Spline Functions on Triangulations

Abstract

Spline functions are universally recognized as highly effective tools in approximation theory, computer-aided geometric design, image analysis, and numerical analysis. The theory of univariate splines is well known but this text is the first comprehensive treatment of the analogous bivariate theory. A detailed mathematical treatment of polynomial splines on triangulations is outlined, providing a basis for developing practical methods for using splines in numerous application areas. The detailed treatment of the Bernstein-Bézier representation of polynomials will provide a valuable source for researchers and students in CAGD. Chapters on smooth macro-element spaces will allow engineers and scientists using the FEM method to solve partial differential equations numerically with new tools. Workers in the geosciences will find new tools for approximation and data fitting on the sphere. Ideal as a graduate text in approximation theory, and as a source book for courses in computer-aided geometric design or in finite-element methods.

Interpolation and scattered data fitting on manifolds using projected Powell-Sabin splines

“…Moreover, c.f. the formulae in Theorem 6.11 of [21], the c ε ijk are also analytic functions of the coordinates of the vertices of △ ε P S . Since the incenter of a triangle is an analytic function of its vertices, the vertices of △ ε P S are analytic functions of V ε , and we conclude that s(V, z, u) is an analytic function of V in this case.…”

“…It follows from well-known error bounds for bivariate Powell-Sabin interpolation [30] (see also [21]) that…”

“…Let △ be a regular triangulation of a polygonal domain G in R 2 , and let {v i } n i=1 be its set of vertices, where v i := (x i , y i ). We write △ P S for the well-known Powell-Sabin refinement of △, where the split point v T in each triangle T is taken to be its incenter see Definition 4.18 of [21]. Thus, △ P S is obtained from △ by connecting the incenter of each triangle T to the incenters of all triangles in △ that share an edge with T , and to the vertices of T .…”

“…be the quadratic polynomial that coincides with the spline s ε := s(V ε , z, ·) on t ε , where B ε ijk are the quadratic Bernstein basis polynomials associated with t ε , see [21]. Since B ε ijk = 2 i!j!k!…”

“…Similarly, to obtain C 2 or higher smoothness C r with our scheme, a local bivariate C r interpolant determined by the function and gradient values at vertices is needed. Recall that all known macro-elements of higher smoothness [21] require higher order derivative values.…”