Isothermic triangulated surfaces and conformal deformationslam/slides/birs.pdfIsothermic triangulated surfaces and conformal deformations Wai Yeung Lam Technische Universität Berlin

Isothermic triangulated surfaces andconformal deformations

Wai Yeung Lam

Technische Universität Berlin

Banff, 14 July 2015

SFB/TRR 109 "Discretization in Geometry and Dynamics"

Joint work with Ulrich Pinkall

Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 1 / 38

Table of content

1 Isothermic triangulated surfaces

2 Discrete conformality

3 Examples of isothermic triangulated surfaces

4 Discrete minimal surfaces (?)


Background in the smooth theory

Isothermic surfaces in Euclidean space R3:

Started before 19th century

Examples: surfaces of revolution, quadrics, constant mean curvature surfaces,

minimal surfaces

Building block in classical differential geometry:

isothermic =⇒ minimal surfaces, constant mean curvature surfaces

Related to integrable systems. (Cieslinski, Goldstein, Sym 1995)


Isothermic surfaces in the smooth theory

Smooth surfaces in R3:

1 (Smyth 2004): A surface is (strongly) isothermic if and only if there exists a selfstress T such that Tr T = 0.

2 A simply connected surface is (strongly) isothermic if and only if there exists an

infinitesimal isometric deformation preserving the mean curvature H.

3 The class of (strongly) isothermic surfaces is stable under Möbiustransformations .



Smooth surfaces in R3:

1 (Smyth 2004): A surface is (strongly) isothermic if and only if there exists a selfstress T such that Tr T = 0.



3 The class of (strongly) isothermic surfaces is stable under Möbiustransformations.


Isothermic triangulated surfaces

Definition (L.,Pinkall)

A triangulated surface (with boundary) f : M → R3 is isothermic if there exists

k : Eint → R such that for every interior vertex i∑j

kij(fj − fi) = 0,∑j

kij(|fj |2 − |fi |2) = 0.

Lemma{∑j kij(fj − fi) = 0,∑j kij(|fj |2 − |fi |2) = 0

⇐⇒

{∑j kij(fj − fi) = 0,∑j kij |fj − fi |2 = 0 (Tr T = 0)

Note: Tr T =∑

i〈T(ei), ei〉 and T(e) ∼ kij(fj − fi).



Some properties of isothermic surfaces in the smooth theory:

1 (Smyth 2004): A surface is (strongly) isothermic if and only if there exists a

non-trivial self stress T such that Tr T = 0.





Infinitesimal isometric deformations

Definition

Given f : M → R3. An infinitesimal deformation f : V → R3 is rigid if for e ∈ E

〈fj − fi , fj − fi〉 = 0

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

fj − fi = (fj − fi)× Zijk

fk − fj = (fk − fj)× Zijk

fi − fk = (fi − fk )× Zijk

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

(fj − fi)× (Zijk − Zjil) = 0 ∀e ∈ E


Mean curvature

edge lengths: `dihedral angles: α

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

Hi :=∑

j

Heij =∑

j

αeij ìj .

If f preserves the integrated mean curvature additionally, it implies

0 = Hi =∑

j

αijìj =∑

j

〈fj − fi , Zijk − Zjil〉 ∀vi ∈ V .


Mean curvature

edge lengths: `dihedral angles: α

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

Hi :=∑

j

Heij =∑

j

αeij ìj .

If f preserves the integrated mean curvature additionally, it implies

0 = Hi =∑

j

αijìj =∑

j

〈fj − fi , Zijk − Zjil〉 ∀vi ∈ V .


An infinitesimal rigid deformation f preserving H is represented by Z : F → R3

satisfying

(fj − fi)× (Zijk − Zjil) = 0 ∀eij ∈ E∑j

〈fj − fi , Zijk − Zjil〉 = 0 ∀i ∈ V

Write Zijk − Zjil = kij(fj − fi) for some k : Eint → R. Then for every interior vertex i{∑j kij(fj − fi) = 0∑j kij |fj − fi |2 = 0

Theorem (L.,Pinkall)

Given a simply connected triangulated surface in R3. There exists a non-trivial

infinitesimal rigid deformation preserving the integrated mean curvature H if and only if it

is isothermic .



Some properties of isothermic surfaces in the smooth theory:

1 (Smyth 2004): A surface is (strongly) isothermic if and only if there exists a

non-trivial self stress T such that Tr T = 0.






