discrete riemann surfaces - impa · 2017-02-14 · discrete laplace beltrami operator [b.,...

Discrete Riemann surfacesLinear Theory. Integration

Alexander Bobenko

Technische Universität Berlin

Differential Geometry School, Manaus, Brazil, July 2012

CRC 109 “Discretization in Geometry and Dynamics”

Alexander Bobenko Discrete Riemann Surfaces

Discrete Differential Geometry

I Development of discrete equivalents of notions andmethods of the classical theory. Classical theory as a limitof refinements of the discretization.

I Computational. Applications: Computer graphics, free formarchitecture

I Existence theorems of classical theory can be madeconstructive when the discretization is intelligent

I Discrete models in physics


Riemann Surfaces

I one dimensional complexmanifolds

I compact, genus g,dimmoduli space =3g − 3, (=1 for g = 1)

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

I applications inmathematics and physics


Harmonic and holomorphic on the square lattice

[Ferrand ’44, Duffin ’56]

conjugate harmonicCauchy-Riemann∂u∂x

=∂v∂y

∂u∂y

= −∂v∂x

discrete Cauchy-Riemann

ur −ul = vu− vduu−ud = vl − vr

holomorphicw = u + iv

∂w∂y

= i∂w∂x

iv

uw = w1

w4

w3

w2

w4 − w2 = i(w3 − w1)


Quad-graph

f harmonic on a graph G = (V ,E), ∆f = 0

∆f (x0) =∑

xk∼x0

ν(x0, xk )(f (xk )− f (x0)), ν : E → R+

G

G cell decomposition of C


Quad-graph


∆f (x0) =∑

xk∼x0

ν(x0, xk )(f (xk )− f (x0)), ν : E → R+

GG∗

G cell decomposition of CG∗ dual


Quad-graph


∆f (x0) =∑

xk∼x0

ν(x0, xk )(f (xk )− f (x0)), ν : E → R+

GG∗

DG cell decomposition of CG∗ dualD double - quad-graph


Harmonic and holomorphic on a graph[Mercat ’01]

f : V (D) = V (G) ∪ V (G∗)→ Cdiscrete holomorphic if it satisfiesdiscrete Cauchy-Riemann equations

f (y1)− f (y0)

f (x1)− f (x0)= iν(x0, x1) = − 1

iν(y0, y1)

ν(e) = 1/ν(e∗) discrete complexstructure x0

x1

e∗e

y0y1

I f : V (D)→ C discrete holomorphic⇒ u := f |V (G), v := f |V (G∗) discrete harmonic:∆u = ∆v = 0

∆u(x) :=∑

[xi x ]=e

ν(e)(u(xi)− u(x))


Example. Quads with orthogonal diagonals

I dRS by gluing quads with orthogonal diagonalsI z is discrete holomorphic + linear eqsI real weights

Delaunay circle patternsAlexander Bobenko Discrete Riemann Surfaces

Planar Delaunay Tessellation

I Faces are polygons with empty circumcirclesI Local Delaunay condition holds for all edges

αij + αji < π

I Triangulate non-triangular faces −→ Delaunay triangulation


“Cotan Formula”

[Pinkall & Polthier ’93]

xi

xj

αij

αji

I ∆f (xi) =12

∑xj∈V :(xi ,xj )∈E

wij (f (xi)− f (xj))

wij = cotαij + cotαji


Discrete holomorphic and discrete harmonic

I f : V (D)→ C discrete holomorphic⇒ f |V (G), f |V (G∗) discrete harmonic

I f : V (G)→ C discrete harmonic⇒ there exists unique (up to additive constant) extensionto discrete holomorphic f : V (D)→ C


Dirichlet Energy

Dirichlet energy

xi

xj

αij

αji

I E(f ) =14

∑(xi ,xj )∈E

wij ‖f (xi)− f (xj)‖2,

I ∆f (xi) =12


wij (f (xi)− f (xj)) , “Gradient” of E(f )


