A New Kind of Trivariate C1 Splines

We propose a construction of trivariate C 1 spline functions over a special tetrahedron partition and compare our construction with other known constructions of C 1 spline functions which are summarized in this paper. We improve the Alfeld construction of C 1 quintic spline spaces. For those polygonal domains in R 3 which admit the special tetrahedron partition, our construction provides a more eecient and eeective tool for applications. For other polygonal domains, our construction may combine with the improved Alfeld construction to be more eecient than using the Alfeld construction alone. x1. Introduction Let be a collection of polyhedra in R 3. Suppose that is a polyhedron partition (polyhedrization) of a polygonal domain in R 3. That is, = t2 t and the intersection of t and t 0 is either empty, or their common facet, or their common edge, or their common vertex, where t and t 0 stand for two diierent polyhedra. Let P d be the space of polynomials of total degree d. Note that the dimension of P d is ? d+3 3. Let S r d (() = fs 2 C r ((); sj t 2 P d ; t 2 g be the spline space of smoothness r and degree d. To be more precise what we the numbers of vertices, edges, faces, polyhedra of. For a v 2 V , we use s 2 C r 1 (v) to denote that s is C r 1 smooth at v. Similarly, we use s 2 C r 2 (e) to denote that s is C r 2 around an edge e. Thus, we are able to deene a super spline subspace of S r d (() by