On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean cur-vature) with isothermal parameterization from Pythagorean triples of complex polynomi-als is presented, and it is shown that they are Pythagorean normal (PN) surfaces, i.e., their unit normal vectors have a rational dependence on the surface parameters. This construc-tion generalizes a prior approach based on Pythagorean triples of real polynomials, and yields more free shape parameters for surfaces of a specified degree. Moreover, when one of the complex polynomials is just a constant, the minimal surfaces have the Pythagorean- hodograph (PH) preserving property - a planar PH curve in the parameter domain is mapped to a spatial PH curve on the surface. Cubic, quartic and quintic examples of these minimal PN surfaces are presented, including examples of solutions to the Plateau prob-lem, with boundaries generated by planar PH curve segments in the parameter domain. The construction is also generalized to the case of minimal surfaces with non-isothermal parameterizations. Finally, an application to the problem of interpolating three given points in R 3 as the corners of a triangular cubic minimal surface patch, such that the three patch sides have prescribed lengths, is addressed.(c) 2022 Elsevier Inc. All rights reserved.