Multi-degree B-spline Curves

B-spline model is an important modeling tool for CAGD, however it also has certainflaws. In the actual product design, curve is sometimes composed of polynomial segmentswith various degrees. Using B-spline model to describe this kind of curves, the correspondingcontrol polygon must meet certain requirements, this make the design complex and bigstorage.In this paper we propose multi-degree B-spline by computing the basis functions formulti-degree B-spline. Multi-degree B-splines are B-spline-like curves that are comprised ofpolynomial segments of various degrees. Multi-degree B-splines are a generalization ofB-spline curves in that if all curve segments in an multi-degree B-spline have the same degree, then it degenerates to a B-spline curve. The curves are at least C^n-1, which n is the smaller oftwo adjoining curve segments. Multi-degree B-spline curves can easily get cuspid or linearlike appearance, when there is a line segment in multi-degree B-spline, we can just use twocontrol points to make it, so multi-degree B-spline model can save some storage.The research background is introduced in the chapter one.The construction of the basis functions for multi-degree B-spline and correspondingproof are given in the chapter two. The basis functions for multi-degree B-spline have somegood properties like positivity, weighting, segmented polynomial properties and so on.The construction of multi-degree B-spline curves and corresponding proof are given inthe chapter three. Multi-degree B-spline have local support, obey the convex hull andvariation diminishing properties. Some applications of multi-degree B-spline curves isdiscussed in this chapter.The equivalent representation of multi-degree B-spline as "expanded-form" controlpolygon, which is a conventional B-spline is given in the chapter four. This chapter give thegeneral expressions for the transformation, and also proves that transformation is a series ofcomer cutting.In the chapter five we draw the conclusion and talk about some prospects.