Parametric polynomial minimal surfaces of arbitrary degree

Parametric polynomial minimal surfaces of arbitrary degree

Gang Xu gxu@sophia.inria.fr Guozhao Wang Galaad, INRIA Sophia-Antipolis, 2004 Route des Lucioles, 06902 Cedex, France Department of Mathematics, Zhejiang University, Hangzhou, China
Abstract

Weierstrass representation is a classical parameterization of minimal surfaces. However, two functions should be specified to construct the parametric form in Weierestrass representation. In this paper, we propose an explicit parametric form for a class of parametric polynomial minimal surfaces of arbitrary degree. It includes the classical Enneper surface for cubic case. The proposed minimal surfaces also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into four categories with respect to , and . The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented.

keywords:
minimal surface; parametric polynomial minimal surface; Enneper surface ; conjugate minimal surface
journal: Elsevier Science

1 Introduction

Minimal surface is a kind of surface with vanishing mean curvature Nitsche:lecture (). As the mean curvature is the variation of area functional, minimal surfaces include the surfaces minimizing the area with a fixed boundary Morgan1 (); Morgan2 (). There have been many literatures on minimal surface in classical differential geometry Osserman (); Meeks1 (); Meeks2 (). 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 Blet:minimal (). Furthermore, triply periodic minimal surfaces naturally arise in a variety of systems, including nano-composites, lipid-water systems and certain cell membranes Jung:levelset ().

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. In this paper, we discuss the answer to the following questions: What are the possible explicit parametric form of polynomial minimal surface of arbitrary degree and how about their properties? The proposed minimal surfaces include the classical Enneper surface for cubic case, and also have some interesting properties such as symmetry, containing straight lines and self-intersections. According to the shape properties, the proposed minimal surface can be classified into four categories with respect to , and . The explicit parametric form of corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented.

The paper includes five sections. Preliminary introduces some notations and lemmas. Main Results presents the explicit parametric formula of parametric polynomial minimal surface of arbitrary degree. The next section, Properties and Classification presents the corresponding properties and classification of the proposed minimal surfaces. The following section focuses on corresponding conjugate counterpart of proposed minimal surface. Finally, in Conclusions, we summarize the main results.

2 Preliminary

In this section, we introduce the following two notations.

(1)
(2)

and have the following properties:

Lemma 1

Lemma 2

Lemma 2 can be proved by using the following equation:

3 Main Results

Theorem 1 If the parametric representation of polynomial surface with arbitrary degree , is given by , where

(3)

then is a minimal surface.

Proof of Theorem 1. From Lemma 2, we have

Hence, is harmonic surface.

By using Lemma 2, we have

From Lemma 2, we have

(5)
(6)
(7)
(8)

Substituting (5)(6)(7)(8) into (4), we can obtain . Similarly, we have

Hence, r(u,v) is a parametric surface with isothermal parameterization. From Nitsche:lecture (), if a parametric surface with isothermal parameterization is harmonic, then it is a minimal surface. The proof is completed.

4 Properties and Classification

From Theorem 1, if , we can get the Enneper surface, which is the unique cubic parametric polynomial minimal surface. It has the following parametric form

Enneper surface has several interesting properties, such as symmetry, self-intersection, and containing orthogonal straight lines on it. For the new proposed minimal surface, we can also prove that it has also has these properties.

If , a kind of quintic polynomial minimal surface proposed in Gang1 () can be obtained as follows

(9)

where

According to the shape properties, the proposed minimal surface in Theorem 1 can be classified into four classes with , , .

Proposition 1. In case of , the corresponding proposed minimal surface has the following properties:

  • is symmetric about the plane and the plane ,

  • contains two orthogonal straight lines on the plane

Fig. 1(a) shows an example of Enneper surface, Fig. 1 (b) shows an example of proposed minimal surface with . The symmetry plane and straight lines of minimal surface in Fig. 1 (b) are shown in Fig. 1 (c) and Fig. 1 (d).

Proposition 2. In case of , the corresponding proposed minimal surface is symmetric about the plane and the plane .

(a) Ennerper surface (b) Polynomial surface of degree 7
(a) Symmetry plane (b) Straight lines
Figure 1: Enneper surface and minimal surface of degree seven. .

Fig. 2 (a) present an example of proposed quartic minimal surface and the corresponding symmetry planes are shown in Fig. 2 (b).

Proposition 3. In case of , the corresponding proposed minimal surface has the following properties:

  • is symmetric about the plane , the plane , the plane and the plane .

