1. Siggraph Course Mesh Parameterization: Theory and Practice Barycentric Mappings

2. Triangle Mesh Parameterization triangle mesh – vertices – triangles parameter mesh – parameter points – parameter triangles parameterization – piecewise linear map

3. The Spring Model replace edges by springs fix boundary vertices relaxation process energy of spring between and : – spring constant – spring length total energy

4. Energy Minimization interior vertices ’s neighbours overall spring energy partial derivative

5. Energy Minimization minimum of spring energy for all interior points is a convex combination of its neighbors with weights

6. The Linear System separation of variables unknown parameter points fixed linear system

7. The Linear System solve system twice for and coordinates of interior parameter points matrix is – sparse – diagonally dominant – nonsingular as long as all

8. Choice of Weights uniform spring constants –, chordal spring constants –, no fold-overs for convex boundary no linear reproduction – planar meshes are distorted

9. Choice of Weights suppose is a planar mesh specify weights such that barycentric coordinates of then solving reproduces

10. Barycentric Coordinates Wachspress coordinates discrete harmonic coordinates mean value coordinates normalization

11. fold-overs for negative coordinates – affine combinations, numerically unstable if mean value coordinates guaranteed to be positive Example – Pyramid Wachspressdiscrete harmonicmean value

12. The Boundary Mapping chordal parameterization around convex shape – circle – rectangle projection into least squares plane – may lead to fold-overs