1. Discrete Geometry Tutorial 2 1
© Maks Ovsjanikov, Alex & Michael Bronstein
tosca.cs.technion.ac.il/book
Numerical geometry of non-rigid shapes
Stanford University, Winter 2009 1

2. Neighborhood Neighborhood of point in
Discrete equivalent of metric ball

3. Nearest neighbors
K nearest neighbors of

4. Connectivity Neighborhood is a topological definition
independent of a metric
Two points are adjacent (directly connected)
if they belong to the same neighborhood
The connectivity structure can be
represented as an undirected graph
with vertices and
edges
Vertices
Edges
Connectivity graph can be represented as a matrix

5. Shape representation
Graph
Cloud of points

6. Connectivity in the plane
Four-neighbor
Six-neighbor
Eight-neighbor

7. Delaunay tessellation
Define connectivity as follows: a pair of points whose Voronoi cells are adjacent are connected
The obtained connectivity graph is dual to the Voronoi diagram and is called Delaunay tesselation
Boris Delaunay
( )
Voronoi regions
Connectivity
Delaunay tesselation

8. Delaunay tessellation
For surfaces, the triangles are replaced by geodesic triangles
[Leibon & Letscher]: under conditions that guarantee the existence of Voronoi tessellation, Delaunay triangles form a valid tessellation
Replacing geodesic triangles by planar ones gives Delaunay triangulation
Geodesic triangles
Euclidean triangles

9. Shape representation
Graph
Cloud of points
Triangular mesh

10. Triangular meshes A structure of the form consisting of Vertices Edges
Faces
is called a triangular mesh
The mesh is a purely topological object and does not contain any geometric properties
The faces can be represented as an matrix of indices, where each row is a vector of the form , and

11. Triangular meshes
The geometric realization of the mesh is defined by specifying the coordinates of the vertices for all
The coordinates can be represented as an matrix
The mesh is a piece-wise planar approximation obtained by gluing the triangular faces together,
Triangular face

12. Example of a triangular mesh
Vertices
Coordinates
Edges
Faces
Topological
Geometric

13. Barycentric coordinates
Any point on the mesh can be represented providing
index of the triangle enclosing it;
coefficients of the convex combination of the triangle vertices
Vector is called barycentric coordinates

14. Manifold meshes is a manifold Neighborhood of each interior
vertex is homeomorphic to a disc
Neighborhood of each boundary
vertex is homeomorphic to a half-disc
Each interior edge belongs to two triangles
Each boundary edge belongs to one triangle

15. Non-manifold meshes
Edge shared by
four triangles
Non-manifold mesh

16. Geometry images Surface Geometry image Global parametrization
Sampling of parametrization domain on a Cartesian grid

17. Geometry images Six-neighbor Eight-neighbor Manifold mesh
Non-manifold mesh

18. Topologically valid Geometrically invalid
Geometric validity
Topologically valid Geometrically invalid
Topological validity (manifold mesh) is insufficient!
Geometric validity means that the realization of the triangular mesh does not contain self-intersections

19. Skeleton
For a smooth compact surface , there exists an envelope (open set in
containing ) such that every point is continuously mappable to a unique point on
The mapping is realized as the closest point on from
Problem when is equidistant from two points on
(such points are called medial axis or skeleton of )
If the mesh is contained in the envelope (does not intersect the medial axis), it is valid

20. Points equidistant from the boundary form the skeleton of a shape

21. Local feature size
Distance from point on to the medial axis of is called the local feature size, denoted
Local feature size related to curvature
(not an intrinsic property!)
[Amenta&Bern, Leibon&Letscher]: if the surface is sampled such that for every an open ball of radius contains a point of , it is guaranteed that does not intersect the medial axis of
Conclusion: there exists sufficiently dense sampling guaranteeing that
is geometrically valid

22. Geometric validity Insufficient density Sufficient density
Invalid mesh
Sufficient density
Valid mesh

23. Approximation quality
How well does the mesh approximate the underlying surface ?
Sampling quality
Topological equivalence (manifold meshes)
Geometric properties such as area, normals, etc. computed on
are close to those computed on

24. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)
Integral
Differential

25. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

26. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

27. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

28. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

29. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

30. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)

31. Discretization of geometric quantities
Given a the mesh evaluate quantities:
Areas, Lengths
Normals
Curvatures (principal, Gauss, mean, etc.)
A whole envelope of possibilities. Which one is the right one?

32. Discretization of geometric quantities
Hundreds of methods for estimating differential information.
Case specific.
Try to preserve essential properties of these quantities.
Express them in terms of integral ones. Averaging.
Local Regions
smooth surface
mesh

33. Discretization of geometric quantities
Estimating Normals.
Well defined on the triangles
On edges and vertices can be defined by averaging:
Scale each normal by the area of corresponding triangle
Integral:
mesh

34. Discretization of geometric quantities
Estimating Normals.
Well defined on the triangles
On edges and vertices can be defined by averaging:
Important: scale each normal by the area of corresponding triangle
Integral:
mesh 34

35. Discretization of geometric quantities
Estimating Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case. 35

36. Discretization of geometric quantities
Estimating Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case. 36

37. Discretization of geometric quantities
Estimating Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case.
Can be extended to 3D as well. [Taubin ’95] 37

38. Discretization of geometric quantities
Estimating Gaussian Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case. 38

39. Discretization of geometric quantities
Estimating Gaussian Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case.
Task: Compute area of spherical polygon. 39

40. Discretization of geometric quantities
Estimating Gaussian Curvature.
Equals to 0 on triangles/edges.
Cannot average.
Limits are often dropped in the discrete case.
Task: Compute area of spherical polygon. 40

41. Approximation quality
How well does the mesh approximate the underlying surface ?
Sampling quality
Topological equivalence (manifold meshes)
Geometric properties such as area, normals, etc. computed on
are close to those computed on 41

42. Schwarz lantern
M=10, N=10
M=20, N=20
M=20, N=10

43. Schwarz lantern Set . Then, in the limit Set . Then, in the limit
Reason: triangles become infinitely thin
[Morvan & Thibert 2002]: if the triangulation is sufficiently fat, the area approximation error is bounded
Hermann Schwarz
( )

44. Conclusion Sampling Voronoi tessellation Connectivity Farthest point
Delaunay
tessellation
Triangular meshes
Geometric validity
Topological validity
Sufficiently dense
sampling
Manifold meshes
Schwarz lantern