A new method to construct polynomial minimal surfaces

Abstract

Minimal surface is an important type of surface with zero mean curvature. It exists widely in nature. The problem of finding all minimal surfaces presented in parametric form as polynomials is discussed by many authors. However, most of the constructions are based on the theorem that a harmonic surface with isothermal parameterization is minimal. As we all know, Weierstrass representation is a classical parameterization of minimal surfaces. Therefore, in this paper, we consider to construct polynomial minimal surfaces of arbitrary degree by Weierstrass representation. Moreover, there is a correspondence between our constructed polynomial minimal surfaces and Pythagorean hodograph curves. Several numerical examples are demonstrated to illustrate our results.

On the construction of polynomial minimal surfaces with Pythagorean normals

“…By the same theorem and Definition 2 the polynomials u, v, w -which we call the preimage polynomials -define a planar PH curve p(t) = t α h(u)du + const by prescribing its hodograph as h(t) = (w(t)(u 2 (t) − v 2 (t), 2w(t)u(t)v(t)), and vice-versa, i.e., for any planar PH curve there exist three preimage polynomials u, v, w that satisfy (4) for (a, b) = p . Therefore, any planar PH curve generates one polynomial minimal surface, and it is further shown in [4] that this generating curve lies on the minimal surface. Let us call the set of all minimal surfaces obtained from planar PH curves through Enneper-Weierstrass parameterization the class 1 minimal surfaces.…”

“…In [4] a special class of parametric polynomial minimal surfaces is derived, based on the connection of (2) with Pythagorean triples and planar Pythagorean-hodograph curves. The main idea relies on the following theorem and subsequent definition.…”

“…Remark 2. The construction of minimal surfaces by Hao in [4] follows from our construction by choosing complex polynomials u, v and w, such that their coefficients are real, i.e., Im(u j ) = Im(v j ) = 0, j = 0, 1, . .…”

“…A PH curve has the distinctive property that its unit tangent has a rational dependence on the curve parameter [3], and a PN surface possesses a unit normal with a rational dependence [9,12] on the surface parameters. Hao [4] demonstrated the construction of minimal surfaces from a given planar PH curve, the plane projection of a surface isoparametric curve being coincident with the chosen PH curve. Ueda [15] noted that an isothermal parameterization of the Enneper surface (a well-known minimal surface [11]) admits a PN representation, and isothermal parameterizations are closely related [6] to mappings that preserve the PH structure.…”

“…The plan for the remainder of the paper is as follows. Section 2 reviews some basic properties of minimal surfaces, and introduces the construction of polynomial minimal surfaces from triples of complex polynomials as a generalization of the approach in [4]. This methodology is elaborated in Section 3, with a focus on surfaces with isothermal parameterizations.…”