Fitting Polynomial Surfaces to Triangular Meshes with Voronoi ...

10 Vincent Nivoliers, Dong-Ming Yan, and Bruno Lévy
Fig. 6: Computing the gradient of the symmetric term ∇ ˜F S→T : configurations of
the vertices of Vor(Y)| S .
where dA/dB = (∂a i /∂b j ) i,j denotes the Jacobian matrix of A.
The set of possible configurations for a vertex c k is similar to the combinatorial
structure of the L p -CVT function [11], with the exception that the roles of
the variables and constants are exchanged. Each configuration yields a Jacobian
matrix that propagates the derivatives of ˜F T S→T from the c k’s to the x i ’s.
There are 3 possible configurations (see overview in Figure 6) :
→ c is a vertex x i of S (then dc/dx i = I 3×3 ) ;
→ c has configuration (a) :
c corresponds to the intersection between the bisector of [y 1 , y 2 ] (left,
plane shown in blue) and an edge [x 1 , x 2 ] of S (right). The Jacobian matrices
dc/dx 1 and dc/dx 2 are given in Appendix B, Equation 13;
→ c has configuration (b) :
c corresponds to the intersection between the three bisectors of [y 1 , y 2 ],
[y 2 , y 3 ], [y 3 , y 1 ] (left) and a facet (x 1 , x 2 , x 3 ) of S (right). The Jacobian matrices
dc/dx 1 , dc/dx 2 and dc/dx 3 are given in Appendix B, Equation 14.