The class of isothermic triangulated surfaces is stable under Möbius transformation.

Möbius transformations = Euclidean transformations + Inversions

= angle-preserving diffeomorphism of R3 ∪ {∞}

Inversion in the unit sphere at the origin:

f 7→ f = − f

||f ||2

points→ points

spheres→ spheres






Inversion in the unit sphere at the origin:

f 7→ f = − f

||f ||2

points→ points

spheres→ spheres
















Möbius−−−→






Möbius−−−→




Proof.

L := {x ∈ R5 | x21 + x2

2 + x23 + x2

4 − x25 = 0}

Given f : M → R3. Consider f : M → L ⊂ R5 defined by

fi := (fi ,1− |fi |2

2,

1 + |fi |2

2) ∈ L ⊂ R5.

Then {∑j kij(fj − fi) = 0,∑j kij(|fj |2 − |fi |2) = 0

⇐⇒∑

j

kij (fj − fi) = 0

Klein’s Erlangen program:

Möbius geometry of R3 ∪ {∞} = subgeometry of the projective geometry of RP4.

Möbius transformations∼= projective transformations preserving L.

Fact: self stress→ self stress under projective transformations.


Table of content





Conformality in the smooth theory

Recall:

Given two metrics 〈, 〉 and 〈, 〉 on a manifold M. They are conformally equivalent if

there exists u : M → R such that

〈, 〉 = eu〈, 〉

Or equivalently they are angle-preserving.

Surface deformations in R3:

{rigid} ⊂ {conformal (angle-preserving)} ⊂ {arbitrary}


Conformality in the smooth theory

Recall:

Given two metrics 〈, 〉 and 〈, 〉 on a manifold M. They are conformally equivalent if

there exists u : M → R such that

〈, 〉 = eu〈, 〉

Or equivalently they are angle-preserving.

Surface deformations in R3:

{rigid} ⊂ {conformal (angle-preserving)} ⊂ {arbitrary}


Conformally equivalence

Definition (Luo,2004)

Given a triangulated surface M. Two length functions `, ˜ : E → R are conformally

equivalent if there exists u : V → R such that for e ∈ E

˜ij = e

ui+uj2 ìj .

Definition

Given two realizations f , f : M → R3. Then f , f are conformally equivalent if their

induced edge lengths are conformally equivalent.

same edge lengths =⇒ u ≡ 0 =⇒ conformally equivalent


Inversion in the unit sphere:

f 7→ f = − f

||f ||2

Then,

||fj − fi ||2 =1

||fi ||2||fj ||2||fj − fi ||2

u := −2 ln ||f || =⇒ f , f conformally equivalent

Theorem

Given f : M → R3. Then for any Möbius transformation Φ

f is conformally equivalent to Φ ◦ f .


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

Definition

Given a length function ` : E → R on a (oriented) triangulated surface. Its length cross

ratio lcrs : Eint → R is defined by

lcrsij :=`jkìl

`ki`lj

Remark: Length cross ratio = norm of cross ratio

i

j

kl

TheoremTwo length functions are conformally equivalent if and only if their length cross ratios are

identical.


Infinitesimal conformal deformations

Definition

Given f : M → R3. An infinitesimal deformation f : V → R3 is conformal if there exists

u : V → R satisfying for e ∈ E

〈fj − fi , fj − fi〉 =ui + uj

2|fj − fi |2.

Remark: Infinitesimal Möbius transformations are always conformal.


Denote TfM = {infinitesimal conformal deformations of f}.


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.


Denote TfM = {infinitesimal conformal deformations of f}.


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.


Table of content





Example 1: Inscribed Triangulated Surfaces

Recall:

Definition

f : M → R3 isothermic if ∃k : Eint → R such that for every interior vertex i∑j

kij(fj − fi) = 0, (1)∑j

kij(|fj |2 − |fi |2) = 0. (2)

If f(V) ⊂ S2 =⇒ |f | ≡ 1 =⇒ (2) is trivial =⇒ self stress k

Smooth theory: Given f : M → S2. All self stresses T of f satisfy Tr T = 0.


