Some Research On Curve Subdivision

With the prevalence and progressively broad application of computer technology, Subdivision method has recently become a focus of study in geometric modeling, computer aided geometric design and computer graphics in the world. Besides the aim of preserving geometric properties become popular. However, subdivision as an excellent tool was not used in fields of geometric restriction.This paper first sums up some classical and developing algorithms of subdivision. Next, the paper[28] proposed new four kinds of approach subdivision schemes with a additional parameter was introduced firstly. The new four kinds of subdivision schemes is not only to keep smooth of limit curve, but also has excellent quality of keeping shape of initial polygon. Every scheme has a free variable. These characters led us to consider some problems of design with geometric restriction. After the author studied these four kinds of subdivision schemes, found that they can preserve arc and proposed some properties of parameter and subdivision curves are proposed based on these new subdivision schemes with geometric restriction. Author did consider the subdivision scheme under the constraint of not only fixed arc length but also other arc length. The method to calculate the parameter and some examples are given. From lots of numerical tests, we get to the conclusion that the parameter in each scheme converges to a const under the constraint of arc length, and that the limit curve will self- joint with sufficiently large arc length constraint except for scheme 1. The results of this paper implies that these subdivision schemes can be used in solving some geometric constraint problems, and some further mathematical fundaments are proposed in the end of the paper.