siggraph course mesh parameterization: theory and practice barycentric mappings
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Siggraph Course
Mesh Parameterization: Theory and Practice
Siggraph Course
Mesh Parameterization: Theory and Practice
Barycentric MappingsBarycentric Mappings
Triangle Mesh ParameterizationTriangle Mesh Parameterization
• triangle mesh
– vertices
– triangles
• parameter mesh
– parameter points
– parameter triangles
• parameterization
– piecewise linear map
The Spring ModelThe Spring Model
• replace edges by springs
• fix boundary vertices
• relaxation process
• energy of spring between and :
– spring constant
– spring length
• total energy
Energy MinimizationEnergy Minimization
• interior vertices
• ’s neighbours
• overall spring energy
• partial derivative
Energy MinimizationEnergy Minimization
• minimum of spring energy
for all interior points
• is a convex combination of its neighbors
with weights
The Linear SystemThe Linear System
• separation of variables
unknown parameter points fixed
• linear system
The Linear SystemThe Linear System
• solve system twice
for and coordinates of interior parameter points
• matrix is
– sparse
– diagonally dominant
– nonsingular
as long as all
Choice of WeightsChoice of Weights
• uniform spring constants
– ,
• chordal spring constants
– ,
• no fold-overs for convex boundary
• no linear reproduction
– planar meshes are distorted
Choice of WeightsChoice of Weights
• suppose is a planar mesh
• specify weights such that
• barycentric coordinates of
• then solving
reproduces
Barycentric CoordinatesBarycentric Coordinates
• Wachspress coordinates
• discrete harmonic coordinates
• mean value coordinates normalization
• fold-overs for negative coordinates
– affine combinations ,
• numerically unstable if
• mean value coordinates guaranteed to be positive
Example – PyramidExample – Pyramid
Wachspressdiscrete harmonicmean value
The Boundary MappingThe Boundary Mapping
• chordal parameterization around convex shape
– circle
– rectangle
• projection into least squares plane
– may lead to fold-overs
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