Boundary value problem for harmonic functions

I Minimize the Dirichlet energy (convex) E(f ) for f withboundary conditions

I at minimum 0 = ddt E(f + th)|t=0 = ∆(f )


Dirichlet Energy and Laplace-Beltrami Operator

I M, N compact Riemannian manifold (with boundary)I f : M → NI Dirichlet Energy E(f ) = 1

2

∫M ‖df‖2

I “Gradient” of E is the Laplace-Beltrami operator.

ddt

E(f + t h)

∣∣∣∣t=0

=

∫M

h∆f (h|∂M = 0)

I Harmonic function: ∆f = 0I Harmonic functions are the critical points of the Dirichlet

energy.I If M,N 2-dim, f (M) = N

E(f ) ≥ Area(N),

“=” iff f conformal map M → N.


PL Functions

I S simplicial surface in 3-spaceI Vertex set V = x1, . . . , x|V |, edge set E , face set FI f : S → Rn piecewise linearI Gradient of f constant on trianglesI PL functions on S ↔ functions V → Rn


“Cotan Formula”

[Pinkall & Polthier ’93] Dirichlet energy of PL functions

xi

xj

αij

αji

I E(f ) =14

∑(xi ,xj )∈E

wij ‖f (xi)− f (xj)‖2,

wij =

cotαij + cotαji for interior edgescotαij for boundary edges

I Very important in geometry processing: [Desbrun et al.’02], [Meyer et al. ’03], [Botsch et al. ’04]...


Negative Weights

I wij = cotαij + cotαji =sin(αij + αji)

sinαij sinαji

I wij < 0 iff αij + αji > π

αjiαij

I Maximum principle: Harmonic function attains maximumon boundary.

I Does not hold for graph Laplacian with weights wij < 0.I More important: Laplace operator is not intrinsic, i.e. does

not depend on intrinsic geometry (metric) alone.


Discrete complex structure via Delaunay triangulation

I Use intrinsic Delaunay triangulation, (determined by themetric⇒ preserved by isometric deformations)


Delaunay Tessellations of Piecewise Flat Surfaces

I PF surface (S,g)I 2-dim manifold S (with boundary)I metric g flat with cone singularitiesI boundary is piecewise geodesic

I PF surfaces←→ abstract triangulations with edge lengthsI Define Delaunay triangulation almost as before

I set of marked points contains cone pointsI disks −→ isometrically immersed empty disks

I Theorem. On every PF surface there exists a uniqueDelaunay tessellation


Edge Flipping Algorithm

flatten resultflip

I Start with triangulation of PF surfaceI Find non-Delaunay edge. Flip it. The flipped edge is a

geodesic segment, and the surface remains unchanged.I Iterate. Algorithm terminates

[Rivin ’94; Indermitte et al. ’01]Proof: Define function that decreases with each flip.

I Yields (locally) Delaunay triangulationI locally Delaunay⇐⇒ Delaunay


Nonregular Delaunay Triangulation

identified edge

unfold

valence 3 vertex valence 2 valence 1

cut here to

unfoldflip flip result

I Delaunay tessellation may not be regular (double edges,cells with identifications on the boundary)

I A vertex of valence 3 is reduced to a valence 1 vertexthrough two intrinsic flips


Dependence on the Triangulation. Rippa’s Theorem

I f : V → RI fT linear interpolation for triangulation T on VI E(fT ) goes down with every edge flip

(except when fT is linear on quad)I min

TE(fT ) is attained for Delaunay triangulation

31

r2

r3r1

θ

2

4

r4

E(f123,241)−E(f124,234) =(f0 − f0)2

sin θ(r1 + r3)(r2 + r4)

r1r2r3r4(r1r3−r2r4)


Discrete complex structure

I Define discrete complex structure on a polyhedral surfaceby extending its triangulation through the circumcirlecenters. Real weights ν : E → R

I Delaunay triangulationI IntrinsicI Delaunay⇔ positive weights ν : E → R+


Discrete Laplace Beltrami Operator

[B., Springborn ’06]I discrete Laplace Beltrami operator on a simplicial surface

by cotan-formula with respect to the intrinsic Delaunaytriangulation.

