- Email: [email protected]
Explicit form of parametric polynomial minimal surfaces with arbitrary degree
Applied Mathematics and Computation 259 (2015) 124–131
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Explicit form of parametric polynomial minimal surfaces with arbitrary degree Gang Xu a,⇑, Yaguang Zhu a, Guozhao Wang b, André Galligo c, Li Zhang a, Kin-chuen Hui d a
Department of Computer Science, Hangzhou Dianzi University, Hangzhou 310018, PR China Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China c University of Nice Sophia-Antipolis, 06108 Nice Cedex 02, France d Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong b
a r t i c l e
i n f o
Keywords: Minimal surface Parametric polynomial minimal surface Enneper surface Conjugate minimal surface
a b s t r a c t In this paper, from the viewpoint of geometric modeling in CAD, we propose an explicit parametric form of a class of polynomial minimal surfaces with arbitrary degree, which includes the classical Enneper surface for the cubic case. The proposed new minimal surface possesses 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 n ¼ 4k 1, n ¼ 4k; n ¼ 4k þ 1 and n ¼ 4k þ 2, where n is the degree of the coordinate functions in the parametric form of the minimal surface and k is a positive integer. The explicit parametric form of the corresponding conjugate minimal surface is given and the isometric deformation is also implemented. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Minimal surface is a surface with vanishing mean curvature [5]. As mean curvature is the variation of area functional, minimal surfaces include those surfaces with minimum area and fixed boundaries [2]. Because of their attractive properties, minimal surfaces have been extensively employed in areas such as architecture, material science and ship manufacturing. Parametric polynomial representation is a standard form widely used in Computer-aided Design. For 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 have to be specified to construct the parametric form in Weierstrass representation. The Plateau-Bézier problems are investigated in [4], in which the area function is approximated by Dirichlet energy. By using geometric PDE method, the Plateau-Bézier or Plateau-B-spline problems are studied in [9]. The modeling methods of minimal subdivision surfaces are proposed in [6]. Some examples of parametric polynomial minimal surface of lower degree is proposed in [7,8]. Hao et.al studied a method to determine the quasi-Bézier surface of minimal area among all the quasi-Bézier surfaces with prescribed borders [1]. Li et al. studied the construction of approximate minimal surface with geodesic constraints [3].
⇑ Corresponding author. E-mail addresses: [email protected] (G. Xu), [email protected] (G. Wang), [email protected] (A. Galligo), [email protected] (K.-c. Hui). http://dx.doi.org/10.1016/j.amc.2015.02.065 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
In this paper, from the viewpoint of geometric modeling, 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 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 n ¼ 4k 1; n ¼ 4k, n ¼ 4k þ 1 and n ¼ 4k þ 2, where n is the degree of the coordinate functions in the parametric form of the minimal surface and k is a positive integer. The explicit parametric form of the corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented. 2. Preliminary In this section, we review some concepts and results related to minimal surfaces [5]. If the parametric form of a regular patch in R3 is given by
rðu; v Þ ¼ ðxðu; v Þ; yðu; v Þ; zðu; v ÞÞ;
u;
v 2 X;
in which X is the parametric domain. Then the coefficients of the first fundamental form of rðu; v Þ are
E ¼ hr u ; r u i;
F ¼ hr u ; rv i;
G ¼ hr v ; rv i;
where r u ; r v are the first-order partial derivatives of rðu; v Þ with respect to u and v respectively and h; i defines the dot product of the vectors. The coefficients of the second fundamental form of rðu; v Þ are
L ¼ ðru ; r v ; ruu Þ;
M ¼ ðr u ; r v ; r uv Þ;
N ¼ ðr u ; r v ; r vv Þ;
where r uu ; r vv and r uv are the second-order partial derivatives of rðu; v Þ and ð; ;Þ denotes the mixed product of the vectors. Then the mean curvature H and the Gaussian curvature K of rðu; v Þ are
H¼
EN 2FM þ LG 2
2ðEG F Þ
;
K¼
LN M 2 EG F 2
:
Definition 2.1. If rðu; v Þ satisfies E ¼ G; F ¼ 0 for all u; parameterizations. Definition 2.2. If rðu; v Þ satisfies r uu þ r vv ¼ 0 for all u; Definition 2.3. If rðu; v Þ satisfies H ¼ 0 for all u;
v 2X
, then rðu; v Þ is called surface with isothermal
v 2 X, then rðu; v Þ is called harmonic surface.
v 2 X, then rðu; v Þ is called minimal surface.
