1. Parametrizing Triangulated Meshes Chalana Bezawada Kernel Group PRISM 3DK – 3DK – September 15, 2000

2. Parametrization Given a set of data points and their triangulations in 3D, each data point in the triangulation is assigned a unique pair of values (u, v) in the 2D domain by means of projection. 3DK – 3DK – September 15, 2000

3. Projection of data points Used to find parametrization of (u,v) of the domain Methods: -Plane projection -Spherical projection -Cylindrical projection 3DK – 3DK – September 15, 2000

4. Good parametrization Should preserve the geometry of the original mesh. Should not produce folds (or overlapping triangles) in the domain. Classic Methods: -Uniform parametrization – Chord length parametrization – Centripetal parametrization 3DK – 3DK – September 15, 2000

5. Projection of data points Used to find parametrization of (u,v) of the domain Methods: -Plane projection -Spherical projection -Cylindrical projection 3DK – 3DK – September 15, 2000

6. Floater’s technique ( M. S. Floater - CAGD/1997 ) Floater's method finds parametrization based on the concepts of graph theory. Considering points P 0, P 1, …., P n-1 to be the internal nodes and points P n, P n+1, …., P m-1 to be the boundary nodes of the original mesh, floater obtains the parametrization as follows: Choose parameters corresponding to the boundary nodes to be the vertices of any (m-n) sided convex polygon in an anti-clockwise sequence. Write each internal node as a convex combination of its neighboring nodes. One drawback of this approach is that the boundary points are initially mapped onto a closed convex polygon in the domain, irrespective of the geometry of the original mesh boundary. 3DK – 3DK – September 15, 2000

7. New parametrization technique ( Dr. Farin ) Based on the geometry of the original mesh, we consider every pair of neighboring triangles in the original mesh. Find the point x 1 on l 1 that is closest to l 2, and point x 2 on l 2 that is closest to l 1. The computed points x 1 and x 2 will be identical only if the two neighboring triangles are coplanar. Even in the case when l 1 and l 2 do not actually intersect, an equation can be formed by forcing the two points x 1 and x 2 to be identical. 3DK – 3DK – September 15, 2000

8. With this approach one might get overlapping triangles in the domain if the angle between the planes containing the two neighboring triangles is very small, and if either A possible solution to this problem is to get the values of α and β close to the interval [0,1]. This can be achieved by replacing the points P i1 and P i4 with the centroids of the respective triangles for calculating the values of α and β. Special case 3DK – 3DK – September 15, 2000 or

9. Original mesh

10. Simple plane projection Parameter domainResulting surface

11. Our adaptive technique Parameter domainResulting surface

12. More results

14. Without trimming of parameter domain

15. With trimming of parameter domain