∆f (xi) =12


wij (f (xi)− f (xj))

wij =

cotαij + cotαji for interior edgescotαij for boundary edges

I Positive weights (only for the Delaunay triangulation!)→ maximum principle holds

I Depends only on intrinsic geometry.Extra edges (non-unique) of Delaunay triangulation⇐Delaunay tessellation have zero weights


Applications: Intrinsic Delaunay Triangulation

white - original edgesblack - removed edgesred - new (flipped) edges, which are geodesic within the originalsurface


Statistics

Model V flips simple lgst. κi/κeCat hd. 131 45 40 3 0.8114Bny hd. 741 380 275 6 0.6139Bty. Frk. 1042 560 364 12 0.1841Hygeia 8268 4017 2861 6 0.1053Planck 25445 6417 5584 5 0.7746Bunny 34834 2365 2283 4 0.0758Camel 40240 17074 12618 22 0.7218Horse 48485 3581 3127 7 0.6734Feline 49864 12178 10767 7 0.5746

Statistics for some representative meshes.[Fischer et al. ’07]


Texture mapping: iLB versus eLB

Original and iDT (Beatuful Freack dataset): Texture planeimage (Dirichlet boundary conditions) and resulting checkerboard mapping. [Fischer et al. ’07]


Minimal surfaces.

Minimal area


Smooth (minimal) surfaces

I smooth immersed surface f : R2 ⊃ U → R3

I Dirichlet Energy E(f ) = 12

∫U ‖df‖2

I “Gradient” of E is the Laplace-Beltrami operator ∆.I mean curvature vector H = ∆fI minimal surface (equivalent definitions):

I minimal areaI H ≡ 0I minimal Dirichlet energy E(f );

E(f ) ≥ Area(f (U)), “=” iff f conformal.I minimal surfaces satisfy the maximum principle: all points

are hyperbolic, i.e. the tangent plane intersects the surface


Area minimization of simplicial surfaces

I A simplicial surface S ⊂ R3 is area minimizing if itsembedding map f : S → R3 is harmonic ∆f = 0 withrespect to the Laplace operator of S (with the triangulationT of S).

I Maximum principle (Every vertex of a discrete minimalsurface lies in convex hull of its neighbors) does not hold.


Discrete minimal surfaces. Maximum principle

Discrete minimal surfaces violating the maximum principle[Polthier, Rossman ’02]


Discrete Minimal Surfaces

I Discrete minimal surface (wide definition): H = ∆f (x) ≡ 0I Maximum principle: Every vertex of a discrete minimal

surface lies in convex hull of its neighborsI Discrete minimal surface (narrow definition): A simplicial

surface S is minimal if H ≡ 0 and the intrinsic Delaunaytriangulation of the carrier of S coincides with thetriangulation of S.

I Such a minimal surface S is a critical point of the areafunctional


Discrete minimal surfaces satisfy the maximumprinciple

eLB versus iLB. Right image is a discrete minimal surfacedefined with the discrete Laplace-Beltrami operator.


Discrete minimal surfaces. Construction

I Take a simplicial surface S0 ⊂ R3 which respects theboundary conditions. Let T0 be its triangulation.

I Calculate the intrinsic Delaunay triangulation T1 of S0 andits weights.

I Find f : S0 → R3 harmonic with respect to theLaplace-Beltrami of (S0,T1).

I One obtains a simplicial surface S1 with the triangulationT1.

