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 2

Discrete minimal surfaces

Weierstrass representation theorem 3

Discrete holomorphicity

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 R3.

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.

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Isothermic Surfaces in the Smooth Theory

Surfaces in Euclidean space R3.

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 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

1

infinitesimal isometric deformations and

2

mean curvature

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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

1

infinitesimal isometric deformations and

2

mean curvature

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Triangulated Surfaces

Given a triangulated surface f : M = (V, E, F) → R3, 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.

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Infinitesimal isometric deformations

Definition

Given f : M → R3. An infinitesimal deformation ˙ f : V → R3 is isometric if ˙

ℓ ≡ 0.

If ˙ f isometric, on each face △ijk there exists Zijk ∈ R3 as angular velocity: d˙ f(eij) = ˙ fj − ˙ fi = df(eij) × Zijk d˙ f(ejk) = ˙ fk − ˙ fj = df(ejk) × Zijk d˙ f(eki) = ˙ fi − ˙ fk = df(eki) × Zijk If two triangles △ijk and △jil share a common edge eij, compatibility condition: df(eij) × (Zijk − Zjil) = 0

∀e ∈ E

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Integrated mean curvature

A known discrete analogue of mean curvature ˜ H : E → R is defined by

˜

He := αeℓe. But if ˙

ℓ = ˙ ˜

H = 0 =

⇒ ˙ α = 0 = ⇒ trivial

Instead, we consider the integrated mean curvature around vertices H : V → R Hvi :=

• j

˜

Heij =

• j

αeijℓij.

If ˙ f preserves the integrated mean curvature additionally, it implies 0 = ˙ Hvi =

• j

˙ αijℓij =

• j

df(eij), Zijk − Zjil ∀vi ∈ V.

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 → R3 is isothermic if there exists a R3-valued dual 1-form

τ such that

• j

τ(∗eij) = 0 ∀vi ∈ V

df(e) × τ(∗e) = 0

∀e ∈ E

• j

df(eij), τ(∗eij) = 0 ∀vi ∈ V.

If additionally τ exact, i.e. ∃Z : F → R3 such that Zijk − Zjil = τ(∗eij). We call Z a Christoffel dual of f. Write f ∗ := Z from now on...

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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

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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 R3:

1

circle patterns

2

conformal equivalence

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Circle patterns

Circumscribed circles

Given f : M → R3, denote θ : E → (0, π] as the intersection angles of circumcircles.

Definition

We call ˙ f : V → R3 an infinitesimal pattern deformation if

˙ θ ≡ 0

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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.

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Conformal equivalence

Luo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010) i j k

˜

k

Definition

Given f : M → R3. We consider the length cross ratios lcr : E → R defined by lcrij := ℓjkℓil

ℓkiℓlj

Definition

An infinitesimal deformation ˙ f : V → R3 is called conformal if

˙

lcr ≡ 0

Definition (Conformal equivalence of triangulated

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Denote TfM = {infinitesimal conformal deformations of f}.

Theorem

For a closed genus-g triangulated surface f : M → R3, we have dim TfM ≥ |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.

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Example 1: Isothermic Quadrilateral Meshes

Definition (Bobenko and Pinkall, 1996)

A discrete isothermic net is a map f : Z2 → R3, for which all elementary quadrilaterals have cross-ratios q(fm,n, fm+1,n, fm+1,n+1, fm,n+1) = −1 ∀m, n ∈ Z, Known: Existence of a mesh (Christoffel Dual) f ∗ : Z2 → R3 such that for each quad f ∗

m+1,n − f ∗ m,n = −

fm+1,n − fm,n

||fm+1,n − fm,n||2

f ∗

m,n+1 − f ∗ m,n =

fm,n+1 − fm,n

||fm,n+1 − fm,n||2

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Theorem

There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with

˙

fm+1,n − ˙ fm,n = (fm+1,n − fm,n) × (f ∗

m+1,n + f ∗ m,n)/2,

˙

fm,n+1 − ˙ fm,n = (fm,n+1 − fm,n) × (f ∗

m,n+1 + f ∗ m,n)/2.

Compared to the smooth theory: d˙ f = df × f ∗

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Subdivision

− − − − − − →

Theorem

There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with

˙

fm+1,n − ˙ fm,n = (fm+1,n − fm,n) × (f ∗

m+1,n + f ∗ m,n)/2,

˙

fm,n+1 − ˙ fm,n = (fm,n+1 − fm,n) × (f ∗

m,n+1 + f ∗ m,n)/2.

