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Plateau’s Riemann of instance the is gen- in problem equation differential partial When eral. nonlinear functions, and a of (Dirichlet) to set energy corresponds admissible quadratic varia- nonlinear convex a a a is problem with Plateau’s problem way, tional this Formulated 16). ref. (e.g., map ary mappings parameterization a of energy X Dirichlet the minimizing by solved curve prescribed a spanning Γ area minimal with problem surface Plateau’s a case. for 2D seeks the in problem Plateau’s of relaxation and uniform is convergence the Lip- scenario H general this to In extended domains. be schitz can domains polygonal for obtained mapping map- in Riemann orbifold-Tutte the the Finally to of uniformly composition converges the pings that prove closure we the tions domains over polygonal uniform connected also is vergence triangulation triangle the initial and the Delaunay where algorithm, triangulation orbifold-Tutte the of ment owo orsodnesol eadesd mi:[email protected] Email: addressed. be should correspondence work.y whom To this to equally contributed Y.L. and R.S., N.D, eutcnomlmpig a ecmue outyand robustly computed be can efficiently. mappings conformal As result problem. linear a a domain, becomes target mappings suitable a In conformal choosing computing fields. by that related show and we medical geometry, paper vision, morphology, this and differential in graphics discrete applications computer many in imaging, found task has central which a is between domains mappings (angle-preserving) conformal Computing Significance ⊂ 1 1 h iervrainlpicpei eie rmatgtlinear tight a from derived is principle variational linear The : B nalcmatsbesof subsets compact all on ∂ R omt h imn mapping Riemann the to norm Ω ⊂ T d → C , (Ω, stebs oan n emk atclrcoc of choice particular a make we and domain, base the as d R X R ti elkonta lta’ rbe a be can problem Plateau’s that known well is It 3. 2, = d X IAppendix SI ∂ Γ where , 2 | y ∂ T xn h riae ftecresof corners the of preimages the fixing , C ∈ ⊂ B o lta’ aitoa rbe snnier(it nonlinear is problem variational Plateau’s for ), NSlicense.y PNAS R : ( ∂ 2 B ¯ B h nqemnmzro lta’ variational Plateau’s of minimizer unique the , Γ) , B → steoe ntds,aogaladmissible all among disc, unit open the is Γ eso htteecnegneresults convergence these that show we iha(eky oemrhcbound- homeomorphic (weakly) a with xn he onso h boundary the on points three fixing utemr,i h aeo the of case the in Furthermore, M. 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APPLIED MATHEMATICS 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. The method (FEM) (33). Since Riemann mappings consist of two right-hand side of Fig. 1 visualizes the high resolution of the map conjugate harmonic functions, researchers have constructed dis- near the boundary. crete conformal mappings by pairs of conjugate discrete har- monic functions defined via the Dirichlet integral (34–36). These Related Work algorithms are linear but do not satisfy any prescribed bound- The notion of a discrete conformal mapping of a triangulation ary conditions and are not known to converge to the Rie- M is a rather well-researched area. It is rich with constructions mann mapping. Convergence to the Riemann mapping, or more and algorithms, each with its own definition of discrete confor- generally the solution of Plateau’s problem, can be obtained mality, often inspired by some property of smooth conformal by minimizing the Dirichlet energy (37, 38) or a conformal mappings. Although we focus here on discrete conformal map- energy (39), while imposing nonlinear boundary conditions. Solv- pings, we note that there are other numerical algorithms with ing these nonconvex variational problems is a computational convergence guarantees to the Riemann mapping based on the challenge.

Fig. 1. (Left) An approximation of the notorious Riemann map from a polygonal approximation of the Koch snowflake (computed with six recursions) to a 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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APPLIED MATHEMATICS right-angled isosceles triangle; ξ :Ω → Ω0 be the unique Riemann where (∗) follows from pointwise application of the Cauchy– 0 0 2 mapping satisfying ξ(pi ) = pi , i = 1, 2, 3; and Mh , Mh be two Schwarz inequality in R , assuming without loss of generality 0 ∗ sequences of 3-connected Delaunay regular triangulations of Ω, Ω (w.l.o.g.) that |ai | = 1; (∗∗) follows from X ∈ V (Ω, ∂T ) and con- 0 2×2 (resp.) with maximal edge length ≤ h; and let Ψh ,Ψh be the discrete dition iii in Definition 2; and A ∈ R is the invertible matrix 0 −1 p T conformal maps from these triangulations to T . Then, (Ψh ) ◦ Ψh with rows |Γi |ai . Finally, converges to ξ uniformly. −1 −1 |x| = |A Ax| ≤ A |Ax|. [7] Proof of Tightness (Theorem 1 ) 2,2 In this section we prove Theorem 1, that is, show that problem Eq. Using Eqs. 6 and 7, we achieve our goal stated in Eq. 5, where 2 has a unique solution and this solution is the conformal map Φ:Ω → T . We start with Lemma 1, showing that ED restricted to −1 ∗ C1 = A CTrace(1 + CPoincar´e). C (Ω, T ) is coercive. Uniqueness of the minimizer then follows. 2,2 ∗ Let us denote by V (Ω, ∂T ) the vector space which consists the 1 ∗ 2 linear part of C (Ω, ∂T ). That is, Hence, the constant C = (CPoincar´e + C1vol(Ω) 2 ) and there- fore also the constant c = c(Ω, ∂T ) in the theorem formulation