I Iterate

[Pinkall, Polthier ’93] + Delaunay step


Fixed versus intrinsic Delaunay triangulation

Computation K. Josefsson, S. Sechelmann



For a fixed triangulation (without the iDT step) the iteration maynot converge (left image). With the iDT step the iterationconverges to a smooth surface and the triangulation of theresulting discrete minimal surface is Delaunay.Computation K. Josefsson, S. Sechelmann


Riemann Surfaces

I one dimensional complexmanifolds

I compact, genus g,dimmoduli space =3g − 3, (=1 for g = 1)

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

I applications inmathematics and physics


Polyhedral surfaces as RS

I polyhedral metric→ RSza-coordinate at conicalsingularities, planez-coordinate at regularpoints

I Every RS can be induced by an abstract polyhedral metric(flat metric with conical singularities). Troyanov [’86]

I Every abstract polyhedral metric can be realized as apolyhedral surface embedded in R3. Burago-Zalgaller [’60]

I Every RS can be realized as a polyhedral surfaceembedded in R3.


Period matrix

I compact genus gI canonical homology basis

ai ,bi , i = 1, . . . ,gI dual basis of holo

differentialsωi , i = 1, . . . ,g,

∫aiωj = δij

I period matrix

Πij =

∫bj

ωi , Π = ΠT , Im Π > 0

I Torelli theorem. Π determines its RSI How to compute Π for a given RS?


Discrete complex structure from a polyhedral surface

Quads are identified with planar quads in the complex plane C

z4

z2

z1

z3

e

e

1

c(e) = −iz4 − z2

z3 − z1


Discrete complex structure from Delaunay tesselation

Orthogonal diagonals, real c


Multivalued functions with periods

I canonical homology basis a1,b1, . . . ,ag ,bgI A multivalued function with periods A1, . . . ,Ag ,

B1, . . .Bg ∈ C is a pair of functions f = (Ref : V → R,Imf : F → R) such that for any x ∈ V , y ∈ F

Ref (ak x)− Ref (x) = ReAk , Ref (bk x)− Ref (x) = ReBk

Imf (ak y)− Ref (y) = ImAk , Imf (bk x)− Ref (y) = ImBk ,

where ak x is a deck transformation of x


Discrete period matrix

multi-valued discrete holomorphic functions are called discreteAbelian integral of the first kind.

TheoremThere exist normalized discrete Abelian integrals of the first kindΩd

k : D → C with ∆aj Ωdk = δjk

Definition

The matrix (Πd )ij = ∆bj Ωdi is called the discrete period matrix


Convergence of discrete period matrix [B., Skopenkov]

TheoremConsider a polyhedral surface R of genus g. Then there existtwo constants ConstR, constR (depending on R only) such that forany triangulation T of R which vertices include all conical singu-larities of R, and the maximal circumradius of triangles satisfiesr < constR, there holds

‖Πd − Π‖ < ConstR r .


Idea of the proof

I Consider (discrete) harmonic differentials u with prescribedperiods Ai ,Bi

I Minimize the Dirichlet energy Ed (convex) for given Ai ,Bi∑[xi xj ]=e∈E

ν(e)(ui − uj)2

I min Ed is a quadratic form of Ai ,Bi , coefficients give Πd(Re Π(Im Π)−1Re Π + Im Π −Re Π(Im Π)−1

−Re Π(Im Π)−1 (Im Π)−1

)I same in the smooth case with E =

∫|∇u|2

I show that min Ed → min E


Convergence of the Abel map

TheoremLet Tn be a non-degenerate uniform sequence of Delaunay tri-angulations of R with maximal edge length approaching zero asn → ∞. Then the normalized discrete Abelian integrals of thefirst kind Ωd

k converge to their smooth counterparts Ωk uniformlyon each compact subset

I nondegenerate Delaunay - intersection angles separatedfrom zero

I uniform - number of vertices in a disc of radius of maximaledge is bounded

I defined up to constant, normalized = 0 at “same” points


Computational RS.Tori with constant mean curvature (CMC)

