The Dimensions of Bivariate Spline Spaces Over Triangulations

Abstract

Let D be a polygon and Δ be a triangulation of D such that any vertex of a triangle does not lie in the interior (open line segment) of any edge of another triangle. Denoted by ^Pk, the collection of all polynomials with real coefficients and total degree k, that is, each p Î ^Pk has the form

$$p\left( {x,y} \right) = \sum\limits_{i = o}^k {} \sum\limits_{j = 0}^{k - i} {{C_{ij}}{x^i}{y^j}} ,$$

where C_ij are real numbers.