Surface Parameterization Dr Scott Schaefer 1 Parameterization and

Surface Parameterization Dr. Scott Schaefer 1

Parameterization and Texturing 2/32

Mercator Projection 3/32

Mercator Projection Image taken from http: //idvux. spaces. live. com 4/32

Types of Distortion Isometric Mappings u Preserve lengths n Conformal Mappings u Preserve angles n Equiareal Mappings u Preserve area n 5/32

Mathematics of Parameterization 6/32

Mathematics of Parameterization 7/32

Mathematics of Parameterization singular values 8/32

Mathematics of Parameterization conformal 9/32

Mathematics of Parameterization equiareal 10/32

Mathematics of Parameterization isometric 11/32

Conformal and Harmonic Mappings 12/32

Conformal and Harmonic Mappings 13/32

Conformal and Harmonic Mappings Conformal => Harmonic 14/32

Harmonic Maps 15/32

Harmonic Maps ith row contains cotan weights for ith vertex 16/32

Mean Value Map 17/32

Mean Value Map Always positive! 18/32

Mean Value Map ith row contains mean value weights for ith vertex 19/32

Comparison Original Barycenter Harmonic Mean value 20/32

Conformal Maps 21/32

Conformal Maps 22/32

Conformal Maps 23/32

Conformal Maps 24/32

Conformal Maps 25/32

Conformal Maps 26/32

Conformal Maps 27/32

Conformal Maps 28/32

Conformal Maps Sum over all triangles Linear system of equations! 29/32

Conformal Maps Part of discrete Laplacian 30/32

Iterative Optimization St itc hg lob a lly Input 3 D mesh Image taken from A Local/Global Approach to Mesh Parameterization Output 2 D parameterization 31/32

Conclusions Many, many methods exist for parameterization n Not all guarantee 1 -to-1 mappings (foldovers) n Most measure some degree of conformality n Need a balance between distortions n 32/32