sobolev space of pairs offunctions (i.e., X = (x , y)T ) with finite Sobolev norm

‖X ‖2 =1

2‖X ‖2L2(Ω) +ED(X ).

In Plateau’s problem it is vital to consider boundary val-ues of mappings X ∈H 1(Ω,R2). This is normally done byconsidering the trace operator, T :H 1(Ω,R2)→L2(∂Ω,R2),that extends the boundary operator, TX =X |∂Ω, defined onmappings X which are continuous on Ω, to the entirety ofH 1(Ω,R2) (e.g., theorem 1.6.6 in ref. 40). We are now readyto define the set of admissible mappings in Plateau’s variationalproblem (16):

Definition 1. The admissible function set C(Ω, ∂T ) is defined asfollows:

i) X ∈H 1(Ω,R2)∩C (Ω).ii) TX is a homeomorphism between the boundaries ∂Ω and ∂T .

iii) TX takes three fixed, positively oriented points p1, p2, p3 ∈ ∂Ωto the corners of the triangle c1, c2, c3 ∈T .

The unique minimizer of Eq. 1 is the unique Riemannmapping Φ : Ω→T which satisfies condition iii.

We relax Eq. 1 by relaxing the homeomorphic condition iiin C(Ω, ∂T ). Let Γi ⊂ ∂Ω, i = 1, 2, 3, denote the closed bound-ary arc connecting pi→ pi+1 (for i = 3 set p4 = p1 and c4 = c1)(Fig. 2A). Then, consider the relaxed admissible mapping space:

Definition 2. The relaxed set of admissible mappings, C∗(Ω, ∂T ), isdefined to be the closure in H 1(Ω,R2) of the mappings satisfyingthe following conditions:

i) X ∈H 1(Ω,R2).ii) TX ∈C (∂Ω,R2).

A B

Fig. 2. (A) The admissible set C and its relaxation using unordered infiniteline constraints, C*. (B) H1 convergence requires only regular triangulations,which can be easily achieved by midedge subdivision.

iii) aTi TX (p) + bi = 0, ∀p ∈Γi , ∀i = 1, 2, 3,

where the infinite line `i = Z ∈R2 | aTi Z + bi = 0 supports

and infinitely extends the edge [ci , ci+1] in the triangle T .

Fig. 2A illustrates one of the lines `i . Note that condition iiirequires only images of points in ∂Ω to lie on the respectivelines `i , and nothing prevents the boundary map TX from beingnoninjective or nonsurjective onto ∂T . Also note that we nowrequire X only to be continuous on the boundary.

The first main result of this paper claims that the relaxation

minX

ED(X ) [2a]

s.t. X ∈C∗(Ω, T ) [2b]

is tight, that is, as in the following:

Theorem 1. The relaxed Plateau’s problem Eq. 2 has the Rie-mann map Φ : Ω→T satisfying Φ(pi) = ci , i = 1, 2, 3, as a uniqueminimizer.

In the second part of the paper we utilize Theorem 1 to com-pute biconformal mappings from a polygonal Lipschitz domainΩ to T . For polygonal Ω, we show that a piecewise-linear FEMapproximation to the minimum of Eq. 2 converges in the H 1

norm to the Riemann map under refinement of the triangula-tionM. Refinement is a sequence of regular triangulationsMh

triangulating a polygonal domain Ω, where the maximal edge sizeh→ 0. By regular triangulation we mean that all angles of the tri-angulations are in some interval [0 + ε,π− ε] for some constantε> 0 (ref. 33, p. 124). One simple subdivision rule that preservesregularity of triangulation is the 1→ 4 shown in Fig. 2B. We fur-ther show that if allMh are 3-connected and Delaunay (i.e., sumof opposite angles is less than π) and T is an Euclidean orbifold,then the convergence is also uniform. Such triangulation fam-ilies can be computed efficiently by the incremental Delaunayalgorithm, for example.

