Polynomial minimal surfaces of degree five

arXiv:1502.07474v2 [math.DG] 31 May 2016 POLYNOMIAL MINIMAL SURFACES OF DEGREE FIVE OGNIAN KASSABOV Abstract. The problem of finding all minimal surfaces presented in parametric form as polynomials is discussed by many authors. It is known that the classical Enneper surface is (up to position in space and homothety) the only polynomial minimal surface of degree 3 in isothermal parameters. In higher degrees the problem is quite more complicated. Here we find a general form for the functions that generate a polynomial minimal surface of arbitrary degree via the Weierstrass formula and prove that any polynomial minimal surface of degree 5 in isothermal parameters may be considered as belonging to one of three special families. 1. Introduction The minimal surfaces are a topic of great interest in many areas as mathematics, computer science, physics, medicine, architecture. The reason is that in small areas they have a minimizing property. For the applications of minimal surfaces, in particular in computer graphic research, it is important to use minimal surfaces in polynomial form and hence to know all such surfaces in small degrees. In this direction Cosı́n and Monterde [1] proved that up to position in space and homothety the only polynomial minimal surface of degree three in isothermal parameters is the classical Enneper surface. The case of degree four is considered in [5]. Polynomial minimal surfaces of degrees five and six are studied in [7] and [6], respectively. Theorems about their coefficients-vectors are found and some examples are considered. Unfortunately the systems for the coefficients are very complicated and the general solution is difficult to be found. In [8], polynomial minimal surfaces of arbitrary degree constructed on some special functions are studied, and thus some special surfaces are proposed. It is remarked that in degrees 3 and 5 these surfaces coincide with the Enneper surface and some of the surfaces from [7], respectively. In the present paper we first show that a polynomial minimal surface in isothermal parameters must be generated via the Weierstrass formula with a polynomial and a rational function (Section 3). Then in Section 4 we determine all polynomial minimal surfaces of degree five but we do not try to solve the system for the coefficients. Instead we use the result from Section 3 and we obtain a list of functions that generate via the Weierstrass formula all such surfaces. The surfaces introduced in [7] belong to one of the obtained families, but these families contain many other surfaces as well. It is natural to ask whether these families contain different surfaces. In general the problem of comparing surfaces given in different parametric form is very complicated. For minimal surfaces such a method is proposed in [4]. It is based on the canonical Key words and phrases. Minimal surface, isothermal parameters, canonical principal parameters, parametric polynomial surface. 2010 Mathematics Subject Classification: 53A10. 