Lemma 2.4. A surface with isothermal parameterization is a minimal surface if and only if it is a harmonic surface. Proof. Suppose that rðu; v Þ is a surface with isothermal parameterization, that is, rðu; v Þ satisfies E ¼ G and F ¼ 0. Then we have,
H¼
EN 2FM þ LG 2ðEG F 2 Þ
¼
N þ L ðru r v Þðr uu þ r vv Þ ¼ : 2E 2E
As r u r v – 0 and E ¼ ðr v ; r u Þ – 0, hence H ¼ 0 if and only if r uu þ r vv ¼ 0. Thus, the proof is completed.
h
Definition 2.5. If two differentiable functions pðu; v Þ; qðu; v Þ : U # R satisfy the Cauchy–Riemann equations
@p @q ¼ ; @u @ v
@p @q ¼ @v @u
and both are harmonic, then the functions are said to be harmonic conjugate. Definition 2.6. If P ¼ ðp1 ; p2 ; p3 Þ and Q ¼ ðq1 ; q2 ; q3 Þ are with isothermal parameterizations such that pk and qk are harmonic conjugate for k ¼ 1; 2; 3, then P and Q are said to be parametric conjugate minimal surfaces. For example, helicoid and catenoid are a pair of conjugate minimal surface. A pair of conjugate minimal surfaces satisfy the following lemma. Lemma 2.7. Given two conjugate minimal surfaces P and Q and a real number t, all surfaces of the one-parameter family
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Pt ¼ ðcos tÞP þ ðsin tÞQ satisfy (a) Pt are minimal surfaces for all t 2 R; (b) Pt have the same first fundamental forms for all t 2 R. From Lemma 2.7, any pair of conjugate minimal surfaces can be joined through a one-parameter family of minimal surfaces, and the first fundamental form of this family is independent of t. In other words, these minimal surfaces are isometric and have the same Gaussian curvatures at corresponding points. 3. Main results 3.1. Explicit form of parametric polynomial minimal surfaces In this section, we firstly introduce the following two notations. n1 dX 2 e n un2k v 2k ; Pn ¼ ð1Þk 2k k¼0
Qn ¼
n1 bX 2 c
ð1Þk
k¼0
n 2k þ 1
un2k1 v 2kþ1 ;
in which n P 3, dxe is the smallest integer not less than x, and bxc is the largest integer not greater than x. P n and Q n have the following properties: Lemma 3.1.
@Pn @P n ¼ nPn1 ; ¼ nQ n1 @u @v @Q n @Q n ¼ nQ n1 ; ¼ nPn1 @u @v Lemma 3.2.
Pn ¼ uPn1 v Q n1 Q n ¼ v Pn1 þ uQ n1 Lemma 3.2 can be proved by using the identity:
n
2k
þ
n
2k þ 1
¼
nþ1 2k þ 1
:
Theorem 3.3. If the parametric representation of polynomial surface rðu; v Þ is given by rðu; v Þ ¼ ðXðu; v Þ; Yðu; v Þ; Zðu; v ÞÞ, where
Xðu; v Þ ¼ Pn þ xP n2 ; Yðu; v Þ ¼ Q n þ xQ n2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nðn 2Þx Zðu; v Þ ¼ Pn1 ; n1 then rðu; v Þ is a minimal surface, in which n is the degree of the coordinate functions, and x is a positive shape parameter. Proof. From Lemma 3.1, we have
@ 2 rðu; v Þ 2
@ u
þ
@ 2 rðu; v Þ @2v
¼0
Hence, rðu; v Þ is harmonic surface. By using Lemma 3.1, we have
F¼
@rðu; v Þ @rðu; v Þ ¼ 2nðn 2ÞxðQ n3 P n1 þ Pn3 Q n1 2Q n2 Pn2 Þ @u @v
ð1Þ
G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
(a) Ennerper surface
(b) Polynomial surface of degree 7
(c) Symmetry plane
(d) Straight lines
127
Fig. 1. Enneper surface and minimal surface of degree seven. x ¼ 1; ½1; 1 ½1; 1.
