Non-Rational and Rational Transfinite Interpolation Using Bernstein Polynomials

Abstract

It is shown that the standard transfinite interpolation in quadrilateral patches may be written in terms of two sets of Bernstein polynomials. The former set is of the same degree with the corresponding blending functions while the latter is of the same degree with the Lagrange polynomials which operate like trial functions along each of the four edges as well as the additional inter-boundaries of the patch. The replacement of the Lagrange polynomials by the Bernstein ones allows the use of control points useful for design. Also it allows the determination of weights that may ensure accurate representation of quadric surfaces including those of revolution (spheres, ellipsoids, hyperboloids, etc). The presentation restricts to the standard Gordon formulation, which refers to structured stencils, similar to rectangular plates reinforced with longitudinal and transverse stiffeners. As an example, the proposed formula is applied to the geometric representation of a cylindrical and a spherical patch. In the latter case a nonlinear programming optimization technique was applied, starting with initial data pertinent to a tensor product surface, from which original weights and control points were obtained for the accurate representation of a spherical cap. In addition, the involved shape functions were successfully applied to the numerical solution of a boundary value problem.