Gang XU

GALAAD Team, INRIA Sophia Antipolis

Sophia Antipolis, France

Minimal surfaces in CAGD
Minimal surface is a kind of surface with vanishing mean curvature. As the mean curvature is the variation of area functional, minimal surfaces include the surfaces minimizing the area with a fixed boundary. There have been many literatures on minimal surface in classical differential geometry. Because of their attractive properties, minimal surfaces have been extensively employed in many areas such as architecture, material science, aviation, ship manufacture, biology and so on. For instance, the shape of the membrane structure, which has appeared frequently in modern architecture, is mainly based on minimal surfaces. Furthermore, triply periodic minimal surfaces naturally arise in a variety of systems, including nano-composites, lipid-water systems and certain cell membranes.
In CAD systems, parametric polynomial representation is the standard form. For parametric polynomial minimal surface, plane is the unique quadratic parametric polynomial minimal surface, Enneper surface is the unique cubic parametric polynomial minimal surface. There are few research work on the parametric form of polynomial minimal surface with higher degree. Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. After some work on parametric polynomial minimal surface of degree five and six[1][2], the explicit formula of arbitrary degree is proposed in [3]. The geometric properties, classification and conjugate formula of the proposed minimal surface are also studied.

[1] Quintic parametric polynomial minimal surfaces and their properties
Gang Xu, Guozhao Wang
Differential Geometry and its Applications, accepted
[2]Parametric polynomial minimal surfaces of degree six with isothermal parameter
Gang Xu, Guozhao Wang
Proc. of Geometric Modeling and Processing (GMP 2008), 2008, LNCS 4975, 329-343 [ pdf ]
[3] Parametric polynomial minimal surfaces of arbitrary degree
Gang Xu, Guozhao Wang
INRIA Research Report 00507790 [ pdf ]