一类具有任意次数的参数多项式极小曲面

极小曲面是在几何造型设计中有着重要应用的一类特殊曲面.本文从几何造型的视角提出一类次数任意的参数多项式极小曲面.所提出的极小曲面具有显式的参数表示,并具有一些重要的几何性质,如对称性、包含直线和自交性.根据几何性质,本文将该参数多项式极小曲面划分为4类:n=4k+1,n=4k+2,n=4k+3,n=4+4,其中n是极小曲面的次数,k是正整数.本文给出与之相对应的共轭极小曲面的显式参数形式,并实现其等距变形.

A class of parametric polynomial minimal surfaces with arbitrary degree

Minimal surfaces is a kind of special surface in differential geometry with important applications in geometric design. In this paper, we propose a new class of parametric polynomial minimal surfaces with arbitrary degree from the viewpoint of geometric modeling. The proposed minimal surface has explicit parametric form, and some interesting geometric properties such as symmetry, containing straight lines and self-intersections. From the geometric properties, the proposed minimal surface can be classified into four categories with respect to n = 4k＋ 1, n = 4k ＋ 2, n = 4k ＋ 3 and n = 4k ＋ 4, where n is the degree of minimal surface and k is a positive integral number. The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented.