Minimal surfaces associated with orthogonal polynomials

TL;DR

This paper explores the relationship between minimal surfaces in Euclidean and hyperbolic spaces and classical orthogonal polynomials, providing numerical representations for various polynomial families.

Contribution

It establishes a connection between soliton surface theory and orthogonal polynomials, offering a new approach to visualize these surfaces using computational methods.

Findings

Numerical representations for Bessel, Legendre, Laguerre, Chebyshev, and Jacobi surfaces.

Application of the Enneper-Weierstrass formula to orthogonal polynomials.

Visualization of minimal surfaces via Mathematica for multiple polynomial families.

Abstract

This paper is devoted to a study of the connection between the immersion functions of two-dimensional surfaces in Euclidean or hyperbolic spaces and classical orthogonal polynomials. After a brief description of the soliton surfaces approach defined by the Enneper-Weierstrass formula for immersion and the solutions of the Gauss-Weingarten equations for moving frames, we derive the three-dimensional numerical representation for these polynomials. We illustrate the theoretical results for several examples, including the Bessel, Legendre, Laguerre, Chebyshev and Jacobi functions. In each case, we generate a numerical representation of the surface using the Mathematica symbolic software.