Discrete Differential Geometry. Discrete Laplace-Beltrami operator and discrete conformal mappings.

Transcription

1 Discrete differential geometry. Discrete Laplace-Beltrami operator and discrete conformal mappings. Technische Universität Berlin Geometric Methods in Classical and Quantum Lattice Systems, Caputh, September 2014 CRC 109 Discretization in Geometry and Dynamics

2 Aim: Development of discrete equivalents of the geometric notions and methods of differential geometry. The latter appears then as a limit of refinements of the discretization. Question: Which discretization is the best one? Main messages: Discretize the whole theory, not just the equations. The discrete geometric theory is as rich as the analogous theory for the smooth problem. Existence theorems of classical theory can be made constructive when the discretization is proper Important for applications: computer graphics, architecture Same models in physics

3 This talk. Papers A.I. Bobenko, B. Springborn, A discrete Laplace-Beltrami operator for simplicial surfaces, Discrete and Computational Geometry 38:4 (2007) A.I. Bobenko, U. Pinkall, B. Springborn, Discrete conformal maps and ideal hyperbolic polyhedra, Geometry and Topology (to appear), arxiv: [math.gt]

4 Dirichlet Energy M, N compact Riemannian manifold (with boundary) f : M N Dirichlet Energy E(f ) = 1 2 M df 2 Gradient of E is the Laplace-Beltrami operator. d E(f + t h) dt = h f (h M = 0) t=0 Harmonic function: f = 0 M

5 PL Functions S simplicial surface in 3-space Vertex set V = {x 1,..., x V }, edge set E, face set F f : S R n piecewise linear Gradient of f constant on triangles {PL functions on S} {functions V R n }

6 Cotan Formula [Pinkall & Polthier 93]: Finite element approach α ij x i α ji x j E(f ) = 1 w ij f (x i ) f (x j ) 2, where 4 (x i,x j ) E { cot αij + cot α w ij = ji for interior edges cot α ij for boundary edges f (x i ) = 1 w ij (f (x i ) f (x j )) 2 x j V :(x i,x j ) E

7 Negative Weights w ij = cot α ij + cot α ji = sin(α ij + α ji ) sin α ij sin α ji w ij < 0 iff α ij + α ji > π α ij α ji Maximum principle: Harmonic function attains maximum on boundary. Does not hold for graph Laplacian with weights w ij < 0. More important: Laplace operator is not intrinsic, i.e. does not depend on intrinsic geometry (metric) alone.

8 Main Idea Use intrinsic Delaunay triangulation, (determined by the metric preserved by isometric deformations) flatten flip result

9 Planar Delaunay Tessellation Local Delaunay condition holds for all edges Delaunay locally Delaunay α ij + α ji < π

10 Planar Delaunay Tessellation Local Delaunay condition holds for all edges Delaunay locally Delaunay α ij + α ji < π

11 Planar Delaunay Tessellation Local Delaunay condition holds for all edges Delaunay locally Delaunay α ij + α ji < π

12 Piecewise Flat Surfaces PF surface (S, g) 2-dim manifold S (with boundary) metric g flat with cone singularities boundary is piecewise geodesic PF surfaces abstract triangulations with edge lengths Define Delaunay triangulation almost as before set of marked points contains cone points disks isometrically immersed disks

13 Discrete Laplace Beltrami Operator [B., Springborn 06] Define discrete Laplace Beltrami operator on a simplicial surface by cotan-formula with respect to the intrinsic Delaunay triangulation. f (x i ) = 1 2 x j V :(x i,x j ) E w ij (f (x i ) f (x j )) { cot αij + cot α w ij = ji for interior edges cot α ij for boundary edges Positive weights (only for the Delaunay triangulation!) maximum principle holds Depends only on intrinsic geometry

14 Applications: Intrinsic Delaunay Triangulation white - original edges black - removed edges red - new (flipped) edges, which are geodesic within the original surface

15 Texture mapping: ilb versus elb Original and idt (Beatuful Freack dataset): Texture plane image (Dirichlet boundary conditions) and resulting checker board mapping. [Fischer, Springborn, Schröder, B. 07]

16 Conformal maps conformal means angle preserving infinitesimal lengths scaled by conformal factor df = e u dx independent of direction in the small like similarity transformations Problem: surface in space conformally plane

17 Analytic description Definition Two Riemannian metrics g, g on a smooth manifold M are called conformally equivalent, if for some function u : M R Gaussian curvatures g = e 2u g e 2u K = K + g u mapping problem Given surface (M, g), find conformally equivalent flat metric g Poisson problem g u = K

18 Discrete conformal equivalence abstract surface triangulation M = (V, E, T ) Definition A discrete metric on M is a function l : E R >0, ij l ij satifying all triangle inequalities: ijk T : l ij < l jk + l ki l jk < l ki + l ij l ki < l ij + l jk

19 Discrete conformal equivalence Definition Two discrete metrics l, l on M are (discretely) conformally equivalent if l ij = e 1 2 (u i +u j ) l ij for some function u : V R use λ ij = 2 log l ij so l ij = e λ ij /2 and λij = λ ij + u i + u j

20 Variational principle S(u) def = ijk T Milnor s Lobachevsky function L(α) = ( α k ij λ ij + α i jk λ jk + α j ki λ ki π/2( λ ij + λ jk + λ ki ) α 0 +2 L( α k ij ) + 2 L( αi jk ) + 2 L( αj ki ) ) + i V log 2 sin t dt Θ i u i l ij = e 1 2 (λ ij +u i +u j ) solves mapping problem u = (u 1,..., u n ) is critical of S(u) S convex

21 AB conformal drawing

22 Intrinsic geometry of surfaces Intrinsic symmetry - Thiesen

23 Books in DDG