1. Mesh Parameterization: Theory and Practice Barycentric Mappings

2. Mesh Parameterization: Theory and Practice Barycentric Mappings Triangle Mesh Parameterization triangle mesh – vertices – triangles parameter mesh – parameter points – parameter triangles parameterization – piecewise linear map

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

4. Mesh Parameterization: Theory and Practice Barycentric Mappings Energy Minimization interior vertices ’s neighbours overall spring energy partial derivative

5. Mesh Parameterization: Theory and Practice Barycentric Mappings Energy Minimization minimum of spring energy for all interior points is a convex combination of its neighbors with weights

6. Mesh Parameterization: Theory and Practice Barycentric Mappings The Linear System separation of variables unknown parameter points fixed linear system

7. Mesh Parameterization: Theory and Practice Barycentric Mappings The Linear System solve system twice for and coordinates of interior parameter points matrix is – sparse – diagonally dominant – nonsingular as long as all

8. Mesh Parameterization: Theory and Practice Barycentric Mappings Choice of Weights uniform spring constants –, chordal spring constants –, no fold-overs for convex boundary no linear reproduction – planar meshes are distorted

9. Mesh Parameterization: Theory and Practice Barycentric Mappings Choice of Weights suppose is a planar mesh specify weights such that barycentric coordinates of then solving reproduces

10. Mesh Parameterization: Theory and Practice Barycentric Mappings Barycentric Coordinates Wachspress coordinates discrete harmonic coordinates mean value coordinates normalization

11. Mesh Parameterization: Theory and Practice Barycentric Mappings fold-overs for negative coordinates – affine combinations, numerically unstable if mean value coordinates guaranteed to be positive Example – Pyramid Wachspressdiscrete harmonicmean value

12. Mesh Parameterization: Theory and Practice Barycentric Mappings The Boundary Mapping chordal parameterization around convex shape – circle – rectangle projection into least squares plane – may lead to fold-overs