Discrete Riemann surfacesNonlinear Theory. Uniformization

Alexander Bobenko

Technische Universität Berlin

Differential Geometry School, Manaus, Brazil, July 2012

CRC 109 “Discretization in Geometry and Dynamics”

Alexander Bobenko Discrete Riemann Surfaces

Riemann Surfaces

I one dimensional complexmanifolds

I compact, genus g,dim{moduli space} =3g − 3, (=1 for g = 1)

I many different realizations:algebraic curves,equivalence classes ofconformal metrics onsurfaces,...

I applications inmathematics and physics

Alexander Bobenko Discrete Riemann Surfaces

Uniformization

TheoremAny compact Riemann surface of genus g possesses a confor-mal metric with constant curvature

I g = 0, K = 1I g = 1, K = 0I g > 1, K = −1

I How to construct? Discretize

Alexander Bobenko Discrete Riemann Surfaces

Conformal maps

I conformal means anglepreserving

I infinitesimal lengths scaled byconformal factor

|df | = eu |dx |

independent of direction

I in the small like similaritytransformations

I Problem:surface in space

conformally−−−−−−−→ plane

Alexander Bobenko Discrete Riemann Surfaces

Smooth theoryDefinitionTwo Riemannian metrics g, g̃ on a smooth manifold M are calledconformally equivalent, if

g̃ = e2u g

for some function u : M → R

I Gaussian curvatures

e2u K̃ = K + ∆gu

I mapping problem⇔

Given surface (M,g), find conformally equivalent flat metric g̃

I Poisson problem ∆gu = −K

Alexander Bobenko Discrete Riemann Surfaces

Discrete conformal texture mapping

I Springborn, Pinkall, Schröder. Conformal equivalence oftriangle meshes. ACM Transactions on Graphics 27:3(2008)

Alexander Bobenko Discrete Riemann Surfaces

Discrete

I abstract surface triangulationM = (V ,E ,T )

DefinitionA discrete metric on M is a function

` : E → R>0, ij 7→ `ij

satifying all triangle inequalities:

∀ ijk ∈ T : `ij < `jk + `ki

`jk < `ki + `ij

`ki < `ij + `jk

Alexander Bobenko Discrete Riemann Surfaces

Discrete

Definition

Two discrete metrics `, ˜̀on M are(discretely) conformally equivalent if

˜̀ij = e12 (ui+uj )`ij

for some function u : V → R

I use λij = 2 log `ij

so `ij = eλij/2

and λ̃ij = λij + ui + uj

Alexander Bobenko Discrete Riemann Surfaces

two single triangles

I two single triangles alwaysconformally equivalent

λ̃12 = λ12 + u1 + u2

λ̃23 = λ23 + u2 + u3

λ̃31 = λ31 + u3 + u1

eλ23/2eλ31/2

eλ12/2

eλ̃23/2

eλ̃31/2

u2

u3

u1

eλ̃12/2

Alexander Bobenko Discrete Riemann Surfaces

two single triangles

I two single triangles alwaysconformally equivalent

λ̃12 = λ12 + u1 + u2 +

λ̃23 = λ23 + u2 + u3 +

λ̃31 = λ31 + u3 + u1 −

eλ23/2eλ31/2

eλ12/2

eλ̃23/2

eλ̃31/2

u2

u3

u1

eλ̃12/2

Alexander Bobenko Discrete Riemann Surfaces

Length cross ratio

DefinitionFor interior edges ij definelength cross ratio

lcrij =`ih `jk`hj `ki

`, ˜̀discretely conformally equivalentm

l̃crij = lcrij

Alexander Bobenko Discrete Riemann Surfaces

Teichmüller space

I ∀ interior vertices i :∏ij3i

lcrij = 1

I discrete conformal structure onM:equivalence class of discretemetrics

I M closed, compact, genus g:

dim{conformal structures}= |E | − |V | = 6g − 6 + 2|V |= dim Tg,|V |

Tg,n: Teichmüller space for genus g with n puncturesAlexander Bobenko Discrete Riemann Surfaces

Möbius invariance

I immersion V → Rn, i 7→ viinduces discrete metric`ij = ‖vi − vj‖

I Möbius transformation:composition of inversions onspheres

I the only conformaltransformationsin Rn if n ≥ 3

Möbius equivalent immersions induceconformally equivalent discrete met-rics

Alexander Bobenko Discrete Riemann Surfaces

Angles and curvatures

I lengths determine angles

αijk = 2 tan−1

√(−`ij+`jk+`ki )(`ij+`jk−`ki )(`ij−`jk+`ki )(`ij+`jk+`ki )

I angles sum around vertex i

Θi =∑ijk3i

αijk

I curvature at interior vertex i

Ki = 2π −Θi

I boundary curvature at boundaryvertex

κi = π −Θi

jiαi

jk

k

`ij

`ki`jk

Alexander Bobenko Discrete Riemann Surfaces

Mapping problem

Discrete mapping problem

Given mesh M, metric `ij = e12λij , and

desired angle sums Θ̂i

Find conformally equivalent metric ˜̀ijwith

Θ̃i = Θ̂i

I Θ̂i = 2π for interior vertices(except for cone-likesingulatrities)