Let Λh ⊂H 1(Ω,R2) be the finite-dimensional linear space ofpiecewise-linear continuous functions defined over the triangu-lationMh . The Ritz methods for approximating the solution ofEq. 2 are

minX

ED(X ) [3a]

s.t. X ∈C∗(Ω, ∂T )∩Λh . [3b]

This is a finite-dimensional, linearly constrained strictly convexquadratic optimization problem (strict convexity follows fromLemma 1 below) and is uniquely solved via a sparse linear system(the Lagrange multipliers equation). Let Ψh denote this solution.We prove the following:

Theorem 2. Let Ω⊂R2 be a simply connected polygonal domain;T ⊂R2 be a triangle; Φ : Ω→T be the Riemann mapping satisfyingΦ(pi) = ci , i = 1, 2, 3; andMh be a sequence of regular triangula-tions with maximal edge length h . Then the solution Ψh of Eq. 3satisfies

limh→0‖Ψh −Φ‖= 0.

Furthermore, if all Mh are 3-connected Delaunay and T isequilateral or right-angled isosceles, the convergence is also uniform.

In the case that the triangle T is one of the Euclidean orb-ifolds, that is, an equilateral triangle or right-angled isoscelestriangle, then Eq. 3 is exactly the orbifold-Tutte algorithm (15).If Mh is Delaunay and T is an orbifold, it is proved in ref. 15that Ψh is bijective. Since it also converges uniformly by Theo-rem 2, we can approximate the Riemann mapping between twopolygons:

Corollary 1 (Proof in SI Appendix ). Let Ω, Ω′⊂R2 be two sim-ply connected polygonal domains; T ⊂R2 be an equilateral or

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right-angled isosceles triangle; ξ : Ω→Ω′ be the unique Riemannmapping satisfying ξ(pi) = p′i , i = 1, 2, 3; and Mh ,M′h be twosequences of 3-connected Delaunay regular triangulations of Ω, Ω′

(resp.) with maximal edge length≤ h; and let Ψh , Ψ′h be the discreteconformal maps from these triangulations to T . Then, (Ψ′h)−1 Ψh

converges to ξ uniformly.

Proof of Tightness (Theorem 1 )In this section we prove Theorem 1, that is, show that problem Eq.2 has a unique solution and this solution is the conformal mapΦ : Ω→T . We start with Lemma 1, showing that ED restricted toC∗(Ω,T ) is coercive. Uniqueness of the minimizer then follows.Let us denote by V∗(Ω, ∂T ) the vector space which consists thelinear part of C∗(Ω, ∂T ). That is,

V∗(Ω, ∂T ) = X −Y∣∣∣ X ,Y ∈C∗(Ω, ∂T ). [4]

Lemma 1. The Dirichlet energy satisfies ED(X )≥ c ‖X ‖2 for someconstant c> 0, and any X ∈V∗(Ω, ∂T ). The constant c dependsonly on Ω, T , and the choice of the three points p1, p2, p3 ∈ ∂Ω.

Proof: Let X ∈V∗(Ω, ∂T ). Since ‖X ‖2 =ED(X ) +

‖X ‖2L2(Ω,R2), it is enough to show a bound of the form

‖X ‖2L2(Ω,R2)≤CED(X ),

for some C > 0. Denote R2 3 x = 1|Ω|

∫ΩX the average of X .

Denote |X |D =ED(X )1/2. We claim that it is sufficient to showthat for some C1 > 0,

|x | ≤C1|X |D . [5]

To see this, use the triangle inequality followed by Poincareinequality [ref. 1, theorem 12.23; note that Ω is in particular aconnected extension domain for H 1(Ω,R)],

‖X ‖L2(Ω,R2)≤‖X − x‖L2(Ω,R2) + ‖x‖L2(Ω,R2)

≤CPoincare|X |D + vol(Ω)12 |x |

≤(CPoincare +C1vol(Ω)

12

)|X |D ,

where for the last inequality we used Eq. 5 and in the second tolast inequality

‖x‖L2(Ω,R2) = vol(Ω)12 |x |.

