From isothermic triangulated surfaces to discrete holomorphicity

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1 From isothermic triangulated surfaces to discrete holomorphicity Wai Yeung Lam TU Berlin Oberwolfach, 2 March 2015 Joint work with Ulrich Pinkall Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

2 Table of Content 1 Isothermic triangulated surfaces Discrete conformality: circle patterns, conformal equivalence 2 Discrete minimal surfaces Weierstrass representation theorem 3 Discrete holomorphicity Planar triangular meshes Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

3 Isothermic Surfaces in the Smooth Theory Surfaces in Euclidean space R 3. 1 Definition: Isothermic if there exists a conformal curvature line parametrization. 2 Examples: surfaces of revolution, quadrics, constant mean curvature surfaces, minimal surfaces. 3 Related to integrable systems. Enneper s Minimal Surface Aim: a discrete analogue without conformal curvature line parametrizations. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

Isothermic Surfaces in the Smooth Theory Surfaces in Euclidean space R 3. 1 Definition: Isothermic if there exists a conformal curvature line parametrization.

4 Isothermic Surfaces in the Smooth Theory Surfaces in Euclidean space R 3. 1 Definition: Isothermic if there exists a conformal curvature line parametrization. 2 Examples: surfaces of revolution, quadrics, constant mean curvature surfaces, minimal surfaces. 3 Related to integrable systems. Enneper s Minimal Surface Aim: a discrete analogue without conformal curvature line parametrizations. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

5 Isothermic Surfaces in the Smooth Theory Theorem A surface in Euclidean space is isothermic if and only if locally there exists a non-trivial infinitesimal isometric deformation preserving the mean curvature. Cieśliński, Goldstein, Sym (1995) Discrete analogues of 1 infinitesimal isometric deformations and 2 mean curvature Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

6 Isothermic Surfaces in the Smooth Theory Theorem A surface in Euclidean space is isothermic if and only if locally there exists a non-trivial infinitesimal isometric deformation preserving the mean curvature. Cieśliński, Goldstein, Sym (1995) Discrete analogues of 1 infinitesimal isometric deformations and 2 mean curvature Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

7 Triangulated Surfaces Given a triangulated surface f : M = (V, E, F) R 3, we can measure 1 edge lengths l : E R, 2 dihedral angles of neighboring triangles α : E R and 3 deform it by moving the vertices. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

8 Infinitesimal isometric deformations Definition Given f : M R 3. An infinitesimal deformation ḟ : V R 3 is isometric if l 0. If ḟ isometric, on each face ijk there exists Z ijk R 3 as angular velocity: dḟ(e ij ) = ḟ j ḟ i = df(e ij ) Z ijk dḟ(e jk ) = ḟ k ḟ j = df(e jk ) Z ijk dḟ(e ki ) = ḟ i ḟ k = df(e ki ) Z ijk If two triangles ijk and jil share a common edge e ij, compatibility condition: df(e ij ) (Z ijk Z jil ) = 0 e E Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

9 Integrated mean curvature A known discrete analogue of mean curvature H : E R is defined by H e := α e l e. But if l = H = 0 = α = 0 = trivial Instead, we consider the integrated mean curvature around vertices H : V R H vi := j H eij = j α eij l ij. If ḟ preserves the integrated mean curvature additionally, it implies 0 = Ḣ vi = j α ij l ij = j df(e ij ), Z ijk Z jil v i V. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

10 M = combinatorial dual graph of M e = dual edge of e. Definition A triangulated surface f : M R 3 is isothermic if there exists a R 3 -valued dual 1-form τ such that τ( e ij ) = 0 v i V j df(e) τ( e) = 0 e E df(e ij ), τ( e ij ) = 0 v i V. If additionally τ exact, i.e. Z : F R 3 such that j Z ijk Z jil = τ( e ij ). We call Z a Christoffel dual of f. Write f := Z from now on... Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

11 The previous argument gives Corollary A simply connected triangulated surface is isothermic if and only if there exists a non-trivial infinitesimal isometric deformation preserving H. As in the smooth theory, we proved Theorem The class of isothermic triangulated surfaces is invariant under Möbius transformations. We can transform τ explicitly under Möbius transformations Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

