# The classification of bi-quintic parametric polynomial minimal surfaces

• Cai-yun Li
• Chun-gang Zhu
Article

## Abstract

Parametric polynomial surface is a fundamental element in CAD systems. Since the most of the classic minimal surfaces are represented by non-parametric polynomial, it is interesting to study the minimal surfaces represented in parametric polynomial form. Recently, Ganchev presented the canonical principal parameters for minimal surfaces. The normal curvature of a minimal surface expressed in these parameters determines completely the surface up to a position in the space. Based on this result, in this paper, we study the bi-quintic isothermal minimal surfaces. According to the condition that any minimal isothermal surface is harmonic, we can acquire the relationship of some control points must satisfy. Follow up, we obtain two holomorphic functions f(z) and g(z) which give the Weierstrass representation of the minimal surface. Under the constrains that the minimal surface is bi-quintic, f(z) and g(z) can be divided into two cases. One case is that f(z) is a constant and g(z) is a quadratic polynomial, and another case is that the degree of f(z) and g(z) are 2 and 1 respectively. For these two cases, we transfer the isothermal parameter to canonical principal parameter, and then compute their normal curvatures and analyze the properties of the corresponding minimal surfaces. Moreover, we study some geometric properties of the bi-quintic harmonic surfaces based on the Bézier representation. Finally, some numerical examples are demonstrated to verify our results.

## Keywords

minimal surface parametric polynomial form canonical principal parameter normal curvature

## References

1. [1]
C Cosin, J Monterde. Bézier surfaces of minimal area, In: Computational Science ICCS 2002, Lecture Notes in Comput Sci, Vol 2330, Springer-Verlag, Amsterdam, 2002, 72–81.
2. [2]
G Ganchev. Canonical Weierstrass representation of minimal surfaces in Euclidean space, arXiv: 1609.01606.Google Scholar
3. [3]
O Kassabov. Transition to canonical principal parameters on minimal surfaces, Comput Aided Geom Design, 2014, 31: 441–450.
4. [4]
J Man, G Wang. Approximating to nonparameterzied minimal surface with B-spline surface, J Softw, 2003, 14(4): 824–829.
5. [5]
J Man, G Wang. Minimal surface modeling using finite element method, Chinese J Comput, 2003, 26(4): 507–510.
6. [6]
J Monterde. Bézier surface of minimal area: the Dirichlet approach, Comput Aided Geom Design, 2004, 21: 117–136.
7. [7]
J Monterde, H Ugail. On harmonic and biharmonic Bézier surfaces, Comput Aided Geom Design, 2004, 21: 697–715.
8. [8]
J C C Nitsche. Lectures on Minimal surfaces, Cambridge Univ Press, Cambridge, 1989, Vol 1.Google Scholar
9. [9]
G Xu, G Wang. Parametric polynomial minimal surfaces of degree six with isothermal parameter, In: Advances in Geometric Modeling and Processing, Lecture Notes in Comput Sci, Vol 4975, Springer, Berlin Heidelberg, 2008, 329–343.
10. [10]
G Xu, G Wang. Quintic parametric polynomial minimal surfaces and their properties, Differential Geom Appl, 2010, 28(6): 697–704.