Applications Of Rational Blossoming In Computer Aided Geometry Design

This paper mainly discusses rational Blossoming in Computer Aided Geometry Design. Specifically, we apply the fact that the binomial theorem is valid for negative integer and fractional exponents, introduce the rational Bernstein bases and fractional Bernstein bases, discuss the properties of RB curves and Poisson curves, give the rational blossom and analytic blossom. These problems help to us understand the theory of rational blossoming in a primary way.In the first chapter, we introduce the concept and properties of the rational Bernstein bases. Furthermore, the definitions of the multirational blossom for analytic functions and divided difference are introduced, and some interrelated propositions are given.In the second chapter, we introduce fractional Bernstein bases by observing that the binomial theorem is valid for fractional exponents. Bernstein bases are extended to fractional degree. Many similar properties and identities, which are valid for integral Bernstein bases, are got. Moreover, Marsden identity and its blossoming form for fractional Bernstein bases are given and the example shows that fractional Bernstein bases are more flexible than integral Bernstein bases.In the third chapter, we introduce the definitions and properties of RB curves and Poisson curves. The relation between the Poisson bases and the rational Bernstein bases is given. We compare the approximation of an analytic function F by its Taylor polynomial and its Poisson partial sum with the same number of terms and illustrate that for functions with limit zero at infinity and for bounded functions the Poisson expansion provides a better approximation to the function than the Taylor expansion.In the forth chapter, we define a notion of the blossom for rational functional. Existence and uniqueness property and dual functional property of the rational blossom are given. Specially, the blossoming form for rational B(?)zier curves of n degree is given. We introduce the definition of the analytic blossom for Poisson curves. At the same time, its existence and uniqueness property and some related propositions are discussed.