1 2 OGNIAN KASSABOV parameters introduced in [2] and then solving an ordinary differential equation for finding these parameters. When trying to investigate the relation between the families obtained in Section 4 we cannot use directly the method from [4], because we cannot find a simple form of the surfaces in canonical parameters. So we change a little the approach and we escape solving the differential equation for the transition to canonical parameters. As a result we find that the obtained three families contain different minimal surfaces except in a special case. 2. Preliminaries Let S be a regular surface in the Euclidean space defined by the parametric equation (u, v) ∈ U ⊂ R2 . x = x(u, v) = (x1 (u, v), x2 (u, v), x3(u, v)), The derivatives of the vector function x = x(u, v) are usualy denoted by xu , xv , xuu , etc. Then the coefficients of the first fundamental form are equal to the scalar products E = x2u , F = xu xv , G = x2v , The unit normal to the surface is the vector field xu × xv xu × xv U= =√ . |xu × xv | EG − F 2 In particular, if E = G, F = 0, then the parameters (u, v) of the surface are called isothermal. The coefficients of the second fundamental form of S are given by L = U xuu , M = U xuv , N = U xvv . The Gauss curvature K and the mean curvature H of S are defined respectively by LN − M 2 K= , EG − F 2 H= EN − 2F M + GL . 2(EG − F 2 ) Recall that the surface S is called minimal if its mean curvature vanishes identically. In this case it follows easily that the Gauss curvature is nonpositive. The study of minimal surfaces is closely related with some complex curves – those with isotropic tangent vectors. They are called minimal curves. Indeed we have the following construction. Let S be a minimal surface defined in isothermal parameters. Then it can be considered as the real part of a minimal curve. More precisely, let f (z) and g(z) be two holomorphic functions. Define the Weierstrass complex curve Ψ(z) by  Z z 1 i 2 2 (2.1) Ψ(z) = f (z)(1 − g (z)), f (z)(1 + g (z)), f (z)g(z) dz . 2 2 z0 Then Ψ(z) is a minimal curve and its real and imaginary parts x(u, v) and y(u, v) are harmonic functions that define two minimal surfaces in isothermal parametrizations. We say that these two minimal surfaces are conjugate. Moreover, every minimal surface can be obtained at least locally as the real (as well as the imaginary) part of a Weierstrass minimal curve. POLYNOMIAL MINIMAL SURFACES OF DEGREE FIVE 3 For any two conjugate minimal surfaces x(u, v) and y(u, v) it is defined the associated family {St }, where St : xt (u, v) = x(u, v) cos t + y(u, v) sin t . Then for any real number t the surface St is also minimal and has the same first fundamental form as S = S0 . Example. Taking f (z) = 1, g(z) = z, we obtain a Weierstrass minimal curve whose real part is the classical Enneper surface   u u2
v v2
1 2 2 2 2 x(u, v) = 1+v − ,− 1 + u − , (u − v ) . 2 3 2 3 2 It is well known that the Enneper surface coincide (up to position in space) with any surface in its associated family, see e.g. [3]. In [2] Ganchev introduces the canonical principal parameters. If a surface is parametrized with them, the coefficients of its fundamental forms are given by 1 1 F =0, G= E= , ν ν L=1, M =0, N = −1 , √ where ν = −K is the normal curvature of the surface. Actually a surface in canonical principal parametrization is the real part of a Weierstrass minimal curve generated by some functions f (z), g(z) with f (z) = −1/g ′ (z), i.e. it is the real part of the special Weierstrass curve  Z z 1 − g 2 (z) i(1 + g 2 (z)) g(z) Φ(z) = − dz . , , ′ 2g ′ (z) 2g ′(z) g (z) z0 The canonical principal parameters and the normal curvature play a role similar to that of the natural parameters and the curvature and torsion of a space curve. Namely the