Parametric Polynomial Minimal Surfaces of Degree Six with Isothermal Parameter
In this paper, parametric polynomial minimal surfaces of degree six with isothermal parameter are discussed. We firstly propose the sufficient and necessary condition of a harmonic polynomial parametric surface of degree six being a minimal surface. Then
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Minimal surface is an important class of surfaces in differential geometry. Since Lagrange derived the minimal surface equation in R3 in 1762, minimal surfaces have a long history of over 200 years. Because of their attractive properties, the minimal surfaces have been extensively employed in many areas such as architecture, material science, aviation, ship manufacture, biology, crystallogeny and so on. For instance, the shape of the membrane structure, which has appeared frequently in modern architecture, is mainly based on the minimal surfaces [1]. Furthermore, triply periodic minimal surfaces naturally arise in a variety of systems, including block copolymers, nanocomposites , micellar materials, lipidwater systems and certain cell membranes[11]. So it is meaningful to introduce the minimal surfaces into CAGD/CAD systems. However, most of the classic minimal surfaces, such as helicoid and catenoid, can not be represented by B´ezier surface or B-spline surface, which are the basic modeling tools in CAGD/CAD systems. In order to introduce the minimal surfaces into CAGD/CAD systems, we must find some minimal surfaces in the parametric polynomial form. In practice, the highest degree of parametric surface used in CAD systems is six, hence, polynomial minimal surface of degree six with F. Chen and B. J¨ uttler (Eds.): GMP 2008, LNCS 4975, pp. 329–343, 2008. c Springer-Verlag Berlin Heidelberg 2008
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G. Xu and G. Wang
isothermal parameter is discussed in this paper. The new minimal surfaces have elegant properties and are valuable for architecture design. 1.1
Related Work
There has been many literatures on the minimal surface in the field of classical differential geometry [19,20]. The discrete minimal surface has been introduced in recent years in [2,4,12,21,22,27,30]. As the topics which are related with the minimal surface, the computational algorithms for conformal structure on discrete surface are presented in [7,8,10]; and some discrete approximation of smooth differential operators are proposed in[31,32]. Cos´ın and Monterde proved that Enneper’s surface is the unique cubic parametric polynomial minimal surface [3]. Based on the nonlinear programming and the FEM(finite element method), the approximation to the solution of the minimal surface equation bounded by B´ezier or B-spline curves is investigated in [14]. Monterde obtained the approximation solution of the Plateau-B´ezier problem by replacing the area functional with the Dirichlet functional in [15,16]. The modeling schemes of harmonic and biharmonic B´ezier surfaces to approximate the minimal surface are presented in [3,17,18,29]. The applications of minimal surface in aesthetic design, aviation and nano structures modeling have been presented in [6,25,26,28]. 1.2
Contributions and Overview
In this paper, we employ the classical theory of minimal surfaces to obtain parametric polynomial minimal surfaces of degree six. Our main contribution are: • We propose the sufficient and necessary condition of a harmonic polynomial parametric surface of
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