G¹ boundary construction

Abstract

This chapter is concerned with the construction of a G ¹-continuous object boundary. For 2D, a straightforward construction of a closed piecewise cubic Bézier curve passing through given vertices with prescribed tangent vectors is presented. For 3D, an analysis of the total degree required to solve several interpolation problems using polynomial patches is given. The attention is then focused on the construction of a closed piecewise triangular cubic Bézier surface, that interpolates given vertices with prescribed normal vectors. In order to get sufficient degrees of freedom to define the control points, a triangle three-split, a two-split and a six-split scheme are developed. The split into six sub-triangles results in a surface that is G ¹-continuous as well as visually pleasing.