I first example. Wente [’86]I all tori, description as

integrable systems.Hitchin, Pinkall, Sterling[’89]

I explicit formulas in termsof RS (theta functions,Abelian integrals).Bobenko [’91] Heil [’95]


Computational RS.Tori with constant mean curvature (CMC)

I parameter - hyperellipticRS (branch points xi )

y2 =∏

i

(x − xi)(x − 1/xi)

I software. Knöppel [’10]

Heil [’95]


Wente torus

Grid 10× 10 20× 20 40× 40 80× 80‖Πd − Π‖ 5.69 · 10−3 2.00 · 10−3 5.11 · 10−4 2.41 · 10−4

Π = 0.41300 + i0.91073

B., Mercat, Schmies [’11]


Lawson surface

Π =i√3

(2 −1−1 2

)y2 = x6 − 1

#vertices ‖Πd − Π‖1162 1.68 · 10−3

2498 3.01 · 10−3

B., Mercat, Schmies [’11]


Hyperelliptic curve

Π =

(−7.70− i0.17 3.72− i2.003.72− i2.00 −6.61 + i2.70

)Πd =

(−7.70− i0.15 3.73− i2.003.73− i2.00 −6.62 + i2.70

)

Knöppel, SechelmannAlexander Bobenko Discrete Riemann Surfaces

Discrete holomorphic and discrete harmonic forarbitrary quads

I f : V (D)→ C discrete holo if,Mercat [’08]

f (z4)− f (z2)

z4 − z2=

f (z3)− f (z1)

z3 − z1

z4

z2

z1

z3

1

I Real part h = Re f is called discrete harmonicI h discrete harmonic⇔ discrete Laplace operator vanishes

∆h(z1) =∑ 1

Re c

(|c|2(h(z1)− h(z3)) + Im c(h(z2)− h(z4))

)sum is over the quads incident to z1

I c ∈ R - cotan-Laplace operatorI ∆h = 0⇒ there exists a unique up to constant holo f with

h = Re fAlexander Bobenko Discrete Riemann Surfaces

Discrete holomorphic and discrete harmonic forarbitrary quads

I f : V (D)→ C discrete holo if,Mercat [’08]

f (z4)− f (z2)

z4 − z2=

f (z3)− f (z1)

z3 − z1

z4

z2

z1

z3

e

e

1

I Real part h = Re f is called discrete harmonicI h discrete harmonic⇔ discrete Laplace operator vanishes

∆h(z1) =∑ 1

Re c

(|c|2(h(z1)− h(z3)) + Im c(h(z2)− h(z4))

)sum is over the quads incident to z1

I c ∈ R - cotan-Laplace operatorI ∆h = 0⇒ there exists a unique up to constant holo f with

h = Re fAlexander Bobenko Discrete Riemann Surfaces

References

I C. Mercat, Discrete Riemann surfaces. In: Papadopoulos,A. (ed), Handbook of TeichmüÌller Theory, vol. I, IRMALect. Math. Theor. Phys., 11, 541â575, 2007

I A. Bobenko, C. Mercat, M. Schmies. Period matrices ofpolyhedral surfaces, In: Bobenko, C. Klein (eds.),Computational approach to Riemann surfaces, Lect. NotesMath, v. 2013, (2011) arXiv:0909.1305

I A. Bobenko, M. Skopenkov. Discrete Riemann surfaces:linear discretization and its convergence (preprint 2012)

I A. Bobenko, B. Springborn. A discrete LaplaceâBeltramioperator for simplicial surfaces, Discrete Comp. Geom.(2007) 38: 740â756

I A. Bobenko, C. Mercat, Yu. Suris. Linear and nonlineartheories of discrete analytic functions. Integrable structureand isomonodromic Green’s function, J. reine und angew.Math. 583 (2005)


discrete riemann surfaces - impa · 2017-02-14 · discrete laplace beltrami operator [b.,...

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