Discrete conformal mappings via circle patterns

1. Discrete conformal mappings via circle patterns ACM Transactions on Graphics 2006, 25 Liliya Kharevych Boris Springborn Peter Schröder Speaker: CAI Hongjie Date: Nov 22, 2007

2. Authors • Liliya Kharevych Graduate student in Multi-Res Modeling group Computer Science, Caltech • Peter Schröder Professor of Computer Science, Applied and Computational Mathematics Director of Multi-Res Modeling group Interested in multiresolution methods

3. Outline • Related concepts • Surface parameterization • Isometric & conformal maps • Voronoi Diagram & Delaunay triangulation • Sketch of paper • Basic algorithm • Mapping to the sphere and the disk

4. S Bijective map f (u,v) Surface Parameterization • Definition A one-to-one mapping from the surface to a suitable domain,especially to a regoin of the plane.

5. Surface Parameterization • Mesh case piecewise linear

6. Surface Parameterization • Applications • Scattered data fitting • Repair of CAD models • Texture mapping

7. Texture mapping

8. Isometric & Conformal maps • A surface to surface regular map f: S→S* is • Isometric, if it preserves lengths of curves • Conformal, if it preserves intersection angles of any two curves • Regular map f: S→S*, I and I* be the first foundamental forms of S and S* respectively then

9. Stereographic Projection Circle-preserving angle-preserving

10. Voronoi Diagram & Delaunay Triangulation • Voronoi Diagram Given points P1, P2,…, Pn in a plane, voronoi region Boundaries of V(Pi) form the Voronoi diagram • Delaunay Triangulation Dual graph of Voronoi diagram with straight lines

11. Voronoi Diagram & Delaunay Triangulation dualize Voronoi region

12. Angles assignment on Edges • A Delaunay triangulation (V,E,T) of finite points in a plane, V={vi}, E={eij}, T={tijk} be the sets of vertices,edges, and triangles, for

13. Interpretation of • Intersection angle of circumcircles

14. Restrictions of Delaunay triangulation Circle patterns

15. Basic Algorithm of Discrete Conformal Mappings • Angles assignment A similar angle system is needed • Minimizing the energy The energy is defined on radiuses of triangles and minimized by sofeware Mosek • Generating the layout

16. Step 1: Angles Assignment • Given a mesh with E,V, T, for edges, Vertices, triangles and angles • Firstly find corresponding angles such that • is minimized

17. Step 1: Angles Assignment • Secondly, for The right graph is probably not a planar graph, but abstract graph with assigned angles

18. Counterexample • A graph with assigned angles may not be lay out as a planar graph Treat the inner square as a point, coherent angle property is satisfied So more restriction is needed for graph layout

19. Characterization of Flat Faces • Below is a kite formed by two triangles yellow region magnify Every triangular face is flat iff

20. Step 2: Minimizes the Energy • Energy is defined as (Bobenko, Springborn [2004]) • Then Minimizes energy S

21. Step 3: Generating Layout • When radiuses of triangles are achieved from the above energy minimized step, we can obtain go around every triangle, and lay out the graph

22. Review of the Basic Algorithm • Step 1: angles assignment on edges • A similar angle system is found • Angles assigment • Step 2: minimizes the energy S(ρ) • Energy defined on the log of radiuses of triangles • Radiuses is achieved after this step • Step 3: generating the layout

23. Mapping to the Sphere • Algorithm: input is a triangle mesh of genus 0 • Remove one vertex together with incident faces • Generate a parameterization by the basic algorithm

24. Mapping to the Sphere • Algorithm: input is a triangle mesh of genus 0 • Project it stereographically to the sphere • Adding a vertex at the north pole and fill the hole

25. Results for Mapping to Sphere

26. Mapping to the Disk • Algorithm: input is a triangle mesh that is a topological disk • Remove a boundary vertex together with incident edges (red objects) • Fix the curvature angles of the other boundary vertices to be 0 (blue vertices)

27. Restrictions of Delaunay triangulation Circle patterns

28. Mapping to the Disk • Algorithm: input is a triangle mesh that is a topological disk • Remove a boundary vertex together with incident edges (red objects) • Fix the curvature angles of the other boundary vertices to be 0 (blue vertices)

29. Mapping to the disk • Generate a parameterization by basic algorithm with boundary restrictions • Inversion transformation • Fill back the removed vertex and edges

30. Other Boundary Controls

31. More results Free boundary Disk boundary

32. Advantages • Flexible boundary control • Angle preserved • Robustness by Delaunay triangulation Different sample rates but result in almost the same shape

33. Disavantage and Remedy • Large area distortion • Cone singularities

34. Reference • L. Kharevych, B. Springborn, P. Schröder Discrete conformal mappings via circle patterns • A. Bobenko, B. Springborn Variational principles for circle patterns and Koebe’s theorem • A. Sheffer, E. de Sturler Surface paramterization for meshing by triangulation flattening • Joseph O’Rourke Computational geometry in C

35. Thanks!Q&A