Compared to the smooth theory: d˙ f = df × f ∗

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Subdivision

− − − − − − →

Theorem

There exists an infinitesimal deformation ˙ f preserving the edge lengths and the integrated mean curvature with

˙

fm+1,n − ˙ fm,n = (fm+1,n − fm,n) × (f ∗

m+1,n + f ∗ m,n)/2,

˙

fm,n+1 − ˙ fm,n = (fm,n+1 − fm,n) × (f ∗

m,n+1 + f ∗ m,n)/2.

Compared to the smooth theory: d˙ f = df × f ∗

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Example 2: Homogeneous cyclinders

Pick g1, g2 ∈ Eucl(R3) which fix z-axis: gi(p) =

 

cos θi sin θi

− sin θi

cos θi 1

  p +  

hi

 

for some θi, hi ∈ R3. Note g1, g2 ∼

= Z2.

Together with an initial point p0 ∈ R3 gives

A strip of an isothermic triangulated cylinder Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 19 / 33

Example 3: Inscribed Triangulated Surfaces

Theorem

For a surface with vertices on a sphere, a R3-valued dual 1-form τ satisfying

• j

τ(∗eij) = 0 ∀vi ∈ V

df(e) × τ(∗e) = 0

∀e ∈ E,

implies

• j

df(eij), τ(∗eij) = 0 ∀vi ∈ V.

Corollary

For triangulated surfaces with vertices on a sphere, any infinitesimal deformation preserving the edge lengths will preserve the integrated mean curvature.

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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 2015 21 / 33

Table of Content

1

Isothermic triangulated surfaces

2

Discrete minimal surfaces

3

Discrete holomorphicity

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Discrete minimal surfaces

Smooth theory: minimal surfaces are Christoffel duals of their Gauss images.

Definition

Given f : M → R3, a surface f ∗ : M∗ → R3 is called a Christoffel dual of f if df(e) × df ∗(∗e) = 0 ∀e ∈ E, (1)

• j

df(eij), df ∗(∗eij) = 0 ∀vi ∈ V,

(2)

Definition

f ∗ : M∗ → R3 is called a discrete minimal surface if f : M → S2 is inscribed on the unit sphere. Note: if f is inscribed, then

(1) holds = ⇒ (2) holds

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Equivalently, discrete minimal surfaces = reciprocal-parallel meshes of inscribed triangulated surfaces

1

f ∗ defined on dual vertices

2

dual edges parallel to primal edges

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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: ˙ f = uN,

3

Preserving the integrated mean curvature =

⇒

j(cot β + cot ˜

β)(uj − ui) = 0.

4

Inverse of stereographic projection

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Weierstrass representation theorem

Recall in the smooth theory

Theorem

Given holomorphic functions f, h : U ⊂ C → C such that f 2h is holomorphic. Then f ∗ : U → R3 defined by df ∗ = Re

• h(z)

 

f

(1 − f 2)/2 (1 + f 2)/2   dz

• is a minimal surface.

In our setting : f(z) = z, h = 2iuzz where u : U → R is harmonic.

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Weierstrass representation theorem

Data: A planar triangular mesh f : M → R2 + a discrete harmonic function u : V → R.

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Table of Content

1

Isothermic triangulated surfaces

2

Discrete minimal surfaces

3

Discrete holomorphicity

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Triangular meshes on C

Luo(2004);Springborn,Schröder,Pinkall(2008);Bobenko et al.(2010)

Theorem

An infinitesimal deformation ˙ z : M → C is conformal if there exists u : V → R such that

˙ |zj − zi| =

ui + uj 2

|zj − zi|.

We call u the scaling factors.

Theorem

An infinitesimal deformation ˙ z : M → C is a pattern deformation if there exists

α : V → R such that ˙ (

zj − zi

|zj − zi|) =

iαi + iαj 2 zj − zi

|zj − zi|.

We call iα the rotation factors.

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Theorem

An infinitesimal deformation ˙ z : V → C is conformal if and only if i ˙ z is a pattern deformation.

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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

• j

(cot βk + cot β˜

k)(hj − hi) = 0

∀i ∈ V.

2

There exists pattern deformation i ˙ z with rotation factors ih. It is unique up to infinitesimal scalings and translations.

3

There exists ˙ z conformal with scaling factors h. It is unique up to infinitesimal rotations and translations.

(1) ⇐ ⇒ (2) in Bobenko, Mercat, Suris (2005)

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Pick a Möbius transformation φ : ˆ

C → ˆ C

z w := φ ◦ z u harmonic

∃ ˜

u harmonic

˙

f conformal dφ(˙ f) conformal

φ

dφ

˜

u unique up to a linear function.

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Thank you!

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