The Alfeld split is obtained by subdividing a simplex in Rs into s+1 subsimplices with the barycenter as one of their vertices. On this split, we consider the space of C1 splines of degree d (d >= s+1), for which we construct a basis of simplex-splines with knots at the barycenter and the vertices of the simplex. The basis consists of two types of simplex-splines: firstly Bernstein polynomials with domain points on the facets of the simplex and secondly certain simplex-splines with at least one knot at the barycenter. Partition of unity, Marsden-like identities, and domain points are shown. We also provide C1 smoothness conditions across a facet between two simplices.

A typical NURBS surface model has a large percentage of superfluous control points that significantly interfere with the design process. This paper presents an algorithm for eliminating such superfluous control points, producing a T-spline. The algorithm can remove substantially more control points than competing methods such as B-spline wavelet decomposition. The paper also presents a new T-spline local refinement algorithm and answers two fundamental open questions on T-spline theory.

We present a solution to the general problem of characterizing shadows in scenes involving a uniform polygonal area emitter and a polygonal occluder in arbitrary position by manifesting shadow irradiance as a spline function.Studying generalized prism-like constructions generated by the emitter and the occluder in a fourdimensional (shadow) space reveals a simpler intrinsic structure of the shadow as compared to the more complicated 2D projection onto a receiver.A closed form expression for the spline shadow irradiance function is derived by twice applying Stokes' theorem to reduce an evaluation over a 4D domain to an explicit formula involving only 2D faces on the receiver, derived from the scene geometry.This leads to a straightforward computational algorithm and an interactive implementation.Moreover, this approach can be extended to scenes involving multiple emitters and occluders, as well as curved emitters, occluders, and receivers.Spline functions are constructed from these prism-like objects.We call them generalized polyhedral splines because they extend the classical polyhedral splines to include curved boundaries and a density function.The approach can be applied to more general problems such as some of those occurring in radiosity, and other related topics.

Stable and efficient algorithms for degree-raising of curves (or surfaces) represented as arbitrary B-splines are presented as a application of the solution to the theoretical problem of rewriting a curve written as a linear combination of m th order B-splines as a linear combination of ( m + 1)st order B-splines with a minimal number of knot insertions. This approach can be used to introduce additional degrees of freedom to a curve (or surface), a problem which naturally arises in certain circumstances in constructing mathematical models for computer-aided geometric design.