I non-linear equations for ui

Alexander Bobenko Discrete Riemann Surfaces

Variational principle

I S(u)def=∑ijk∈T

(α̃k

ij λ̃ij + α̃ijk λ̃jk + α̃j

ki λ̃ki −π

2(λ̃ij + λ̃jk + λ̃ki)

+2L(α̃kij ) + 2L(α̃i

jk ) + 2L(α̃jki))

+∑i∈V

Θ̂iui

I Milnor’s Lobachevsky function

L(α) = −∫ α

0log |2 sin t |dt

I∂S∂ui

= Θ̂i − Θ̃i

˜̀ij = e12 (λij+ui+uj ) solves mapping problem

mu = (u1, . . . ,un) is critical point of S(u)

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

I xi = log ai =⇒(xi − log(2 sinαi)

)= log

ai

2 sinαi

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

I xi = log ai =⇒(xi − log(2 sinαi)

)= log

ai

2 sinαi

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

I xi = log ai =⇒(xi − log(2 sinαi)

)= log

ai

2 sinαi

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

I xi = log ai =⇒(xi − log(2 sinαi)

)= log

ai

2 sinαi

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

I xi = log ai =⇒(xi − log(2 sinαi)

)= log

ai

2 sinαi

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2 a1 = ex1

a3 = ex3R

I xi = log ai =⇒(xi−log(2 sinαi)

)= log

ai

2 sinαi= log R

I∂f∂x1

= α1

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2 a1 = ex1

a3 = ex3R

I xi = log ai =⇒(xi−log(2 sinαi)

)= log

ai

2 sinαi= log R

I∂f∂x1

= α1 + log R · ∂∂x1

(α1 + α2 + α3)

Alexander Bobenko Discrete Riemann Surfaces

How does it work?

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

I L′(α) = − log |2 sinα|

I∂f∂x1

= α1 +(x1 − log(2 sinα1)

)∂α1

∂x1

+(x2 − log(2 sinα2)

)∂α2

∂x1

+(x3 − log(2 sinα3)

)∂α3

∂x1

1

3

2

α3

α1 α2

a2 = ex2 a1 = ex1

a3 = ex3R

I xi = log ai =⇒(xi−log(2 sinαi)

)= log

ai

2 sinαi= log R

I∂f∂x1

= α1 + log R ·���

������

�:0∂∂x1

(α1 + α2 + α3)

Alexander Bobenko Discrete Riemann Surfaces

Convexity

I S(u) =∑ijk∈T

(2f (

λ̃ij2 ,

λ̃jk2 ,

λ̃ki2 )−π/2(λ̃ij + λ̃jk + λ̃ki)

)+∑i∈V

Θ̂iui

I f (x1, x2, x3) = α1 x1 + α2 x3 + α3 x3+L(α1) + L(α2) + L(α3)

1

3

2

α3

α1 α2

a2 = ex2

a3 = ex3

a1 = ex1

∑ ∂2S∂ui∂ui

=12

∑wij(dui − duj)

2, wij =12

(cotαk

ij + cotαlij

)

Alexander Bobenko Discrete Riemann Surfaces

boundary conditions

I Neumannfix angle sums at boundary

I Dirichletfix ui at boundary

I u = 0→ isometry onboundary

I diskmixed boundary conditions,exploit Möbius invariance

Alexander Bobenko Discrete Riemann Surfaces

boundary conditions

I Neumannfix angle sums at boundary

I Dirichletfix ui at boundary

I u = 0→ isometry onboundary

I diskmixed boundary conditions,exploit Möbius invariance

Alexander Bobenko Discrete Riemann Surfaces

Discrete version of conformal maps

Alexander Bobenko Discrete Riemann Surfaces

Discrete version of conformal maps

Alexander Bobenko Discrete Riemann Surfaces

Discrete version of conformal maps

Alexander Bobenko Discrete Riemann Surfaces

Uniformization of the Wente torus

Period Π

explicit 0.4130 + i 0.9107nonlinear 0.4134 + i 0.9106

linear 0.4133 + i 0.9106

Knöppel, Sechelmann

Alexander Bobenko Discrete Riemann Surfaces

Circular domains

−→

I Logarithmic edge lengths λ̃ij for additional (red) edges arefree variables

I Functional S(u, λ̃)

Alexander Bobenko Discrete Riemann Surfaces

induced hyperbolic metric

I log lcrij = shear coordinates

I λij = Penner coordinates k

l

λij

λik

λkj

λjl

λli

Alexander Bobenko Discrete Riemann Surfaces

3D building block: ideal tetrahedra

˜̀ij = e12 (λij+ui+u)

Alexander Bobenko Discrete Riemann Surfaces

decorated ideal triangles

I p3 = 12(−λ12 + λ23 + λ31) ⇒ c3 = e

12 (λ12−λ23−λ31)

Alexander Bobenko Discrete Riemann Surfaces

polyhedal realization of hyperbolic cusp metrics

ProblemGiven an ideal traingulation of a punc-tured sphere with hyperbolic metricwith cusps,find an isometric ideal polyhedron.

