From isothermic triangulated surfaces to discrete holomorphicity - PowerPoint PPT Presentation

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 2015 1 / 33

Table of Content 1 Isothermic triangulated surfaces Discrete conformality: circle patterns, conformal equivalence Discrete minimal surfaces 2 Weierstrass representation theorem Discrete holomorphicity 3 Planar triangular meshes Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 2 / 33

Isothermic Surfaces in the Smooth Theory Surfaces in Euclidean space R 3 . 1 Definition: Isothermic if there exists a conformal curvature line parametrization . Examples: surfaces of revolution, quadrics, constant mean curvature surfaces, 2 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 2015 3 / 33

Isothermic Surfaces in the Smooth Theory Surfaces in Euclidean space R 3 . 1 Definition: Isothermic if there exists a conformal curvature line parametrization . Examples: surfaces of revolution, quadrics, constant mean curvature surfaces, 2 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 2015 3 / 33

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 . Cie´ sli´ nski, Goldstein, Sym (1995) Discrete analogues of infinitesimal isometric deformations and 1 2 mean curvature Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 4 / 33

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 . Cie´ sli´ nski, Goldstein, Sym (1995) Discrete analogues of infinitesimal isometric deformations and 1 2 mean curvature Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 4 / 33

Triangulated Surfaces Given a triangulated surface f : M = ( V , E , F ) → R 3 , we can measure edge lengths ℓ : E → R , 1 dihedral angles of neighboring triangles α : E → R and 2 3 deform it by moving the vertices. Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 5 / 33

Infinitesimal isometric deformations Definition f : V → R 3 is isometric if ˙ Given f : M → R 3 . An infinitesimal deformation ˙ ℓ ≡ 0. If ˙ f isometric, on each face △ ijk there exists Z ijk ∈ R 3 as angular velocity: d ˙ f ( e ij ) = ˙ f j − ˙ f i = df ( e ij ) × Z ijk d ˙ f ( e jk ) = ˙ f k − ˙ f j = df ( e jk ) × Z ijk d ˙ f ( e ki ) = ˙ f i − ˙ f 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 2015 6 / 33

Integrated mean curvature A known discrete analogue of mean curvature ˜ H : E → R is defined by ˜ H e := α e ℓ e . ℓ = ˙ But if ˙ ˜ H = 0 = ⇒ ˙ α = 0 = ⇒ trivial Instead, we consider the integrated mean curvature around vertices H : V → R � � ˜ H v i := H e ij = α e ij ℓ ij . j j If ˙ f preserves the integrated mean curvature additionally, it implies � � 0 = ˙ H v i = α ij ℓ ij = ˙ � df ( e ij ) , Z ijk − Z jil � ∀ v i ∈ V . j j Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 7 / 33

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 . j If additionally τ exact, i.e. ∃ Z : F → R 3 such that 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 2015 8 / 33

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 2015 9 / 33

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 2015 9 / 33

Discrete conformality Two notions of discrete conformality of a triangular mesh in R 3 : circle patterns 1 2 conformal equivalence Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 10 / 33

Circle patterns Circumscribed circles Given f : M → R 3 , denote θ : E → ( 0 , π ] as the intersection angles of circumcircles. Definition f : V → R 3 an infinitesimal pattern deformation if We call ˙ ˙ θ ≡ 0 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 11 / 33

Circumscribed spheres Circumscribed circles 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 2015 12 / 33

Conformal equivalence Luo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010) j k ˜ k i Definition Given f : M → R 3 . We consider the length cross ratios lcr : E → R defined by lcr ij := ℓ jk ℓ il ℓ ki ℓ lj Definition An infinitesimal deformation ˙ f : V → R 3 is called conformal if ˙ lcr ≡ 0 Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 13 / 33 Definition (Conformal equivalence of triangulated

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 | − 6 g + 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 2015 14 / 33

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 2015 15 / 33

Theorem There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with ˙ 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 . Compared to the smooth theory: d ˙ f = df × f ∗ Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 16 / 33

Subdivision − − − − − − → Theorem There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with ˙ 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 . Compared to the smooth theory: d ˙ f = df × f ∗ Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 17 / 33

Subdivision − − − − − − → Theorem There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with ˙ 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 . Compared to the smooth theory: d ˙ f = df × f ∗ Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 18 / 33