following theorem holds: Theorem 2.1. [2] If a surface is parametrized with canonical principal parameters, then its normal curvature ν satisfies the differential equation ∆ ln ν + 2ν = 0. Conversely, for any solution ν(u, v) of this equation (with νu νv 6= 0), there exists an unique (up to position in space) minimal surface with normal curvature ν(u, v), (u, v) being canonical principal parameters. Moreover, the canonical principal parameters (u, v) are determined uniquely up to the following transformations u = εū + a, v = εv̄ + b, ε = ±1 , a = const., b = const. We will also use the following results: Theorem 2.2. [4] Let the minimal surface S be defined by the real part of the Weierstrass minimal curve (2.1). Any solution of the differential equation 1 (2.2) (z ′ (w))2 = − f (z(w))g ′ (z(w)) 4 OGNIAN KASSABOV defines a transformation of the isothermal parameters of S to canonical principal parameters. Moreover the function g̃(w) that defines S via the Ganchev formula is given by g̃(w) = g(z(w)). Theorem 2.3. [4] Let the holomorphic function g(z) generate a minimal surface in canonical principal parameters, i.e. via the Ganchev formula. Then, for an arbitrary complex number α, and for an arbitrary real number ϕ, any of the functions eiϕ α + g(z) , 1 − ᾱg(z) eiϕ g(z) generates the same surface in canonical principal parameters (up to position in space). Conversely, any function that generates this surface (up to position in space) in canonical principal parameters has one of the above forms. In sections 4 and 5 we shall consider minimal polynomial surfaces of degree five. An interesting study of such surfaces is presented in [7]. First of all it is proved that the harmonic condition implies that such a surface must have the form (2.3) r(u, v) = a (u5 − 10u3 v 2 + 5uv 4 ) + b (v 5 − 10u2 v 3 + 5u4 v) +c (u4 − 6u2 v 2 + v 4 ) + d uv(u2 − v 2 ) + e u(u2 − 3v 2 ) +f v(v 2 − 3u2 ) + g (u2 − v 2 ) + h uv + i u + j v + k where a, b, c, d, e, f, g, h, i, j, k are coefficient vectors. For these coefficients the following holds, see [7]: Theorem 2.4. The harmonic polynomial surface (2.1) is minimal if and only if its coefficient vectors satisfy the following system of equations  2 a = b2     a.b = 0     4 a.c − b.d = 0     a.d + 4b.c = 0     16 c2 − d2 + 30 a.e + 30 b.f = 0     4 d.c + 15 b.e − 15a.f = 0     9 e2 − 9f 2 + 16 c.g − 2 d.h + 10 a.i − 10 b.j = 0    9 e.f − 4 c.h − 2 d.g − 5 b.i − 5 a.j = 0    4 g 2 − h2 + 6 e.i + 6 f .j = 0 (2.4) 2 g.h − 3 f .i + 3 e.j = 0     5 a.h + 10 b.g − 12 c.f + 3 d.e = 0     5 b.h − 10 a.g − 3 d.f − 12 c.e = 0     6 e.g + 3f .h + 4c.i − d.j = 0     6 f .g − 3e.h − d.i − 4c.j = 0     h.i + 2g.j = 0     2 g.i − h.j = 0   2   i = j2   i.j = 0 . Is seems impossible to find the general solution of the system (2.4). So in [7] some special solutions are considered and several interesting properties are proved for the obtained surfaces. Using a different approach we shall find all polynomial minimal surfaces of degree five. POLYNOMIAL MINIMAL SURFACES OF DEGREE FIVE 5 3. Polynomial minimal surfaces of arbitrary degree As is said in Introduction, polynomial minimal surfaces of arbitrary degree are studied in [8]. The following construction is proposed. Consider the functions ⌈ n−1 ⌉ 2  n un−2k v 2k Pn = (−1) 2k k=0 ⌊ n−1 ⌋   2 X n k Qn = un−2k−1 v 2k+1 , (−1) 2k + 1 k=0 X k  where ⌈x⌉ denotes the smallest integer not less than x and ⌊x⌋ denotes the largest integer not greater than x. Then it is proved that for any real number ω the polynomial surface of degree n defined by p