12 The previous argument gives Corollary A simply connected triangulated surface is isothermic if and only if there exists a non-trivial infinitesimal isometric deformation preserving H. As in the smooth theory, we proved Theorem The class of isothermic triangulated surfaces is invariant under Möbius transformations. We can transform τ explicitly under Möbius transformations Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

13 Discrete conformality Two notions of discrete conformality of a triangular mesh in R 3 : 1 circle patterns 2 conformal equivalence Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

14 Circle patterns Circumscribed circles Given f : M R 3, denote θ : E (0, π] as the intersection angles of circumcircles. Definition We call ḟ : V R 3 an infinitesimal pattern deformation if θ 0 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

15 Circumscribed circles Circumscribed spheres Theorem A simply connected triangulated surface is isothermic if and only if there exists a non-trivial infinitesimal pattern deformation preserving the intersection angles of neighboring spheres. Trivial deformations = Möbius deformations Smooth theory: an infinitesimal conformal deformation preserving Hopf differential. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

16 Conformal equivalence Luo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010) j k k Definition Given f : M R 3. We consider the length cross ratios lcr : E R defined by Definition i lcr ij := l jkl il l ki l lj An infinitesimal deformation ḟ : V R 3 is called conformal if lcr 0 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

17 Denote T f M = {infinitesimal conformal deformations of f}. Theorem For a closed genus-g triangulated surface f : M R 3, we have dim T f M V 6g + 6. The inequality is strict if and only if f is isothermic. Smooth Theory: Isothermic surfaces are the singularities of the space of conformal immersions. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

18 Example 1: Isothermic Quadrilateral Meshes Definition (Bobenko and Pinkall, 1996) A discrete isothermic net is a map f : Z 2 R 3, for which all elementary quadrilaterals have cross-ratios q(f m,n, f m+1,n, f m+1,n+1, f m,n+1 ) = 1 m, n Z, Known: Existence of a mesh (Christoffel Dual) f : Z 2 R 3 such that for each quad f m+1,n f m,n f m+1,n f m,n = f m+1,n f m,n 2 f m,n+1 f m,n = f m,n+1 f m,n f m,n+1 f m,n 2 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

19 Theorem There exists an infinitesimal deformation ḟ preserving the edge lengths and the integrated mean curvature with ḟ m+1,n ḟ m,n = (f m+1,n f m,n ) (f m+1,n + fm,n)/2, ḟ m,n+1 ḟ m,n = (f m,n+1 f m,n ) (f m,n+1 + fm,n)/2. Compared to the smooth theory: dḟ = df f Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

20 Subdivision Theorem There exists an infinitesimal deformation ḟ preserving the edge lengths and the integrated mean curvature with ḟ m+1,n ḟ m,n = (f m+1,n f m,n ) (f m+1,n + fm,n)/2, ḟ m,n+1 ḟ m,n = (f m,n+1 f m,n ) (f m,n+1 + fm,n)/2. Compared to the smooth theory: dḟ = df f Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

Subdivision Theorem There exists an infinitesimal deformation ḟ preserving the edge lengths and the integrated mean curvature with ḟ m+1,n ḟ m,n = (f m+1,n f m,n ) (f m+1,n +

21 Subdivision Theorem There exists an infinitesimal deformation ḟ preserving the edge lengths and the integrated mean curvature with ḟ m+1,n ḟ m,n = (f m+1,n f m,n ) (f m+1,n + fm,n)/2, ḟ m,n+1 ḟ m,n = (f m,n+1 f m,n ) (f m,n+1 + fm,n)/2. Compared to the smooth theory: dḟ = df f Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

Example 2: Homogeneous cyclinders Pick g 1, g 2 Eucl(R 3 ) which fix z-axis: g i (p) = cos θ i sin θ i 0 sin θ i cos θ i 0 0 0 1 p + 0 0 h i for some θ i, h i R 3.

