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 (?)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 2 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 3 / 38
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 .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 4 / 38
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 5 / 38
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).
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 6 / 38
Isothermic surfaces in the smooth theory
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.
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 7 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 8 / 38
Mean curvature
edge lengths: `dihedral angles: α
A known discrete analogue of mean curvature H : E → R is defined by
He := αe`e.
But if ˙ = ˙H = 0 =⇒ α = 0 =⇒ trivial
Instead, we consider the integrated mean curvature around vertices H : V → R
Hi :=∑
j
Heij =∑
j
αeij `ij .
If f preserves the integrated mean curvature additionally, it implies
0 = Hi =∑
j
αij`ij =∑
j
〈fj − fi , Zijk − Zjil〉 ∀vi ∈ V .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 9 / 38
Mean curvature
edge lengths: `dihedral angles: α
A known discrete analogue of mean curvature H : E → R is defined by
He := αe`e.
But if ˙ = ˙H = 0 =⇒ α = 0 =⇒ trivial
Instead, we consider the integrated mean curvature around vertices H : V → R
Hi :=∑
j
Heij =∑
j
αeij `ij .
If f preserves the integrated mean curvature additionally, it implies
0 = Hi =∑
j
αij`ij =∑
j
〈fj − fi , Zijk − Zjil〉 ∀vi ∈ V .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 9 / 38
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 .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 10 / 38
Isothermic surfaces in the smooth theory
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.
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 11 / 38
Theorem (L.,Pinkall)
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 12 / 38
Theorem (L.,Pinkall)
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 12 / 38
Theorem (L.,Pinkall)
The class of isothermic triangulated surfaces is stable under Möbius transformation.
Möbius transformations = Euclidean transformations + Inversions
= angle-preserving diffeomorphism of R3 ∪ {∞}
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 13 / 38
Theorem (L.,Pinkall)
The class of isothermic triangulated surfaces is stable under Möbius transformation.
Möbius transformations = Euclidean transformations + Inversions
= angle-preserving diffeomorphism of R3 ∪ {∞}
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 14 / 38
Theorem (L.,Pinkall)
The class of isothermic triangulated surfaces is stable under Möbius transformation.
Möbius transformations = Euclidean transformations + Inversions
= angle-preserving diffeomorphism of R3 ∪ {∞}
Möbius−−−→
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 15 / 38
Theorem (L.,Pinkall)
The class of isothermic triangulated surfaces is stable under Möbius transformation.
Möbius transformations = Euclidean transformations + Inversions
= angle-preserving diffeomorphism of R3 ∪ {∞}
Möbius−−−→
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 16 / 38
Theorem (L.,Pinkall)
The class of isothermic triangulated surfaces is stable under Möbius transformation.
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 17 / 38
Table of content
1 Isothermic triangulated surfaces
2 Discrete conformality
3 Examples of isothermic triangulated surfaces
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 18 / 38
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}
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 19 / 38
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}
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 19 / 38
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 `ij .
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 20 / 38
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 .
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 21 / 38
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 :=`jk`il
`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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 22 / 38
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 23 / 38
Denote TfM = {infinitesimal conformal deformations of f}.
Theorem (L.,Pinkall)
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 24 / 38
Denote TfM = {infinitesimal conformal deformations of f}.
Theorem (L.,Pinkall)
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 24 / 38
Table of content
1 Isothermic triangulated surfaces
2 Discrete conformality
3 Examples of isothermic triangulated surfaces
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 25 / 38
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 26 / 38
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 26 / 38
(a) Inscribed Triangular meshes with boundary (b) Jessen’s Orthogonal Icosahedron
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 27 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 28 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 28 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 28 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 29 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 30 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 31 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 31 / 38
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
Integrable structures (?)
W. Y. Lam and U. Pinkall. Isothermic triangulated surfaces. arXiv, Jan 2015.
-circle patterns
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 31 / 38
Discrete minimal surfaces
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 32 / 38
discrete minimal surfaces
n : V → S2 ⊂ R3 a reciprocal parallel mesh f : F → R3
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 33 / 38
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.
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 34 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 35 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 35 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 35 / 38
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)
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 35 / 38
What does f : F → R3 look like?
f : F → R3
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 36 / 38
What does f : F → R3 look like?
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 `ij tan
αij
2= 0
1-1 correspondence→ a conjugate pair of minimal surfaces
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 37 / 38
What does f : F → R3 look like?
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 `ij tan
αij
2= 0
1-1 correspondence→ a conjugate pair of minimal surfaces
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 37 / 38
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
Wai Yeung Lam (TU Berlin) isothermic triangulated surfaces 2 March 2015 38 / 38