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.

中文翻译：

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构造多项式最小曲面的新方法

最小曲面是具有零平均曲率的重要曲面类型。它在自然界中广泛存在。许多作者讨论了寻找以参数形式表示为多项式的所有最小曲面的问题。但是，大多数构造基于等温参数化极小的谐波曲面的定理。众所周知，Weierstrass表示是最小曲面的经典参数化。因此，在本文中，我们考虑通过Weierstrass表示构造任意次数的多项式极小曲面。而且，我们构造的多项式最小曲面与勾股勾线图曲线之间存在对应关系。几个数值例子说明了我们的结果。