(a) Quartic minimal surface
(b) Symmetry plane
Fig. 2. Quartic minimal surface and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
From Lemma 3.2, we have
Pn2 ¼ uPn3 v Q n3 ; Q n2 ¼ v Pn3 þ uQ n3 ;
ð2Þ ð3Þ
Pn1 ¼ ðu2 v 2 ÞPn3 2uv Q n3 ;
ð4Þ
Q n1 ¼ ðu v ÞQ n3 þ 2uv Pn3 ;
ð5Þ
2
2
Substituting (2)–(5) into (1), we can obtain F ¼ 0. Similarly, we have
EG¼
@rðu; v Þ @rðu; v Þ @rðu; v Þ @rðu; v Þ ¼ 4nðn 2ÞxðQ n1 Q n3 Pn3 Pn1 þ P2n2 Q 2n2 Þ ¼ 0 @u @u @v @v
Hence, rðu; v Þ is a parametric surface with isothermal parameterization. From Lemma 2.4, if a parametric surface with isothermal parameterization is harmonic, then it is a minimal surface. Thus, the proof is completed. h
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
(a) Quintic minimal surface
(b) Symmetry plane
Fig. 3. Quintic minimal surface and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
(a) Minimal surface of degree six
(b) Symmetry plane
Fig. 4. Minimal surface of degree six and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
3.2. Properties and classification From Theorem 3.3, if n ¼ 3, we can obtain the Enneper surface, which is the unique cubic parametric polynomial minimal surface. It has the following parametric form
pffiffiffiffiffiffiffi rðu; v Þ ¼ ðu3 3uv 2 Þ þ xu; ðv 3 3v u2 Þ þ xv ; 3xðu2 v 2 Þ : If n ¼ 5, a quintic polynomial minimal surface proposed in [8] can be obtained. Enneper surface has several interesting properties, such as symmetry, self-intersection, and containing orthogonal straight lines. For the proposed minimal surface, we can prove that it also has these properties. According to the shape properties, the proposed minimal surface in Theorem 3.3 can be classified into four classes with n ¼ 4k 1; n ¼ 4k; n ¼ 4k þ 1; n ¼ 4k þ 2. Proposition 3.4. In case of n ¼ 4k 1, the corresponding proposed minimal surface rðu; v Þ has the following properties: rðu; v Þ is symmetric about the plane X ¼ 0 and Y ¼ 0, rðu; v Þ contains two orthogonal straight lines x ¼ y on the plane Z ¼ 0 Proof. The first property can be proved directly by simple calculations. For the second property, supposing u ¼ v in rðu; v Þ with n ¼ 4k 1 , we have Xðu; v Þ ¼ Yðu; v Þ; Zðu; v Þ ¼ 0. Obviously, they are two orthogonal straight lines x ¼ y on the plane Z ¼ 0. h Fig. 1(a) shows an example of Enneper surface, Fig. 1(b) shows an example of the proposed minimal surface with n ¼ 7. The symmetry plane and straight lines of the minimal surface in Fig. 1(b) are shown in Fig. 1(c) and (d). Proposition 3.5. In case of n ¼ 4k, the corresponding proposed minimal surface rðu; v Þ is symmetric about the plane Z ¼ 0 and Y ¼ 0.
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
Its proof can be done by direct computation. Fig. 2(a) presents an example of the proposed quartic minimal surface and the corresponding symmetry planes are shown in Fig. 2(b). For the case of n ¼ 4k þ 1, it is easy to prove the following proposition with simple calculations. Proposition 3.6. In case of n ¼ 4k þ 1, the corresponding proposed minimal surface rðu; v Þ has the following properties: rðu; v Þ is symmetric about the plane X ¼ 0, Y ¼ 0; X ¼ Y and X ¼ Y. Self-intersection points of rðu; v Þ only lie on the symmetry planes, i.e., there are no other self-intersection points on rðu; v Þ, and the self-intersection curve has the same symmetry plane as the minimal surface.
Fig. 3(a) present an example of the proposed quintic minimal surface and the corresponding symmetry planes are shown in Fig. 3(b). Proposition 3.7. In case of n ¼ 4k þ 2, the corresponding proposed minimal surface rðu; v Þ is symmetric about the plane Z ¼ 0 and Y ¼ 0. We can prove the above proposition by direct computation. For the case of n ¼ 6, it has been studied in [7]. Fig. 4(a) presents an example of the proposed minimal surface with n ¼ 6 and the corresponding symmetry planes are shown in Fig. 4(b). 4. Conjugate minimal surface Helicoid and catenoid are a pair of conjugate minimal surfaces. For rðu; v Þ, we can find a new pair of conjugate minimal surfaces as follows. Theorem 4.1. The conjugate minimal surface of rðu; v Þ has the following parametric form
sðu; v Þ ¼ ðX s ðu; v Þ; Y s ðu; v Þ; Z s ðu; v ÞÞ where
X s ðu; v Þ ¼ Q n þ xQ n2 ; Y s ðu; v Þ ¼ Pn xPn2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nðn 2Þx Q n1 ; Z s ðu; v Þ ¼ n1
ð6Þ
a) t = 0
b) t = π10
c) t = π5
d) t = 3π10
e) t = 2π5
f) t = π2
Fig. 5. Isometric deformation between conjugate minimal surface of degree nine, u;
v 2 ½4; 4.
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
(a)
(b)
(c)
(d)
Fig. 6. Control mesh representation of the minimal surfaces: (a) tensor product representation of quintic minimal surface with x ¼ 2 and X ¼ ½0; 1 ½0; 1; (b) tensor product representation of quintic minimal surface with x ¼ 8 and X ¼ ½0; 1 ½0; 1; (c) triangular Bézier representation of the minimal surface of degree six with x ¼ 2; (d) triangular Bézier representation of the minimal surface of degree six with x ¼ 8.
It can be proved directly by Lemma 3.1. According to the shape properties, the conjugate minimal surface of the proposed minimal surface in Theorem 3.3 can be classified into four classes with n ¼ 4k 1; 4k; 4k þ 1; 4k þ 2. These kinds of conjugate minimal surfaces also have similar properties as described in Propositions 3.4–3.7. Proposition 4.2. The conjugate minimal surfaces sðu; v Þ in Theorem 4.1 has the following properties: when n ¼ 4k 1, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane X ¼ Y and X ¼ Y; when n ¼ 4k, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane Z ¼ 0 and X ¼ 0; when n ¼ 4k þ 1, the corresponding proposed minimal surface sðu; v Þ contains two orthogonal straight lines x ¼ y on the plane Z ¼ 0; when n ¼ 4k þ 2, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane Z ¼ 0 and X ¼ 0. The above properties can be proved by direct calculation method similarly as in subSection 3.2. From Lemma 2.7, the surfaces of one-parametric family
C t ðu; v Þ ¼ ðcos tÞrðu; v Þ þ ðsin tÞsðu; v Þ are minimal surfaces with the same first fundamental form. These minimal surfaces are isometric and have the same Gaussian curvature at corresponding points. Let t 2 ½0; p=2. When t ¼ 0, the minimal surface C t ðu; v Þ reduces to rðu; v Þ; for t ¼ p=2, it reduces to sðu; v Þ. Then when t varies from 0 to p=2; rðu; v Þ can be continuously deformed into sðu; v Þ, and each intermediate surface is also minimal surface. Fig. 5 illustrates the isometric deformation between the conjugate minimal surfaces of degree nine. It is similar to the smooth transition between the helicoid and the catenoid.
G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
131
5. Control mesh representation From the parametric form of these new minimal surfaces, they can be represented by tensor product Bézier surfaces or triangular Bézier surfaces with a control-point based description, which are the standard forms in CAD system. Fig. 6(a) and (b) show the tensor product Bézier representation of quintic minimal surfaces with different shape parameter x. Fig. 6(c) and (d) show the triangular Bézier representation of minimal surfaces of degree six with different shape parameter x. 6. Conclusions In this paper, from the viewpoint of geometric modeling, 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. The proposed parametric form of polynomial minimal surface provides a control mesh representation, which can be integrated into CAD systems directly. In the future, we will explore the applications of the proposed minimal surface in architectural design, ship manufacturing and material science. Acknowledgments The authors wish to thank all anonymous referees for their valuable comments and suggestions. The authors are partially supported by the National Nature Science Foundation of China (Nos. 61472111, 61272300, 51475309, 61102132), and the Open Project Program of the State Key Lab of CAD & CG (No. A1406), Zhejiang University. Kin-chuen Hui is supported by a Direct Grant (No. 2050492) from the Chinese University of Hong Kong, and a Grant from the Research Grants Council of the Hong Kong Special Administration Region (Project No. 412913). References [1] [2] [3] [4] [5] [6] [7]
Y.X. Hao, R.H. Wang, C.J. Li, Minimal quasi-Bézier surface, Appl. Math. Model. 36 (12) (2012) 5751–5757. L.H. Hippolyte, H. William, Meeks III, Classification of doubly-periodic minimal surfaces of genus zero, Inventiones Mathematicae 143 (1) (2001) 1–27. C.Y. Li, R.H. Wang, C.G. Zhu, Designing approximation minimal surfaces with geodesics, Appl. Math. Model. 37 (9) (2013) 6415–6424. J. Monterde, Bézier surfaces of minimal area: the Dirichlet approach, Comput. Aided Geom. Des. 1 (2004) 117–136. J.C.C. Nitsche, Lectures on Minimal Surfaces, vol. 1, Cambridge Univ. Press, Cambridge, 1989. Q. Pan, G. Xu, Construction of minimal subdivision surface with a given boundary, Comput. Aided Des. 43 (4) (2011) 374–380. G. Xu, G.Z. Wang, Parametric polynomial minimal surfaces of degree six with isothermal parameter, Proc. of Geometric Modeling and Processing, 2008, pp. 329–343. [8] G. Xu, G.Z. Wang, Quintic parametric polynomial minimal surfaces and their properties, Diff. Geom. Appl. 28 (6) (2010) 697–704. [9] G. Xu, Q. Zhang, G2 surface modeling using minimal mean-curvature-variation flow, Comput. Aided Des. 5 (2007) 342–351.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Explicit form of parametric polynomial minimal surfaces with arbitrary degree Gang Xu a,⇑, Yaguang Zhu a, Guozhao Wang b, André Galligo c, Li Zhang a, Kin-chuen Hui d a
Department of Computer Science, Hangzhou Dianzi University, Hangzhou 310018, PR China Department of Mathematics, Zhejiang University, Hangzhou 310027, PR China c University of Nice Sophia-Antipolis, 06108 Nice Cedex 02, France d Department of Mechanical and Automation Engineering, The Chinese University of Hong Kong, Hong Kong b
a r t i c l e
i n f o
Keywords: Minimal surface Parametric polynomial minimal surface Enneper surface Conjugate minimal surface
a b s t r a c t In this paper, from the viewpoint of geometric modeling in CAD, we propose an explicit parametric form of a class of polynomial minimal surfaces with arbitrary degree, which includes the classical Enneper surface for the cubic case. The proposed new minimal surface possesses 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 n ¼ 4k 1, n ¼ 4k; n ¼ 4k þ 1 and n ¼ 4k þ 2, where n is the degree of the coordinate functions in the parametric form of the minimal surface and k is a positive integer. The explicit parametric form of the corresponding conjugate minimal surface is given and the isometric deformation is also implemented. Ó 2015 Elsevier Inc. All rights reserved.
1. Introduction Minimal surface is a surface with vanishing mean curvature [5]. As mean curvature is the variation of area functional, minimal surfaces include those surfaces with minimum area and fixed boundaries [2]. Because of their attractive properties, minimal surfaces have been extensively employed in areas such as architecture, material science and ship manufacturing. Parametric polynomial representation is a standard form widely used in Computer-aided Design. For 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 have to be specified to construct the parametric form in Weierstrass representation. The Plateau-Bézier problems are investigated in [4], in which the area function is approximated by Dirichlet energy. By using geometric PDE method, the Plateau-Bézier or Plateau-B-spline problems are studied in [9]. The modeling methods of minimal subdivision surfaces are proposed in [6]. Some examples of parametric polynomial minimal surface of lower degree is proposed in [7,8]. Hao et.al studied a method to determine the quasi-Bézier surface of minimal area among all the quasi-Bézier surfaces with prescribed borders [1]. Li et al. studied the construction of approximate minimal surface with geodesic constraints [3].
⇑ Corresponding author. E-mail addresses: [email protected] (G. Xu), [email protected] (G. Wang), [email protected] (A. Galligo), [email protected] (K.-c. Hui). http://dx.doi.org/10.1016/j.amc.2015.02.065 0096-3003/Ó 2015 Elsevier Inc. All rights reserved.
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
In this paper, from the viewpoint of geometric modeling, 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 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 n ¼ 4k 1; n ¼ 4k, n ¼ 4k þ 1 and n ¼ 4k þ 2, where n is the degree of the coordinate functions in the parametric form of the minimal surface and k is a positive integer. The explicit parametric form of the corresponding conjugate minimal surfaces is given and the isometric deformation is also implemented. 2. Preliminary In this section, we review some concepts and results related to minimal surfaces [5]. If the parametric form of a regular patch in R3 is given by
rðu; v Þ ¼ ðxðu; v Þ; yðu; v Þ; zðu; v ÞÞ;
u;
v 2 X;
in which X is the parametric domain. Then the coefficients of the first fundamental form of rðu; v Þ are
E ¼ hr u ; r u i;
F ¼ hr u ; rv i;
G ¼ hr v ; rv i;
where r u ; r v are the first-order partial derivatives of rðu; v Þ with respect to u and v respectively and h; i defines the dot product of the vectors. The coefficients of the second fundamental form of rðu; v Þ are
L ¼ ðru ; r v ; ruu Þ;
M ¼ ðr u ; r v ; r uv Þ;
N ¼ ðr u ; r v ; r vv Þ;
where r uu ; r vv and r uv are the second-order partial derivatives of rðu; v Þ and ð; ;Þ denotes the mixed product of the vectors. Then the mean curvature H and the Gaussian curvature K of rðu; v Þ are
H¼
EN 2FM þ LG 2
2ðEG F Þ
;
K¼
LN M 2 EG F 2
:
Definition 2.1. If rðu; v Þ satisfies E ¼ G; F ¼ 0 for all u; parameterizations. Definition 2.2. If rðu; v Þ satisfies r uu þ r vv ¼ 0 for all u; Definition 2.3. If rðu; v Þ satisfies H ¼ 0 for all u;
v 2X
, then rðu; v Þ is called surface with isothermal
v 2 X, then rðu; v Þ is called harmonic surface.
v 2 X, then rðu; v Þ is called minimal surface.
Lemma 2.4. A surface with isothermal parameterization is a minimal surface if and only if it is a harmonic surface. Proof. Suppose that rðu; v Þ is a surface with isothermal parameterization, that is, rðu; v Þ satisfies E ¼ G and F ¼ 0. Then we have,
H¼
EN 2FM þ LG 2ðEG F 2 Þ
¼
N þ L ðru r v Þðr uu þ r vv Þ ¼ : 2E 2E
As r u r v – 0 and E ¼ ðr v ; r u Þ – 0, hence H ¼ 0 if and only if r uu þ r vv ¼ 0. Thus, the proof is completed.
h
Definition 2.5. If two differentiable functions pðu; v Þ; qðu; v Þ : U # R satisfy the Cauchy–Riemann equations
@p @q ¼ ; @u @ v
@p @q ¼ @v @u
and both are harmonic, then the functions are said to be harmonic conjugate. Definition 2.6. If P ¼ ðp1 ; p2 ; p3 Þ and Q ¼ ðq1 ; q2 ; q3 Þ are with isothermal parameterizations such that pk and qk are harmonic conjugate for k ¼ 1; 2; 3, then P and Q are said to be parametric conjugate minimal surfaces. For example, helicoid and catenoid are a pair of conjugate minimal surface. A pair of conjugate minimal surfaces satisfy the following lemma. Lemma 2.7. Given two conjugate minimal surfaces P and Q and a real number t, all surfaces of the one-parameter family
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G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
Pt ¼ ðcos tÞP þ ðsin tÞQ satisfy (a) Pt are minimal surfaces for all t 2 R; (b) Pt have the same first fundamental forms for all t 2 R. From Lemma 2.7, any pair of conjugate minimal surfaces can be joined through a one-parameter family of minimal surfaces, and the first fundamental form of this family is independent of t. In other words, these minimal surfaces are isometric and have the same Gaussian curvatures at corresponding points. 3. Main results 3.1. Explicit form of parametric polynomial minimal surfaces In this section, we firstly introduce the following two notations. n1 dX 2 e n un2k v 2k ; Pn ¼ ð1Þk 2k k¼0
Qn ¼
n1 bX 2 c
ð1Þk
k¼0
n 2k þ 1
un2k1 v 2kþ1 ;
in which n P 3, dxe is the smallest integer not less than x, and bxc is the largest integer not greater than x. P n and Q n have the following properties: Lemma 3.1.
@Pn @P n ¼ nPn1 ; ¼ nQ n1 @u @v @Q n @Q n ¼ nQ n1 ; ¼ nPn1 @u @v Lemma 3.2.
Pn ¼ uPn1 v Q n1 Q n ¼ v Pn1 þ uQ n1 Lemma 3.2 can be proved by using the identity:
n
2k
þ
n
2k þ 1
¼
nþ1 2k þ 1
:
Theorem 3.3. If the parametric representation of polynomial surface rðu; v Þ is given by rðu; v Þ ¼ ðXðu; v Þ; Yðu; v Þ; Zðu; v ÞÞ, where
Xðu; v Þ ¼ Pn þ xP n2 ; Yðu; v Þ ¼ Q n þ xQ n2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nðn 2Þx Zðu; v Þ ¼ Pn1 ; n1 then rðu; v Þ is a minimal surface, in which n is the degree of the coordinate functions, and x is a positive shape parameter. Proof. From Lemma 3.1, we have
@ 2 rðu; v Þ 2
@ u
þ
@ 2 rðu; v Þ @2v
¼0
Hence, rðu; v Þ is harmonic surface. By using Lemma 3.1, we have
F¼
@rðu; v Þ @rðu; v Þ ¼ 2nðn 2ÞxðQ n3 P n1 þ Pn3 Q n1 2Q n2 Pn2 Þ @u @v
ð1Þ
G. Xu et al. / Applied Mathematics and Computation 259 (2015) 124–131
(a) Ennerper surface
(b) Polynomial surface of degree 7
(c) Symmetry plane
(d) Straight lines
127
Fig. 1. Enneper surface and minimal surface of degree seven. x ¼ 1; ½1; 1 ½1; 1.