We now bound |x | as in Eq. 5. Using the trace inequality and thePoincare inequality yet again, we have

‖X − x‖L2(∂Ω,R2)≤Ctrace ‖X − x‖ [6]

≤Ctrace(1 +CPoincare)|X |D .

The square norm of L2(∂Ω,R2) is the sum of the squared normover each boundary arc Γi , i = 1, 2, 3. Denote the length of Γi by|Γi |. So by omitting the last arc we obtain

‖X − x‖2L2(∂Ω,R2) ≥2∑

i=1

‖X − x‖2L2(Γi ,R2)

(∗)≥

2∑i=1

∥∥∥aTi (X − x )

∥∥∥2

L2(Γi )

(∗∗)=

2∑i=1

∥∥∥aTi x∥∥∥2

L2(Γi )

=

2∑i=1

(aTi x )2 |Γi |= |Ax |2,

where (∗) follows from pointwise application of the Cauchy–Schwarz inequality in R2, assuming without loss of generality(w.l.o.g.) that |ai |= 1; (∗∗) follows from X ∈V∗(Ω, ∂T ) and con-dition iii in Definition 2; and A∈R2×2 is the invertible matrixwith rows

√|Γi |aT

i . Finally,

|x |= |A−1Ax | ≤∥∥A−1

∥∥2,2|Ax |. [7]

Using Eqs. 6 and 7, we achieve our goal stated in Eq. 5, where

C1 =∥∥A−1

∥∥2,2

CTrace(1 +CPoincare).

Hence, the constant C = (CPoincare +C1vol(Ω)12 )2 and there-

fore also the constant c = c(Ω, ∂T ) in the theorem formulationare dependent only on Ω, T , and the choice of three pointsp1, p2, p3 ∈ ∂Ω.

An important consequence of the coercivity of ED is theuniqueness of the solution of Eq. 2:

Lemma 2. The relaxed Plateau’s problem Eq. 2 has at most a singlesolution.

Proof: Assume Eq. 2 has two solutions X ,Y ∈C∗(Ω, ∂T ).Restricting ED to the infinite line tX + (1− t)Y , t ∈R, resultsin a coercive 1D quadratic polynomial in t and hence it is strictlyconvex. Thus X =Y .

Lemma 2 implies that if the conformal map Φ is a solution toEq. 2, then it is unique and the relaxation is indeed tight. To showΦ is a solution we first recall that the Dirichlet energy is an upperbound of the area functional,

ED(X )≥EA(X ), [8]

whereEA(X ) =

∫Ω

|det [Xu Xv ]|

is the area functional. This inequality can be proved usingthe inequality | detA| ≤ 1

2|A|2F , where A∈R2×2 and |·|F is the

Frobenious norm of a matrix. When A is a similarity matrix,equality holds. For the conformal map Φ, [Φu , Φv ] is a similaritymatrix everywhere in the open set Ω and therefore

ED(Φ) =EA(Φ) = |T | ,

where |T | denotes the area of the triangle T . It follows that toshow that Φ is a solution to Eq. 2, and thus conclude the proof ofTheorem 1, it is sufficient to prove the following lemmas:

Lemma 3. Every X ∈C∗(Ω, ∂T )∩C (Ω,R2)∩C∞(Ω,R2) satis-fies ED(X )≥ |T |.

Lemma 4 (Proof in SI Appendix ). C∗(Ω, ∂T ) equals the closure inH 1(Ω,R2) of C∗(Ω, ∂T )∩C (Ω,R2)∩C∞(Ω,R2).

