A Computational Approach to a Quasi- Minimal Bezier Surface for Computer Graphics

Abstract

In computer science, the algorithms related to geometry can be exploited in computer aided geometric design, a field in computational geometry. Bézier surfaces are restricted class of surfaces used in computer science, computer graphics and the allied disciplines of science. In this work, a computational approach for finding the Béziersurface related minimal surfaces as the extremal of mean curvature functional is presented. A minimal surface is the surface that locally minimizes its area and has zero mean curvature everywhere. The vanishing mean curvature results in a non-linear partial differential equation for a minimal surface spanned by the boundary of interest and the solution of the partial differential equation does exist for very few special cases whereas, the vanishing condition of the gradient of the area functional is in general not possible as it involves square-root in its integrand. Instead, we find the vanishing condition of the gradient of the mean curvature for a related problem, Plateau-Bézier problem that gives the constraints on the interior control points in terms of boundary control points of the prescribed border. The emerging Bézier surface is the quasi-minimal surface as the extremal of mean curvature.