Polynomial splines over hierarchical T-meshes

Abstract

In this paper, we introduce a new type of splines—polynomial splines over hierarchical T-meshes (called PHT-splines) to model geometric objects. PHT-splines are a generalization of B-splines over hierarchical T-meshes. We present the detailed construction process of spline basis functions over T-meshes which have the same important properties as B-splines do, such as nonnegativity, local support and partition of unity. As two fundamental operations, cross insertion and cross removal of PHT-splines are discussed. With the new splines, surface models can be constructed efficiently and adaptively to fit open or closed mesh models, where only linear systems of equations with a few unknowns are involved. With this approach, a NURBS surface can be efficiently simplified into a PHT-spline which dramatically reduces the superfluous control points of the NURBS surface. Furthermore, PHT-splines allow for several important types of geometry processing in a natural and efficient manner, such as conversion of a PHT-spline into an assembly of tensor-product spline patches, and shape simplification of PHT-splines over a coarser T-mesh. PHT-splines not only inherit many good properties of Sederberg’s T-splines such as adaptivity and locality, but also extend T-splines in several aspects except that they are only C 1 continuous. For example, PHT-splines are polynomial instead of rational; cross insertion/removal of PHT-splines is local and simple.