∗ ∗ are dependent only on Ω, T , and the choice of three points V (Ω, ∂T ) = {X − Y X , Y ∈ C (Ω, ∂T )}. [4] p1, p2, p3 ∈ ∂Ω. 2 An important consequence of the coercivity of ED is the Lemma 1. The Dirichlet energy satisfies ED (X ) ≥ c kX k for some uniqueness of the solution of Eq. 2: constant c > 0, and any X ∈ V∗(Ω, ∂T ). The constant c depends only on Ω, T , and the choice of the three points p1, p2, p3 ∈ ∂Ω. Lemma 2. The relaxed Plateau’s problem Eq. 2 has at most a single ∗ 2 Proof: Let X ∈ V (Ω, ∂T ). Since kX k = ED (X ) + solution. 2 ∗ kX k 2 2 , it is enough to show a bound of the form Proof: Assume Eq. 2 has two solutions X , Y ∈ C (Ω, ∂T ). L (Ω,R ) Restricting ED to the infinite line tX + (1 − t)Y , t ∈ R, results 2 in a coercive 1D quadratic polynomial in t and hence it is strictly kX kL2(Ω, 2) ≤ CED (X ), R convex. Thus X = Y . 2 1 R Lemma 2 implies that if the conformal map Φ is a solution to for some C > 0. Denote R 3 x = |Ω| Ω X the average of X . 1/2 Eq. 2, then it is unique and the relaxation is indeed tight. To show Denote |X |D = ED (X ) . We claim that it is sufficient to show Φ is a solution we first recall that the Dirichlet energy is an upper that for some C1 > 0, bound of the area functional,

|x| ≤ C1|X | . [5] D ED (X ) ≥ EA(X ), [8] To see this, use the triangle inequality followed by Poincar´e where inequality [ref. 1, theorem 12.23; note that Ω is in particular a Z 1 connected extension domain for H (Ω, R)], EA(X ) = |det [Xu Xv ]| Ω

kX k 2 2 ≤kX − xk 2 2 + kxk 2 2 L (Ω,R ) L (Ω,R ) L (Ω,R ) is the area functional. This inequality can be proved using 1 2 2×2 1 the inequality | det A| ≤ |A| , where A ∈ R and |·| is the ≤C |X | + (Ω) 2 |x| 2 F F Poincar´e D vol Frobenious norm of a matrix. When A is a similarity matrix,  1  2 equality holds. For the conformal map Φ, [Φu ,Φv ]is a similarity ≤ CPoincar´e + C1vol(Ω) |X | , D matrix everywhere in the open set Ω and therefore 5 where for the last inequality we used Eq. and in the second to E (Φ) = E (Φ) = |T | , last inequality D A 1 2 kxk 2 2 = vol(Ω) |x|. |T | T L (Ω,R ) where denotes the area of the triangle . It follows that to Φ We now bound |x| as in Eq. 5. Using the trace inequality and the show that is a solution to Eq. 2, and thus conclude the proof of Poincar´e inequality yet again, we have Theorem 1, it is sufficient to prove the following lemmas: Lemma 3. Every X ∈ C∗(Ω, ∂T ) ∩ C (Ω,¯ 2) ∩ C ∞(Ω, 2) satis- kX − xk 2 2 ≤Ctrace kX − xk [6] R R L (∂Ω,R ) fies ED (X ) ≥ |T |. ≤Ctrace(1 + CPoincar´e)|X |D . Lemma 4 (Proof in SI Appendix ). C∗(Ω, ∂T ) equals the closure in 2 2 1 2 ∗ 2 ∞ 2 The square norm of L (∂Ω, R ) is the sum of the squared norm H (Ω, R ) of C (Ω, ∂T ) ∩ C (Ω,¯ R ) ∩ C (Ω, R ). over each boundary arc Γi , i = 1, 2, 3. Denote the length of Γi by Proof of Lemma 3: Take an arbitrary X ∈ C∗(Ω, ∂T ) ∩ 2 ∞ 2 |Γi |. So by omitting the last arc we obtain C (Ω,¯ R ) ∩ C (Ω, R ). We first want to prove that every point q T p ∈ Ω 2 in (the interior of the triangle) has a preimage . 2 X 2 Assume q ∈ T ; the winding number of q w.r.t. the restriction kX − xk 2 2 ≥ kX − xk 2 2 L (∂Ω,R ) L (Γi ,R ) of X to ∂Ω is w(q, TX ) = 1. To see that, consider a home- i=1 omorphism Y : Ω¯ → T¯ satisfying Y (pi ) = ci , i = 1, 2, 3 (e.g., 2 (∗) 2 X T the Riemann mapping). Consider the homotopy H (·, t) = (1 − ≥ ai (X − x) L2(Γ ) t)TX (·) + t TY (·). Note that the image of H (·, t) is contained i=1 i 3 in ∪i=1`i and, since q does not belong to the latter set, the 2 2 winding number g(t) = w(q, H (·, t)) is a continuous function (∗∗)X T = ai x of t. Since TY : ∂Ω → ∂T is a homeomorphism, we have that L2(Γ ) i=1 i g(1) = w(q, H (·, 1)) = w(q, TY ) = 1. We also know that g(t) ∈ 2 and therefore w(q, TX ) = w(q, H (·, 0)) = g(0) = 1. Now to X T 2 2 Z = (ai x) |Γi | = |Ax| , show that p has a preimage under X we use a mapping degree i=1 argument. Assume toward a contradiction that it does not have