(a) Quartic minimal surface
(b) Symmetry plane
Fig. 2. Quartic minimal surface and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
From Lemma 3.2, we have
Pn2 ¼ uPn3 v Q n3 ; Q n2 ¼ v Pn3 þ uQ n3 ;
ð2Þ ð3Þ
Pn1 ¼ ðu2 v 2 ÞPn3 2uv Q n3 ;
ð4Þ
Q n1 ¼ ðu v ÞQ n3 þ 2uv Pn3 ;
ð5Þ
2
2
Substituting (2)–(5) into (1), we can obtain F ¼ 0. Similarly, we have
EG¼
@rðu; v Þ @rðu; v Þ @rðu; v Þ @rðu; v Þ ¼ 4nðn 2ÞxðQ n1 Q n3 Pn3 Pn1 þ P2n2 Q 2n2 Þ ¼ 0 @u @u @v @v
Hence, rðu; v Þ is a parametric surface with isothermal parameterization. From Lemma 2.4, if a parametric surface with isothermal parameterization is harmonic, then it is a minimal surface. Thus, the proof is completed. h
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(a) Quintic minimal surface
(b) Symmetry plane
Fig. 3. Quintic minimal surface and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
(a) Minimal surface of degree six
(b) Symmetry plane
Fig. 4. Minimal surface of degree six and its symmetry plane. x ¼ 1; ½1; 1 ½1; 1.
3.2. Properties and classification From Theorem 3.3, if n ¼ 3, we can obtain the Enneper surface, which is the unique cubic parametric polynomial minimal surface. It has the following parametric form
pffiffiffiffiffiffiffi rðu; v Þ ¼ ðu3 3uv 2 Þ þ xu; ðv 3 3v u2 Þ þ xv ; 3xðu2 v 2 Þ : If n ¼ 5, a quintic polynomial minimal surface proposed in [8] can be obtained. Enneper surface has several interesting properties, such as symmetry, self-intersection, and containing orthogonal straight lines. For the proposed minimal surface, we can prove that it also has these properties. According to the shape properties, the proposed minimal surface in Theorem 3.3 can be classified into four classes with n ¼ 4k 1; n ¼ 4k; n ¼ 4k þ 1; n ¼ 4k þ 2. Proposition 3.4. In case of n ¼ 4k 1, the corresponding proposed minimal surface rðu; v Þ has the following properties: rðu; v Þ is symmetric about the plane X ¼ 0 and Y ¼ 0, rðu; v Þ contains two orthogonal straight lines x ¼ y on the plane Z ¼ 0 Proof. The first property can be proved directly by simple calculations. For the second property, supposing u ¼ v in rðu; v Þ with n ¼ 4k 1 , we have Xðu; v Þ ¼ Yðu; v Þ; Zðu; v Þ ¼ 0. Obviously, they are two orthogonal straight lines x ¼ y on the plane Z ¼ 0. h Fig. 1(a) shows an example of Enneper surface, Fig. 1(b) shows an example of the proposed minimal surface with n ¼ 7. The symmetry plane and straight lines of the minimal surface in Fig. 1(b) are shown in Fig. 1(c) and (d). Proposition 3.5. In case of n ¼ 4k, the corresponding proposed minimal surface rðu; v Þ is symmetric about the plane Z ¼ 0 and Y ¼ 0.
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Its proof can be done by direct computation. Fig. 2(a) presents an example of the proposed quartic minimal surface and the corresponding symmetry planes are shown in Fig. 2(b). For the case of n ¼ 4k þ 1, it is easy to prove the following proposition with simple calculations. Proposition 3.6. In case of n ¼ 4k þ 1, the corresponding proposed minimal surface rðu; v Þ has the following properties: rðu; v Þ is symmetric about the plane X ¼ 0, Y ¼ 0; X ¼ Y and X ¼ Y. Self-intersection points of rðu; v Þ only lie on the symmetry planes, i.e., there are no other self-intersection points on rðu; v Þ, and the self-intersection curve has the same symmetry plane as the minimal surface.
