Constrained approximation of rational Bezier curves based on a matrix expression of its end points continuity condition
For high order interpolations at both end points of two rational Bezier curves, we introduce the concept of C((v,u))-continuity and give a matrix expression of a necessary and sufficient condition for satisfying it. Then we propose three new algorithms, in a unified approach, for the degree reduction of Bezier curves, approximating rational Bezier curves by Bezier curves and the degree reduction of rational Bezier curves respectively; all are in L(2) norm and C((v,u))-continuity is satisfied. The algorithms for the first and second problems can get the best approximation results, and for the third one, resorting to the steepest descent method in numerical optimization obtains a series of degree reduced curves iteratively with decreasing approximation errors. Compared to some well-known algorithms for the degree reduction of rational Bezier curves, such as the uniformizing weights algorithm, canceling the best linear common divisor algorithm and shifted Chebyshev polynomials algorithm, the new one presented here can give a better approximation error, do multiple degrees of reduction at a time and preserve high order interpolations at both end points. (C) 2010 Elsevier Ltd. All rights reserved.
constrained approximation rational bezier curves c((v,u))-continuity degree reduction numerical optimization