Optimal Derivative Bounds of Rational Quadratic Bèzier Curves

For the sake of simplification and convenience, the derivative bound estimation problem was usually turned into another estimation problem of parameter λ such that||R'(t)|| ≤ λ \mathop \max \limits_i ||P_i-P_i+1||, where P_i is the i-th control point of a rational Bèzier curve R(t). This paper focuses on the estimation of the derivative bounds of a rational quadratic Bèzier curve, and provides the optimal low bound of the parameter λ. Firstly, it divides all of the cases of the three weights of R(t) into eight cases; secondly, it explicitly expresses the optimal bound of λ in the three weights for each case; finally, it leads to a general conclusion for all of the cases. Numerical examples are also given to illustrate that the bounds of the new method are better than those of prevailing methods.