I equivalent to a discreteconformal mapping problem

Alexander Bobenko Discrete Riemann Surfaces

(Origin of) variational principles

Schläfli, Milnor

dV = −12

∑λijdαij

W = 2V +∑

αijλij

dW =∑

αijdλij

I S =∑

W (λij) + linear terms

I convexity of V ⇒ convexity of S

Alexander Bobenko Discrete Riemann Surfaces

Meshes in hyperbolic space

Hyperboloid model of the hyperbolic plane

H2 ={

x ∈ R2,1 ∣∣ ‖x‖h = x21 + x2

2 − x23 = −1, x3 > 0

},

dx

y`

Lorentz d and hyperbolic distances ` between two pointsx , y ∈ H2 are related by

d = ‖x − y‖h = 2 sinh `(x ,y)2

Alexander Bobenko Discrete Riemann Surfaces

Hyperbolic triangulations

DefinitionTwo hyperbolic triangulations are discretely conformally equiva-lent if the edge lengths `, ˜̀are related by

sinh( ˜̀ij

2

)= e

12 (ui+uj ) sinh

(`ij2

)for some function ui : V → R.

DefinitionA euclidean triangulation (T , `) and a hyperbolic triangulation(T , ˜̀)h are discretely conformally equivalent if the edge lengths`, ˜̀are related by

sinh˜̀ij

2= e

12 (ui+uj )`ij

for some function ui : V → R.

Alexander Bobenko Discrete Riemann Surfaces

Origin of discrete conformal hyperbolic triangulations

a different building block

˜̀ij = e12 (λij+ui+uj )

sinh( ˜̀ij

2

)= e

12 (λij+ui+uj )

Alexander Bobenko Discrete Riemann Surfaces

Variational principle

dV = −12

∑λdα

W =∑

λα + 2V

dW =∑

αijdλij −∑

αidui

I S(u) =∑

W (λ,u) + linear terms

I convexity of V (Leibon)⇒ convexity of S∑ ∂2S∂ui∂ui

=∑

wij

((dui − duj)

2 + tanh2(˜̀ij2

)(dui + duj)2

)

wij =12

(cotαij + cotα′ij)

Alexander Bobenko Discrete Riemann Surfaces

Discrete hyperbolic uniformization

−→

Alexander Bobenko Discrete Riemann Surfaces

Discrete hyperbolic uniformization

lengths⇒discrete conformally equivalent hyperbolic mesh⇒fundamental domain of the Fuchsian uniformization group G

Alexander Bobenko Discrete Riemann Surfaces

Canonical fundamental domain, group generators

canonical fundamental domain of the Fuchsian uniformizationgroup G

Alexander Bobenko Discrete Riemann Surfaces

Convergence

ProblemGonvergence of the uniformization group for polyhedral surfaces

Alexander Bobenko Discrete Riemann Surfaces

Hyperelliptic Fuchsian uniformization group

I Uniformization of a hyperelliptic RS of genus g > 2I Fuchsian group G is a subgroup of index 2 of a group G̃

generated by the involutions h1, . . . ,hg , π

I Fundamental polygon with identified opposite edges. Axesof generators πhi of G intersect

I Hyperellipticity criteria

Alexander Bobenko Discrete Riemann Surfaces

Schottky uniformization

I Schottky group GS

I Fundamental domainI Generators σi ∈ GS i = 1, . . . ,g

σiz − Bi

σiz − Ai= µi

z − Bi

z − Ai

I Riemann surface of genus g

A1 B1

B2A2

σ1

σ1

Alexander Bobenko Discrete Riemann Surfaces

Lengths from cross-ratios

B2

A2

A1 B1

Problem:Identified edges havedifferent lengths

Given cross-ratios lcr satisfying∏

ij3i lcrij = 1there exist up to a common multiple factor unique lengths lij with

lcrij =lkj llilik ljl

Alexander Bobenko Discrete Riemann Surfaces

From Schottky to Fuchsian uniformization

Alexander Bobenko Discrete Riemann Surfaces

References

I A. Bobenko, U. Pinkall, B. Springborn, Discrete conformalmaps and ideal hyperbolic polyhedra, arXiv:1005.2698(2010)

I B. Springborn, U. Pinkall, P. Schröder, Conformalequivalence of triangle meshes. ACM Transactions onGraphics 27:3, (2008)

I A. Bobenko, S. Sechelmann, B. Springborn, Discreteuniformization of Riemann surfaces, (Preprint)

Alexander Bobenko Discrete Riemann Surfaces