Constrained approximation of rational Bzier curves based on a matrix expression of its end points continuity condition
For high order interpolations at both end points of two rational Bzier curves, we introduce the concept of C(v,u) 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 Bzier curves, approximating rational Bzier curves by Bzier curves and the degree reduction of rational Bzier curves respectively; all are in L2 L 2 norm and C(v,u) 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 Bzier 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.
constrained approximation rational bzier curves c(v u) c ( v u) c ( v u) c ( v u) c ( v u ) degree reduction numerical optimization u ) -continuity c(v u ) c(v u ) c ( v