Numerical Calculation of a Minimal Surface Using Bilinear Interpolations and Chebyshev Polynomials

Abstract

We calculate the minimal surface bounded by four-sided figures whose projection on a plane is a rectangle, starting with the bilinear interpolation and using, for smoothness, the Chebyshev polynomial expansion in our discretized numerical algorithm to get closer to satisfying the zero mean curvature condition. We report values for both the bilinear and improved areas, suggesting a quantitative evaluation of the bilinear interpolation. An analytical expression of the Schwarz minimal surface with polygonal boundaries and its 3-dimensional plot is also given.