Proof of Lemma 3: Take an arbitrary X ∈C∗(Ω, ∂T )∩C (Ω,R2)∩C∞(Ω,R2). We first want to prove that every pointq in T (the interior of the triangle) has a preimage p ∈Ω.Assume q ∈T ; the winding number of q w.r.t. the restrictionof X to ∂Ω is w(q ,TX ) = 1. To see that, consider a home-omorphism Y : Ω→T satisfying Y (pi) = ci , i = 1, 2, 3 (e.g.,the Riemann mapping). Consider the homotopy H (·, t) = (1−t)TX (·) + t TY (·). Note that the image of H (·, t) is containedin ∪3

i=1`i and, since q does not belong to the latter set, thewinding number g(t) =w(q ,H (·, t)) is a continuous functionof t . Since TY : ∂Ω→ ∂T is a homeomorphism, we have thatg(1) =w(q ,H (·, 1)) =w(q ,TY ) = 1. We also know that g(t)∈Z and therefore w(q ,TX ) =w(q ,H (·, 0)) = g(0) = 1. Now toshow that p has a preimage under X we use a mapping degreeargument. Assume toward a contradiction that it does not have

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a preimage. Then by the boundary theorem (ref. 41, proposition4.4) since q /∈X (Ω), we have w(q ,TX ) = 0, a contradiction. Thelemma now follows from the following intuitive lemma.

Lemma 5 (Proof in SI Appendix ). If the image of X ∈C (Ω,R2)∩C∞(Ω,R2) contains T , then ED(X )≥EA(T )≥ |T |.

Convergence of Finite-Element ApproximationsWe now approximate the Riemann mapping Φ : Ω→T , using theRitz–Galerkin method. Given a series of regular triangulationsMh = (Vh , Eh ,Fh) of the polygonal domain Ω, with maximaledge length h→ 0 we denote by Λh ⊂C∗(Ω, T ) the set of con-tinuous piecewise-linear mappings defined over Mh . That is,Ψ∈Λh is a continuous function that is affine when restricted toeach triangle face, Ψ|f , f ∈Fh . We approximate the Riemannmapping Φ, by solving Eq. 3. Restricted to the finite-dimensionalspace Λh , Eq. 3 is a linearly constrained quadratic minimiza-tion problem. Indeed, letϕ1, . . . ,ϕ|Vh | denote the standard FEMbasis of the continuous piecewise-linear scalar functions overMh defined by ϕj (vk ) = δjk , for all j and vertices vk ∈Vh . Thenthe admissible set Eq. 3b, C∗(Ω, ∂T )∩Λh , can be written as thefollowing affine set in R2|Vh |:

X =∑j

xjϕj , xj|Vh |j=1 ⊂R2

aTi xk + bi = 0, ∀vk ∈Vh ∩Γi , i = 1, 2, 3.

The Dirichlet energy is

ED(X ) =∑kl

xkxlWkl , Wkl =

∫Ω

〈∇ϕk ,∇ϕl〉 .

Lemma 1 implies that this quadratic form is strictly positive def-inite over C∗(Ω, ∂T )∩Λh and therefore Eq. 3 has a uniquesolution. This solution is computed by solving the correspond-ing sparse linear Lagrange multiplier system which can be solvedefficiently with, e.g., a direct linear solver. Denoting the solutionto Eq. 3 by Ψh , we want to prove convergence of Ψh to Φ as themaximal edge length of the triangulationsMh goes to zero. Forthat end, we use Theorem 1 that identifies Φ as the unique mini-mum of the relaxed Plateau’s problem Eq. 2, the coercivity of ED

over C∗(Ω, ∂T ), and employ a result from a FEM called Cea’slemma (40) (proved in SI Appendix for the sake of completeness):

Lemma 6 (Cea). Let Φ be the unique Riemann map in C∗(Ω, ∂T )and Ψ be the solution of Eq. 3. Then,

‖Φ−Ψh‖≤C ‖Φ−X ‖ , ∀X ∈C∗(Ω, ∂T )∩Λh ,

where C is a constant independent of the choice of Λh .Proof of Theorem 2: Cea’s lemma reduces the problem of

showing that ‖Φ−Ψh‖→ 0 to an approximation problem; i.e.,

infX∈C∗(Ω,∂T )∩Λh

‖Φ−X ‖→ 0, [9]

as h→ 0. We prove Eq. 9 using the following lemma:

Lemma 7 (Proof in SI Appendix ). There is a sequence of functionsΦε⊆C∗(Ω, ∂T )∩C∞(Ω,R2) which converges to the Riemannmapping Φ in H 1(Ω,R2).

