Minimizing the maximal ratio of weights of rational Bézier curves and surfaces

Applying the Möbius transformation to rational Bézier curves and surfaces, the weights can be modified whereas the control points remain unchanged. With appropriate transformation parameters, the maximal ratio of the weights of rational Bézier curves and surfaces can be minimized, which have applications in improving the bounds of derivatives, optimizing degree reduction of rational Bézier curves. In the surface case, there has not yet been a solution for the problem of finding transformation parameters such that the maximal ratio of the weights reaches its minimum. In this paper, a new method for the problem in the curve case is presented, and the uniqueness of the solution can be easily proved; then the method is generalized to the surface case with geometric perception. Some numerical examples are given for showing our results in improving the bounds of derivatives of rational Bézier curves and surfaces.

Research highlights
► Solution for minimizing the maximal ratio of weights of rational Bézier curves.
► Solutions for minimizing the maximal ratio of weights of rational Bézier surfaces.
► Geometric perceptions for these solutions.
► Applications of these solutions.