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G 2 Tensor Product Splines over Extraordinary Vertices
Author(s) -
Loop Charles,
Schaefer Scott
Publication year - 2008
Publication title -
computer graphics forum
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.578
H-Index - 120
eISSN - 1467-8659
pISSN - 0167-7055
DOI - 10.1111/j.1467-8659.2008.01277.x
Subject(s) - mathematics , tensor product , spline (mechanical) , bicubic interpolation , minification , product (mathematics) , quadratic equation , ring (chemistry) , pure mathematics , combinatorics , geometry , mathematical analysis , polynomial , mathematical optimization , physics , linear interpolation , thermodynamics , chemistry , organic chemistry
We present a second order smooth filling of an n‐valent Catmull‐Clark spline ring with n biseptic patches. While an underdetermined biseptic solution to this problem has appeared previously, we make several advances in this paper. Most notably, we cast the problem as a constrained minimization and introduce a novel quadratic energy functional whose absolute minimum of zero is achieved for bicubic polynomials. This means that for the regular 4‐valent case, we reproduce the bicubic B‐splines. In other cases, the resulting surfaces are aesthetically well behaved. We extend our constrained minimization framework to handle the case of input mesh with boundary.