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(e.g., proposition function continuous-limit lemma 263, Courant–Lebesgue and T the 257 that pp. known is H to consider and ping function continuous-limit disk some to C uniformly converging Φ oni osldtrGat“itac”713 n salSineFounda- Science Israel 1830/17. and Grant 771136 tion “LiftMatch” Grant Consolidator Council ACKNOWLEDGMENTS. that have to uniformly verge change conformal a to all invariant and is coordinates) energy of Dirichlet the that function. zero the to uniformly converge nterm2o e.4 ti hw that where shown variable), is it mapping 43 ref. of Φ 2 theorem In ietcluainsosta foetakes one if lines that polygonal shows > the  calculation of direct angles A the on depending problem: the case this In isosceles. mapping right-angled orbifold-Tutte or equilateral that an namely and Delaunay 3-connected htis, That in that of function vertices unique the the is, that X and h hoyo nt lmns(.. hoe ..0i e.40) ref. in 4.4.20 particular, theorem in kΦ (e.g., since, elements that finite states of theory the .LiL,TiuekdmS agY hmsnP,Ca F(00 piie con- Optimized (2010) TF Chan PM, Thompson Y, Wang S, flattening. Thiruvenkadam colon LM, virtual Lui Conformal (2006) 7. A Kaufman M, Jin F, confor- Qiu surface X, zero Gu Genus W, Hong (2004) ST 6. Yau PM, Thompson TF, Chan Y, Wang X, Gu 5. Ψ (∂ ∈ ∈ 1 Consider that assume we convergence, uniform prove To inequality, triangle The u otetaetermw aethat have we theorem trace the to Due Since enx att hwthat show to want next We L omlsraergsrto ihsaebsdlnmr matching. 3:52–78. landmark shape-based with registration surface formal Modeling Physical and 85–93. Solid on pp York), Symposium ACM 2006 the of Proceedings mapping. surface brain to 23:949–958. application its and mapping mal C ∈ g hoe 2. Theorem  (Ω, 0 h 2 W W − ◦ Ω, B em 7 Lemma X (∂ Ψ Ψ = ssfcetysals that so small sufficiently is ϕ ∗ fta snttecs elet we case the not is that If . p p X h 1 1 R h (Ω, R kΦ Ω, kΦ (Ω, (Ω, iff C ∈ h 2 2 and → k h Φ ) Φ o hspr ecnasm ....that w.l.o.g. assume can we part this For ). 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Λ ⊂ h iihe nryof energy Dirichlet the 0, 2 D Ψ ) (v h X ∈ suiomybudd(remember bounded uniformly is p h 0 2π (0, h (X Ψ g k ≤ h tnadapoiainrsl in result approximation standard A . h Φ(v = ) > Λ ◦ → ∈ Ω : h (v T g h Clearly ϕ. ) h iglrte fteRiemann the of singularities The 2. T C kΦ z = i satisfy 0. Φ T T → Φ = ) α Φ Φ X and ), (∂ T Ψ Φ nfrl.Since uniformly. h > α − h h (here Λ Ω,  h hsipisthat implies This Φ. T ϕ ), − ovreto converge h X ob h nepln of interpolant the be to ∈ Ψ = sahmoopim(15). homeomorphism a is  : R T hscnldsteproof the concludes this Φ, 1 (v kΦ ≤ k → B hc geswith agrees which Ψ sa ulda orbifold, Euclidean an is W Θ 2 − ∀v Ψ g T i h ). z 2 h o all for ), NSLts Articles Latest PNAS . 2 (p h safiiesto angles of set finite a is Ψ p | T = (Ω, ∂ ∈ k + Ω 2 i Φ h 0 Ω C Ψ = ) x ∂ H eaReanmap- Riemann a be − h ovreuniformly converge ( R h a subsequence a has Ω. + o all for ∂ → EETasMdImaging Med Trans IEEE 1 Ω,R Φ 2 ovre to converges c Φ iy p ehv that have we ), omi proved. is norm i  Φ , k + 2 = 2 M nfrl,we uniformly, v IMJIaigSci Imaging J SIAM ) sacomplex a is ∂ i + i nfrl if uniformly Ω → α Ω ,2 3 2, 1, = V ∈ h kΦ T steunit the is ∈ sDelau- is 0, and Φ M AM New (ACM,  Ψ h Φ then Θ,  h Note . | Φ −  where h h [10b] [10a] [10c] − Ψ  f6 of 5 con- with X ∂ T It . g Φ h are Ψ on T k, Φ ∈ ∈  h . ,

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