On the construction of polynomial minimal surfaces with Pythagorean normals

A novel approach to constructing polynomial minimal surfaces (surfaces of zero mean curvature) with isothermal parameterization from Pythagorean triples complex polynomials is presented, and it shown that they are normal (PN) surfaces, i.e., their unit vectors have a rational dependence on the surface parameters. This construction generalizes prior based real polynomials, yields more free shape parameters for specified degree. Moreover, when one just constant, Pythagorean–hodograph (PH) preserving property — planar PH curve in parameter domain mapped spatial surface. Cubic, quartic quintic examples these PN including solutions Plateau problem, boundaries generated by segments domain. The also generalized case non–isothermal parameterizations. Finally, an application problem interpolating three given points R3 as corners triangular cubic patch, such patch sides prescribed lengths, addressed.