From isothermic triangulated surfaces to discrete ...lam/slides/oberwolfach_ddg2015.pdfFrom isothermic triangulated surfaces to discrete holomorphicity Wai Yeung Lam TU Berlin Oberwolfach,

From isothermic triangulated surfaces todiscrete 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


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.



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.



TheoremA surface in Euclidean space is isothermic if and only if locally there exists a non-trivial

infinitesimal isometric deformation preserving the mean curvature.

Cieslinski, Goldstein, Sym (1995)

Discrete analogues of

1 infinitesimal isometric deformations and

2 mean curvature



TheoremA surface in Euclidean space is isothermic if and only if locally there exists a non-trivial

infinitesimal isometric deformation preserving the mean curvature.

Cieslinski, Goldstein, Sym (1995)

Discrete analogues of

1 infinitesimal isometric deformations and

2 mean curvature


Triangulated SurfacesGiven a triangulated surface f : M = (V , E, F)→ R3, we can measure

1 edge lengths ` : E → R,

2 dihedral angles of neighboring triangles α : E → R and

3 deform it by moving the vertices.


Infinitesimal isometric deformations

Definition

Given f : M → R3. An infinitesimal deformation f : V → R3 is isometric if ˙ ≡ 0.

If f isometric, on each face4ijk there exists Zijk ∈ R3 as angular velocity:

df(eij) = fj − fi = df(eij)× Zijk

df(ejk) = fk − fj = df(ejk)× Zijk

df(eki) = fi − fk = df(eki)× Zijk

If two triangles4ijk and4jil share a common edge eij , compatibility condition:

df(eij)× (Zijk − Zjil) = 0 ∀e ∈ E


Integrated mean curvature

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

He := αeè.

But if ˙ = ˙H = 0 =⇒ α = 0 =⇒ trivial

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

Hvi :=∑

j

Heij =∑

j

αeij ìj .

If f preserves the integrated mean curvature additionally, it implies

0 = Hvi =∑

j

αijìj =∑

j

〈df(eij), Zijk − Zjil〉 ∀vi ∈ V .


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...


The previous argument gives

CorollaryA 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

TheoremThe class of isothermic triangulated surfaces is invariant under Möbius transformations.

We can transform τ explicitly under Möbius transformations


The previous argument gives

CorollaryA 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

TheoremThe class of isothermic triangulated surfaces is invariant under Möbius transformations.

We can transform τ explicitly under Möbius transformations


Discrete conformality

Two notions of discrete conformality of a triangular mesh in R3:

1 circle patterns

2 conformal equivalence


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


Circumscribed circles Circumscribed spheres

TheoremA 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.


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

i

j

kk

Definition

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

lcrij :=`jkìl

`ki`lj

Definition

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

˙lcr ≡ 0

Definition (Conformal equivalence of triangulatedsurfaces)

Two edge length functions `, ˜ : E → R are conformally equivalent if there exists

u : V → R such that˜

ij = eui+uj

2 ìj .

Definition

Given f : M → R3, an infinitesimal deformation f : V → R3 is conformal if there exists

u : V → R3 such that˙ij =

ui + uj

2ìj .


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.


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


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:

df = df × f∗


Subdivision−−−−−−→

Theorem






df = df × f∗


Subdivision−−−−−−→

Theorem






df = df × f∗


Example 2: Homogeneous cyclinders

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

gi(p) =

cos θi sin θi 0

− sin θi cos θi 0

0 0 1

p +

0

0

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


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 .

CorollaryFor triangulated surfaces with vertices on a sphere, any infinitesimal deformation

preserving the edge lengths will preserve the integrated mean curvature.


More examples of isothermic surfaces:

(a) Inscribed Triangular meshes with boundary (b) Jessen’s Orthogonal Icosahedron


Table of Content





Discrete minimal surfacesSmooth 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


Equivalently,

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

1 f∗ defined on dual vertices

2 dual edges parallel to primal edges


Constructing discrete minimal surfacesEquivalent 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(cotβ+ cot β)(uj − ui) = 0.

4 Inverse of stereographic projection



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.



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


Table of Content





Triangular meshes on CLuo(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.


Theorem

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

deformation.


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

(cotβk + cotβk)(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)


Pick a Möbius transformation φ : C→ C

z w := φ ◦ z

u harmonic ∃ u harmonic

f conformal dφ(f) conformal

φ

dφ

u unique up to a linear function.


Thank you!


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From isothermic triangulated surfaces to discrete ...lam/slides/oberwolfach_ddg2015.pdfFrom isothermic triangulated surfaces to discrete holomorphicity Wai Yeung Lam TU Berlin Oberwolfach,