Piecewise polynomials on polyhedral complexes
For researchers in approximation theory and geometric modeling, this provides the first complete formula for the dimension of spline spaces on 2D polyhedral complexes.
The paper derives the first three coefficients of the polynomial giving the dimension of spline spaces on polyhedral complexes in 2D, solving a problem for which no formula was previously known.
For a d-dimensional polyhedral complex P, the dimension of the space of piecewise polynomial functions (splines) on P of smoothness r and degree k is given, for k sufficiently large, by a polynomial f(P,r,k) of degree d. When d=2 and P is simplicial, Alfeld and Schumaker determined a formula for all three coefficients of f. However, in the polyhedral case, no formula is known. Using localization techniques and specialized dual graphs associated to codimension--2 linear spaces, we obtain the first three coefficients of f(P,r,k), giving a complete answer when d=2.