The triangle inequality, ‖Φ−X ‖≤‖Φ−Φε‖+ ‖Φε−X ‖,and Lemma 7 imply that it is enough to approximate Φε withX ∈C∗(Ω, ∂T )∩Λh . We take Xh to be the interpolant of Φε,that is, the unique function in Λh which agrees with Φε onthe vertices of Th ; i.e., Xh(vi) = Φε(vi), for all vi ∈Vh . Notethat Xh ∈C∗(Ω, ∂T )∩Λh . A standard approximation result inthe theory of finite elements (e.g., theorem 4.4.20 in ref. 40)states that since, in particular, Φε ∈W 2

2 (Ω,R2), we have that‖Φε−Xh‖→ 0 as h→ 0. So convergence in H 1 norm is proved.That is, ‖Φ−Ψh‖→ 0 as h→ 0.

To prove uniform convergence, we assume that Mh are3-connected Delaunay and that T is an Euclidean orbifold,namely an equilateral or right-angled isosceles. In this case theorbifold-Tutte mapping Ψh : Ω→T is a homeomorphism (15).

Consider Φh to be a solution of the following optimizationproblem:

minX

ED(X ) [10a]

s.t. X ∈Λh [10b]X (v) = Φ(v), ∀v ∈ ∂Ω. [10c]

In theorem 2 of ref. 43 it is shown that Φh→Φ uniformly ifΦ∈W 1

p (Ω,R2) for some p> 2. The singularities of the Riemannmapping Φ are of the form zα (here z = x + iy is a complexvariable), where α∈Θ⊂ (0, 2π), and Θ is a finite set of anglesdepending on the angles of the polygonal lines ∂Ω and ∂T .A direct calculation shows that if one takes p = 2 + ε whereε> 0 is sufficiently small so that α> 1− 2

p+εfor all α∈Θ, then

Φ∈W 1p (Ω,R2). Therefore, it is enough to show that Φh −Ψh

converge uniformly to the zero function.We next want to show that TΨh = Ψh |∂Ω has a subsequence

converging uniformly to some continuous-limit function g ∈C (∂Ω,R2). For this part we can assume w.l.o.g. that Ω is the unitdisk B. If that is not the case we let ϕ :B→Ω be a Riemann map-ping and consider Ψ′h = Ψh ϕ. Clearly TΨ′h converge uniformlyto g ϕ iff TΨh converge uniformly to g .

Since ‖Φ−Ψh‖→ 0, the Dirichlet energy of Ψh ∈H 1(Ω,R2)∩C (Ω,R2) is uniformly bounded (rememberthat the Dirichlet energy is invariant to a conformal changeof coordinates) and all Ψh satisfy Ψh(pi) = ci , i = 1, 2, 3. Itis known that the Courant–Lebesgue lemma (e.g., ref. 16,pp. 257 and 263, proposition 2) implies in this case thatTΨh = Ψh |∂Ω has a subsequence converging uniformly to somecontinuous-limit function g ∈C (∂Ω,R2).

Due to the trace theorem we have that TΨh converges to TΦin L2(∂Ω,R2) and therefore g =TΦ. This implies that TΨh con-verge uniformly to TΦ. Since Φh converge to Φ uniformly, wehave that TΦh converge to TΦ uniformly. SinceMh is Delau-nay, Ψh and Φh satisfy the discrete maximum principle (43).Hence,

‖Φh −Ψh‖C(Ω,R2)≤‖TΦh −TΨh‖C(∂Ω,R2)→ 0,

and since Φh converges uniformly to Φ, this concludes the proofof Theorem 2.

ACKNOWLEDGMENTS. This research was supported by European ResearchCouncil Consolidator Grant “LiftMatch” 771136 and Israel Science Founda-tion Grant 1830/17.

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