A Computational Model for q -Bernstein Quasi-Minimal Bézier Surface

Abstract

A computational model is presented to find the $q$ -Bernstein quasi-minimal Bézier surfaces as the extremal of Dirichlet functional, and the Bézier surfaces are used quite frequently in the literature of computer science for computer graphics and the related disciplines. The recent work [1–5] on $q$ -Bernstein–Bézier surfaces leads the way to the new generalizations of $q$ -Bernstein polynomial Bézier surfaces for the related Plateau–Bézier problem. The $q$ -Bernstein polynomial-based Plateau–Bézier problem is the minimal area surface amongst all the $q$ -Bernstein polynomial-based Bézier surfaces, spanned by the prescribed boundary. Instead of usual area functional that depends on square root of its integrand, we choose the Dirichlet functional. Related Euler–Lagrange equation is a partial differential equation, for which solutions are known for a few special cases to obtain the corresponding minimal surface. Instead of solving the partial differential equation, we can find the optimal conditions for which the surface is the extremal of the Dirichlet functional. We workout the minimal Bézier surface based on the $q$ -Bernstein polynomials as the extremal of Dirichlet functional by determining the vanishing condition for the gradient of the Dirichlet functional for prescribed boundary. The vanishing condition is reduced to a system of algebraic constraints, which can then be solved for unknown control points in terms of known boundary control points. The resulting Bézier surface is $q$ -Bernstein–Bézier minimal surface.