2 n(n − 2)ω Pn−1 x(u, v) = − Pn + ωPn−2 , Qn + ωQn−2 , n−1 is minimal. Of course this large family is very interesting. But it is important also to know are these all the possible polynomial minimal surfaces and if not to find other families. To resolve the last problem we propose the following approach. Let S : x(u, v) = (x1 (u, v), x2 (u, v), x3(u, v)) , be a polynomial minimal surface of degree n in isothermal parameters. Then xi (u, v) are polynomials of degree ≤ n, and at least for one i = 1, 2, 3 there is an equality. Suppose that the parametrization is isothermal and S is defined in an open subset of R2 , containing (0, 0). From Lemma 22.25 in [3] it follows that (up to translation) x(u, v) is the real part of the minimal curve z z  . Ψ(z) = 2x , 2 2i So this minimal curve is also polynomial of degree n. Then   1 i ′ 2 2 Ψ (z) = (φ1 (z), φ2 (z), φ3 (z)) = f (z)(1 − g (z)), f (z)(1 + g (z)), f (z)g(z) 2 2 for some functions f (z), g(z) and the coordinate functions φi (z) are polynomials of degree at least ≤ n − 1 so that at least for one i the degree of φi (z) is exactly n − 1. Hence every one of the functions f (z)(1 − g 2 (z)) = 2φ1 (z) f (z)(1 + g 2 (z)) = −2iφ2 (z) f (z)g(z) = φ3 (z) is a polynomial and so (3.1) f (z) = φ1 (z) − iφ2 (z) f (z)g 2 (z) = −(φ1 (z) + iφ2 (z)) f (z)g(z) = φ3 (z) are polynomials of degree ≤ n − 1 and at least for one i the degree in (3.1) is exactly n − 1. So f (z) is a polynomial of degree ≤ n − 1. Now (3.1)3 implies that g(z) is a rational function of the form Pp (z) , (3.2) g(z) = Qq (z) 6 OGNIAN KASSABOV where Pp (z) and Qq (z) are polynomials (of degrees p and q, respectively) with no common zeros. According to (3.1)2 the function f (z)g 2 (z) is also a polynomial, so f (z) = (Qq (z))2 Rr (z) (3.3) with a polynomial Rr (z) (of degree r). Moreover since the polynomials f (z) = (Qq (z))2 Rr (z) f (z)g 2 (z) = Pp2 (z)Rr (z) f (z)g(z) = Pp (z)Qq (z)Rr (z) are of degree ≤ n − 1, then 2q + r ≤ n − 1, 2p + r ≤ n − 1, p + q + r ≤ n − 1 and at least once there is an equality. Conversely it is easy to see that for arbitrary polynomials Pp (z), Qq (z), Rr (z) with the above restrictions on p, q, r, the functions (3.2), (3.3) generate a minimal polynomial surface of degree n via the Weierstrass formula. So we have Theorem 3.1. Any polynomial minimal surface of degree n in isothermal parameters is generated via the Weierstrass formula by two functions of the form f (z) = (Qq (z))2 Rr (z) g(z) = Pp (z) Qq (z) where Pp (z), Qq (z), Rr (z) are polynomials of degree p, q, r, respectively and 2q+r ≤ n−1, 2p + r ≤ n − 1, p + q + r ≤ n − 1 with at least one equality. Conversely any two functions f (z), g(z) with the above form generate a minimal polynomial surface of degree n via the Weierstrass formula. 4. Consequences for polynomial minimal surfaces of degree five With the notations of the previous section we assume n = 5. Then 2q + r ≤ 4, 2p + r ≤ 4, p+q+r ≤4 with at least one equality. According to the first two equations in Theorem 2.4 we may assume that (up to a change of the coordinate system) a = (a1 , a2 , 0), b = (−a2 , a1 , 0) and hence (4.1) 2q + r ≤ 4, 2p + r ≤ 4, p+q+r ≤3 . Then (4.1)1 and (4.1)2 imply q ≤ 2, p ≤ 2, so the following cases can appear: 1. p = 2. Then r = 0. 