Fig. 3(a) present an example of the proposed quintic minimal surface and the corresponding symmetry planes are shown in Fig. 3(b). Proposition 3.7. In case of n ¼ 4k þ 2, the corresponding proposed minimal surface rðu; v Þ is symmetric about the plane Z ¼ 0 and Y ¼ 0. We can prove the above proposition by direct computation. For the case of n ¼ 6, it has been studied in [7]. Fig. 4(a) presents an example of the proposed minimal surface with n ¼ 6 and the corresponding symmetry planes are shown in Fig. 4(b). 4. Conjugate minimal surface Helicoid and catenoid are a pair of conjugate minimal surfaces. For rðu; v Þ, we can find a new pair of conjugate minimal surfaces as follows. Theorem 4.1. The conjugate minimal surface of rðu; v Þ has the following parametric form
sðu; v Þ ¼ ðX s ðu; v Þ; Y s ðu; v Þ; Z s ðu; v ÞÞ where
X s ðu; v Þ ¼ Q n þ xQ n2 ; Y s ðu; v Þ ¼ Pn xPn2 ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 nðn 2Þx Q n1 ; Z s ðu; v Þ ¼ n1
ð6Þ
a) t = 0
b) t = π10
c) t = π5
d) t = 3π10
e) t = 2π5
f) t = π2
Fig. 5. Isometric deformation between conjugate minimal surface of degree nine, u;
v 2 ½4; 4.
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(a)
(b)
(c)
(d)
Fig. 6. Control mesh representation of the minimal surfaces: (a) tensor product representation of quintic minimal surface with x ¼ 2 and X ¼ ½0; 1 ½0; 1; (b) tensor product representation of quintic minimal surface with x ¼ 8 and X ¼ ½0; 1 ½0; 1; (c) triangular Bézier representation of the minimal surface of degree six with x ¼ 2; (d) triangular Bézier representation of the minimal surface of degree six with x ¼ 8.
It can be proved directly by Lemma 3.1. According to the shape properties, the conjugate minimal surface of the proposed minimal surface in Theorem 3.3 can be classified into four classes with n ¼ 4k 1; 4k; 4k þ 1; 4k þ 2. These kinds of conjugate minimal surfaces also have similar properties as described in Propositions 3.4–3.7. Proposition 4.2. The conjugate minimal surfaces sðu; v Þ in Theorem 4.1 has the following properties: when n ¼ 4k 1, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane X ¼ Y and X ¼ Y; when n ¼ 4k, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane Z ¼ 0 and X ¼ 0; when n ¼ 4k þ 1, the corresponding proposed minimal surface sðu; v Þ contains two orthogonal straight lines x ¼ y on the plane Z ¼ 0; when n ¼ 4k þ 2, the corresponding proposed minimal surface sðu; v Þ is symmetric about the plane Z ¼ 0 and X ¼ 0. The above properties can be proved by direct calculation method similarly as in subSection 3.2. From Lemma 2.7, the surfaces of one-parametric family
C t ðu; v Þ ¼ ðcos tÞrðu; v Þ þ ðsin tÞsðu; v Þ are minimal surfaces with the same first fundamental form. These minimal surfaces are isometric and have the same Gaussian curvature at corresponding points. Let t 2 ½0; p=2. When t ¼ 0, the minimal surface C t ðu; v Þ reduces to rðu; v Þ; for t ¼ p=2, it reduces to sðu; v Þ. Then when t varies from 0 to p=2; rðu; v Þ can be continuously deformed into sðu; v Þ, and each intermediate surface is also minimal surface. Fig. 5 illustrates the isometric deformation between the conjugate minimal surfaces of degree nine. It is similar to the smooth transition between the helicoid and the catenoid.
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5. Control mesh representation From the parametric form of these new minimal surfaces, they can be represented by tensor product Bézier surfaces or triangular Bézier surfaces with a control-point based description, which are the standard forms in CAD system. Fig. 6(a) and (b) show the tensor product Bézier representation of quintic minimal surfaces with different shape parameter x. Fig. 6(c) and (d) show the triangular Bézier representation of minimal surfaces of degree six with different shape parameter x. 6. Conclusions In this paper, from the viewpoint of geometric modeling, 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. The proposed parametric form of polynomial minimal surface provides a control mesh representation, which can be integrated into CAD systems directly. In the future, we will explore the applications of the proposed minimal surface in architectural design, ship manufacturing and material science. Acknowledgments The authors wish to thank all anonymous referees for their valuable comments and suggestions. The authors are partially supported by the National Nature Science Foundation of China (Nos. 61472111, 61272300, 51475309, 61102132), and the Open Project Program of the State Key Lab of CAD & CG (No. A1406), Zhejiang University. Kin-chuen Hui is supported by a Direct Grant (No. 2050492) from the Chinese University of Hong Kong, and a Grant from the Research Grants Council of the Hong Kong Special Administration Region (Project No. 412913). References [1] [2] [3] [4] [5] [6] [7]
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