  • Self-intersection points of are only on the symmetric planes, i.e., there are no other self-intersection points on , and the self-intersection curve has the same symmetric plane as the minimal surface.

Fig. 3 (a) present an example of proposed quintic minimal surface and the corresponding symmetry planes are shown in Fig. 3 (b).

Proposition 4. In case of , the corresponding proposed minimal surface is symmetric about the plane and the plane .

For the case of , it has been studied in Gang2 (). Fig.4 (a) present an example of proposed minimal surface with and the corresponding symmetry planes are shown in Fig. 4 (b).

(a) Quartic minimal surface (b) Symmetry plane
Figure 2: Quartic minimal surface and its symmetry plane. .
(a) Quintic minimal surface (b) Symmetry plane
Figure 3: Quintic minimal surface and its symmetry plane. .
(a) Minimal surface of degree six (b) Symmetry plane
Figure 4: Minimal surface of degree six and its symmetry plane. .
a) b) c)
d) e) f)
Figure 5: Dynamic deformation between and , .

5 Conjugate Minimal Surface

Definition 1. If two differentiable functions satisfy the Cauchy-Riemann equations

and both are harmonic, then the functions are said to be harmonic conjugate.

Definition 2. If and are with isothermal parameterizations such that and are harmonic conjugate for , then P and Q are said to be parametric conjugate minimal surfaces.

Helicoid and catenoid are a pair of conjugate minimal surfaces. For , we can find out a new pair of conjugate minimal surfaces as follows.

Theorem 2 The conjugate minimal surface of has the following parametric form

where

(10)

It can be proved directly by Lemma 2. From [2], the surfaces of one-parametric family

are minimal surfaces with the same first fundamental form. These minimal surfaces are isometric and have the same Gaussian curvature at corresponding points. Fig. 5 illustrates the isometric deformation between and . It is similar with the isometric deformation between helicoid and catenoid.

6 Conclusion

The explicit parametric formula of polynomial minimal surface is presented. It can be considered as the generalization of Enneper surface in cubic case. The corresponding properties and classification of the proposed minimal surface are investigated. The corresponding conjugate minimal surface are constructed and the dynamic isometric deformation between them are also implemented.

Acknowlegments

This work was partially supported by the National Nature Science Foundation of China (No.60970079, 60933008), Foundation of State Key Basic Research 973 Development Programming Item of China (No.2004CB318000), and the Nature Science Foundation of Zhejiang Province, China(No. Y1090718).

References

  • (1) Nitsche J.C.C. 1989. Lectures on minimal surfaces, vol. 1. Cambridge Univ. Press, Cambridge
  • (2) Frank Morgan. Minimal surfaces, crystals, and norms on Rn, Proc. 7th Annual Symp. on Computational Geom., June, 1991, N. Conway, NH.
  • (3) Frank Morgan. Area-minimizing surfaces, faces of Grassmannians, and calibrations. The Am. Math. Monthly 95 (1988), 813-822
  • (4) Osserman R.. 1986. A survey of minimal surfaces, Dover publ. 2nd ed., New York
  • (5) W. H. Meeks III, L. Simon and S. T. Yau. Embedded minimal surfaces, exotic spheres, and manifolds with positive Ricci curvature . Ann. of Math. (2) 116 (1982), no. 3, 621-659.
  • (6) W. H. Meeks III and S. T. Yau. The existence of embedded minimal surfaces and the problem of uniqueness . Math. Z. 179(1982), 151-168.
  • (7) T.H. Colding and W.P.Minicozzi II. Shapes of embedded minimal surfaces, Proc. Nat. Acad. Sciences, July 25, 2006, vol. 103, no. 30, 11106-11111
  • (8) Bletzinger K.W.. 1997. Form finding of membrane structures and minimal surfaces by numerical continuation. Proceeding of the IASS Congress on Structural Morphology: Towards the New Millennium, Nottingham. ISBN 0-85358-064-2, 68-75
  • (9) Jung K., Chu, K.T., Torquato, S.. 2007. A variational level set approach for surface area minimization of triply-periodic surfaces. Journal of Computational Physics, 2, 711-730
  • (10) G. Xu, G.Z Wang. Quintic parametric polynomial minimal surfaces and their properties. Differential Geometry and its Applications, to appear
  • (11) G. Xu, G.Z Wang. Parametric polynomial minimal surfaces of degree six with isothermal parameter Proc. of Geometric Modeling and Processing (GMP 2008), 2008, LNCS 4975, 329-343
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