Surfaces over Arbitrary Triangulations

Recently in [7). a new multivariate B-spline scheme based on blending functions and control vertices was developed . This surface scheme allows CIo-1-continuous piecewise polynomial surfaces of degree k over arbitrary triangulations to be modelled . Actually, piecewise polynomial surfaces over a refined triangulation are produced given an arbitrary triangulation . The scheme exhibits both affine invariance and the convex hull property, and the control points can be used to manipulate the shape of the surface locally. This paper describes a test implementation of the scheme for quadratic and cubic surfaces. Issues such as evaluating points on the surface, evaluating derivatives on the surface and representing piecewise polynomial surfaces as linear combinations of B-splines will be discussed. Several examples illustrate the implementation. The work is incorporated into a surface editor which is currently being developed at the University of Waterloo.