Example 1: Inscribed Triangulated Surfaces

Recall:

Definition

f : M → R3 isothermic if ∃k : Eint → R such that for every interior vertex i∑j

kij(fj − fi) = 0, (1)∑j

kij(|fj |2 − |fi |2) = 0. (2)

If f(V) ⊂ S2 =⇒ |f | ≡ 1 =⇒ (2) is trivial =⇒ self stress k

Smooth theory: Given f : M → S2. All self stresses T of f satisfy Tr T = 0.


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


Example 2: Planar triangular meshes

Look for infinitesimal rigid deformations in R3 preserving H.

Infinitesimal rigid deformation =⇒ u : V → RPreserving H =⇒ u is harmonic, i.e. for every interior vertex i∑

j

(cotβkij + cotβ l

ij)(uj − ui) = 0.

Discrete complex analysis, Ising model in statistical mechanics (Smirnov, 2010)





j

(cotβkij + cotβ l

ij)(uj − ui) = 0.






j

(cotβkij + cotβ l

ij)(uj − ui) = 0.



Example 3: Isothermic Quadrilateral Meshes

Definition (Bobenko and Pinkall, 1996)

A discrete isothermic net is a map f : Z2 → R3, for which all quadrilaterals have

cross-ratios

q(fm,n, fm+1,n, fm+1,n+1, fm,n+1) = −1 ∀m, n ∈ Z,

Subdivision−−−−−−→

Remark:

1 a cornerstone of discrete differential geometry

2 involve discrete integrable systems


Example 3: Isothermic Quadrilateral Meshes

Definition (Bobenko and Pinkall, 1996)

A discrete isothermic net is a map f : Z2 → R3, for which all quadrilaterals have

cross-ratios

q(fm,n, fm+1,n, fm+1,n+1, fm,n+1) = −1 ∀m, n ∈ Z,

Subdivision−−−−−−→

Remark:

1 a cornerstone of discrete differential geometry

2 involve discrete integrable systems


Comparison

M simply connected (smooth) surface in R3.

Infinitesimal flexibleSelf stress T

Infinitesimal rigid deformation

Projective invariant

Singularity of isometric immersions

IsothermicSelf stress T with Tr T = 0

Infinitesimal rigid deformation with H = 0

Möbius invariant

Singularity of conformal immersions

W. Y. Lam and U. Pinkall. Isothermic triangulated surfaces. arXiv, Jan 2015.

-circle patterns


Comparison








Möbius invariant



-circle patterns


Comparison








Möbius invariant


Integrable structures (?)


-circle patterns


Discrete minimal surfaces


discrete minimal surfaces

n : V → S2 ⊂ R3 a reciprocal parallel mesh f : F → R3


Smooth theory:

Given an immersion n : M → S2 and an infinitesimal rigid deformation n, then

∃f : M → R3 such that

dn = dn × f

f is called the rotation field.

Theorem: If f is an immersion, f is a minimal surface with Gauss map n.


Given n : V → S2.

infinitesimal rigid deformation n

=⇒ ∃f : F → R3 infinitesimal rotation of each face with compatibility

=⇒ f is a reciprocal parallel mesh of n

=⇒ n is a reciprocal parallel mesh of f

=⇒ ∃ infinitesimal rigid deformation f : F → R3

Q: What does f : F → R3 look like? (as a realization of the dual graph)


Given n : V → S2.








Given n : V → S2.








Given n : V → S2.








What does f : F → R3 look like?

f : F → R3



f : F → R3

planar vertex stars, reciprocal parallel mesh ofn : V → S2

f : F → R3

planar faces with face normal n, ˙Areai is zero underface offsets: Hi :=

∑ij ìj tan

αij

2= 0

1-1 correspondence→ a conjugate pair of minimal surfaces



f : F → R3

planar vertex stars, reciprocal parallel mesh ofn : V → S2

f : F → R3

planar faces with face normal n, ˙Areai is zero underface offsets: Hi :=

∑ij ìj tan

αij

2= 0

1-1 correspondence→ a conjugate pair of minimal surfaces


Thank you!

W. Y. Lam and U. Pinkall. Holomorphic vector fields and quadratic differentials on planartriangular meshes. arXiv, Jun 2015.

Weierstrass representation

discrete harmonic functions→ conjugate pairs of discrete minimal surfaces


Documents

Isothermic triangulated surfaces and conformal deformationslam/slides/birs.pdfIsothermic triangulated surfaces and conformal deformations Wai Yeung Lam Technische Universität Berlin