22 Example 2: Homogeneous cyclinders Pick g 1, g 2 Eucl(R 3 ) which fix z-axis: g i (p) = cos θ i sin θ i 0 sin θ i cos θ i p h i for some θ i, h i R 3. Note g 1, g 2 = Z 2. Together with an initial point p 0 R 3 gives A strip of an isothermic triangulated cylinder Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

23 Example 3: Inscribed Triangulated Surfaces Theorem For a surface with vertices on a sphere, a R 3 -valued dual 1-form τ satisfying τ( e ij ) = 0 v i V j df(e) τ( e) = 0 e E, implies df(e ij ), τ( e ij ) = 0 v i V. j Corollary For triangulated surfaces with vertices on a sphere, any infinitesimal deformation preserving the edge lengths will preserve the integrated mean curvature. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

24 More examples of isothermic surfaces: (a) Inscribed Triangular meshes with boundary (b) Jessen s Orthogonal Icosahedron Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

25 Table of Content 1 Isothermic triangulated surfaces 2 Discrete minimal surfaces 3 Discrete holomorphicity Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

26 Discrete minimal surfaces Smooth theory: minimal surfaces are Christoffel duals of their Gauss images. Definition Given f : M R 3, a surface f : M R 3 is called a Christoffel dual of f if df(e) df ( e) = 0 e E, (1) df(e ij ), df ( e ij ) = 0 v i V, (2) j Definition f : M R 3 is called a discrete minimal surface if f : M S 2 is inscribed on the unit sphere. Note: if f is inscribed, then (1) holds = (2) holds Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

27 Equivalently, discrete minimal surfaces = reciprocal-parallel meshes of inscribed triangulated surfaces 1 f defined on dual vertices 2 dual edges parallel to primal edges Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

Constructing discrete minimal surfaces Equivalent to find an infinitesimal rigid deformation of a planar triangular mesh preserving the integrated mean curvature.

28 Constructing discrete minimal surfaces Equivalent to find an infinitesimal rigid deformation of a planar triangular mesh preserving the integrated mean curvature. 1 a planar triangular mesh, 2 Infinitesimal rigid deformation of a planar triangular mesh: ḟ = un, 3 Preserving the integrated mean curvature = j (cot β + cot β)(u j u i ) = 0. 4 Inverse of stereographic projection Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

29 Weierstrass representation theorem Recall in the smooth theory Theorem Given holomorphic functions f, h : U C C such that f 2 h is holomorphic. Then f : U R 3 defined by ( df = Re h(z) is a minimal surface. f (1 f 2 )/2 (1 + f 2 )/2 ) dz In our setting : f(z) = z, h = 2iu zz where u : U R is harmonic. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

30 Weierstrass representation theorem Data: A planar triangular mesh f : M R 2 + a discrete harmonic function u : V R. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

31 Table of Content 1 Isothermic triangulated surfaces 2 Discrete minimal surfaces 3 Discrete holomorphicity Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

32 Triangular meshes on C Luo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010) Theorem An infinitesimal deformation ż : M C is conformal if there exists u : V R such that We call u the scaling factors. Theorem z j z i = u i + u j z j z i. An infinitesimal deformation ż : M C is a pattern deformation if there exists α : V R such that We call iα the rotation factors. ( z j z i z j z i ) = iα i + iα j z j z i 2 z j z i. 2 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

33 Theorem An infinitesimal deformation ż : V C is conformal if and only if iż is a pattern deformation. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

34 Theorem Let z : M C be an immersed triangular mesh and h : V R be a function. The following are equivalent. 1 h is a harmonic function (cot β k + cot β k)(h j h i ) = 0 i V. j 2 There exists pattern deformation i ż with rotation factors ih. It is unique up to infinitesimal scalings and translations. 3 There exists ż conformal with scaling factors h. It is unique up to infinitesimal rotations and translations. (1) (2) in Bobenko, Mercat, Suris (2005) Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

35 Pick a Möbius transformation φ : Ĉ Ĉ z w := φ z φ u harmonic ũ harmonic ḟ conformal dφ dφ(ḟ) conformal ũ unique up to a linear function. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

36 Thank you! Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March / 33

Discrete differential geometry: Surfaces made from Circles

Discrete differential geometry: Alexander Bobenko (TU Berlin) with help of Tim Hoffmann, Boris Springborn, Ulrich Pinkall, Ulrike Scheerer, Daniel Matthes, Yuri Suris, Kevin Bauer Papers A.I. Bobenko,