Preliminaries

In this chapter, we consider some results and techniques of multivariate spline theory. Since we are mainly interested in trivariate splines in this work, and as an aid to some extent also bivariate splines, we state most of the results and definitions shown in this chapter combined in the multivariate setting. However, to ease the understanding, we also show some of theses results for the trivariate and bivariate case. In section 2.1, we define tessellations of a domain in ℝ

n

. Furthermore, we show some refinement schemes of triangles and tetrahedral partitions and introduce some notation and the Euler relations for tetrahedra. In section 2.2, we examine multivariate polynomials, which form a basis for the subsequently considered multivariate splines and supersplines. In the next section, we describe the Bernstein-Bézier techniques for multivariate splines. These are based on the barycentric coordinates, which are needed to define Bernstein polynomials and the resulting B-form of polynomials that is used throughout this dissertation.