nonlinear theory. uniformization · 2017-02-14 · nonlinear theory. uniformization alexander...
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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