Discrete Laplace Operators for Polygonal Meshes Δ

2. Discrete Laplace Operatorsfor Polygonal MeshesΔ Marc Alexa Max Wardetzky TU Berlin U Göttingen

3. Laplace Operators • Continuous • Symmetric, PSD, linearly precise, maximum principle • Discrete (weak form) • Cotandiscretization [Pinkall/Polthier,Desbrun et al.] • Linearly precise, PSD, symmetric, NO maximum principle • No discrete Laplace = smooth Laplace [Wardetzky et al.]

4. Geometry Processing • Smoothing / fairing [Desbrun et al. ’99]

5. Geometry Processing • Smoothing / fairing • Parameterization [Gu/Yau ’03]

6. Geometry Processing • Smoothing / fairing • Parameterization • Mesh editing [Sorkine et al. ’04]

7. Geometry Processing • Smoothing / fairing • Parameterization • Mesh editing • Simulation [Bergou et al. ’06]

8. Polygon meshes

9. Polygon meshes

10. Polygon • Polygons are not planar • Not clear what surface the boundary spans • Integration of basis function unclear / slow

11. Laplace on Polygon Meshes

12. Laplace on Polygon Meshes • Triangulating the polygons?

13. Laplace on Polygon Meshes • Goal: ‘cotan-like’ operator for polygons • Symmetric (weak form) • Linearly precise • Positive semidefinite (positive energies) • Reduces to cotan on all-triangle mesh

14. Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangle • cotan

15. Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangle • cotan

16. Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Triangles • Same plane

17. Laplace as Area Gradient • Laplace flow = area gradient [Desbrun et al.] • Flat polygon

18. Non-planar polygons

19. Non-planar polygons • Vector area x2 x1 x0 0

20. Non-planar polygons • Properties of vector area • Projecting in direction yields largest planar polygon • Area is independent of choice of origin or orientation

21. Non-planar polygons • Vector area gradient • Is in the plane of maximalprojection • As before, orthogonal to • Simply use cross product with a

22. Non-planar polygons e1 e0 b0 0

23. Non-planar polygons

24. Non-planar polygons • Differences along oriented edges • “Co-boundary” operator

25. Non-planar polygons

26. Non-planar polygons

27. Properties of • is symmetric by construction as • Consequently, L is symmetric

28. Properties of • L is linearly precise

29. Properties of • Is L PSD with only constants in kernel? • Co-boundary d behaves right • Kernel ofmay be too large • spans kernel of

30. Main result • Laplace operator for any mesh • Symmetric, Linearly precise, PSD • Reduces to standard ‘cotan’ for triangles

31. Implementation • Very simple! • For each face, compute • and (differences, sums of coordinates) • , , (matrix products) • from (SVD)

32. Implementation • Write M into large sparse matrix M1 • M1 has dimension halfedges×halfedges • Build the d-matrices • Have dimension halfedges× vertices • Then L = dT M1d(weak form) • Strong form requires normalization by M0

33. Smoothing

34. Parameterization

35. Parameterization

36. Parameterization

37. Planarization • Planarization

38. Planarization

39. Conclusions / Future work • Laplace operator all meshes • Symmetric, PSD,linear precision • Reduces to cotan • Make non-planar part geometric