1.1. q = 0, i.e. f (z) = a, g(z) = Az 2 + Bz + C, where a, A 6= 0. To obtain another functions defining the surface we can use a consequence of the following assertion: Proposition 4.1. [4] Suppose the pairs (f˜(z), g̃(z)) and (f (w), g(w)) generate two minimal surfaces via the Weierstrass formula. Then these surfaces coincide (up to translation) iff there exists a function w = w(z), such that f˜(z) = f (w(z))w ′ (z) and g̃(z) = g(w(z)) . Corollary 4.2. Suppose the pair (f (z), g(z)) generates a minimal surface via the Weierstrass formula. Then for arbitrary numbers α (α 6= 0), β the pair f˜(z) = αf (αz + β) , g̃(z) = g(αz + β) generates the same minimal surface (up to translation). POLYNOMIAL MINIMAL SURFACES OF DEGREE FIVE 7 √ Using Corollary 4.2 with α = A , β = 2√BA we can say that the surface is generated by two functions of the form 1.1. f (z) = a, g(z) = z 2 + b, with a 6= 0. Analogously we obtain the cases: cz 2 + d 1.2. p = 2, q = 1, r = 0 and f (z) = a(z + b)2 , g(z) = , with a, c 6= 0. z+b 1 2.1. p = 0, q = 2, r = 0 and f (z) = a(z 2 + b)2 , g(z) = 2 , with a 6= 0; z +b z+d 2.2. p = 1, q = 2, r = 0 and f (z) = a(bz 2 + c)2 , g(z) = 2 , with a, b 6= 0; bz + c 3. p = 1, q = 0, r = 2 and f (z) = az 2 + b, g(z) = z + c, with a 6= 0; 1 , with a 6= 0. z+c We will denote the corresponding surfaces r 11 [a, b](u, v), r 12 [a, b, c, d](u, v) etc., respectively. 4. p = 0, q = 1, r = 2 and f (z) = (az 2 + b)(z + c)2 , g(z) = Remark 4.1. The case p = q = r = 1 is not interesting, because in this case the surface is not of degree 5. Now we note that the following can be easily proved: Proposition 4.3. Consider the surfaces  Z z if1 (z) f1 (z) 2 2 (1 − g1 (z)), (1 + g1 (z)), f1 (z)g1 (z) dz , S1 : x1 (u, v) = Re 2 2 z0  Z w f2 (w) if2 (w) 2 2 S2 : x2 (u, v) = Re (1 − g2 (w)), (1 + g2 (w)), f2 (w)g2(w) dw . 2 2 w0 Denote by  Z w f2 (w) if2 (w) s s 2 2 S2 : x2 (u, v) = Re − (1 − g2 (w)), (1 + g2 (w)), f2(w)g2 (w) dw 2 2 w0 the surface, symmetric of S2 about the plane Oyz. Then S1 and S2s coincide if and only if 1 f2 (w) = f1 (Z(w))g12(Z(w))Z ′(w) g2 (w) = g1 (Z(w)) for some function Z(w). Using this proposition we see that the surfaces from cases 2.1, 2.2 and 4 can be viewed as symmetric to those in cases 1.1, 1.2 and 3, respectively. Consequently we have Theorem 4.4. Any polynomial minimal surface of degree 5 in isothermal parameters coincides up to position in space and symmetry with a surface generated via the Weierstrass formula with the pair of functions 1.1. f (z) = a, g(z) = z 2 + b, with a 6= 0. cz 2 + d 1.2. f (z) = a(z + b)2 , g(z) = , with a, c 6= 0. z+b 3. f (z) = az 2 + b, g(z) = z + c, with a 6= 0, where a, b, c are complex numbers. 8 OGNIAN KASSABOV Remark 4.2. The family of surfaces introduced in [7] belongs to the case 1.2. More precisely the family from [7] is defined by
r(u, v) = X(u, v), Y (u, v), Z(u, v) with X(u, v) = a1 (u5 − 10u3v 2 + 5uv 4 ) − a2 (v 5 − 10v 3 u2 + 5vu4) +e1 u(u2 − 3v 2 ) − e2 v(v 2 − 3u2 ) Y (u, v) = a2 (u5 − 10u3 v 2 + 5uv 4) + a1 (v 5 − 10v 3 u2 + 5vu4) +e2 u(u2 − 3v 2 ) + e1 v(v 2 − 3u2) q √ p Z(u, v) = 430 (a21 + a22 )(e21 + e22 ) − (a1 e1 + a2 e2 ) (u4 − 6u2 v 2 + v 4 ) √ qp 2 (a1 + a22 )(e21 + e22 ) + (a1 e1 + a2 e2 ) uv(u2 − v 2 ) , − 30 where a1 , a2 , e1 , e2 are real parameters. For a2 e1 − a1 e2 < 0 the surfaces are minimal. Such a surface is generated by the Weierstrass formula with f (z) = 6(e1 − ie2 )z 2 q q√(a2 +a2 )(e2 +e2 )−a e −a e +iq√(a2 +a2 )(e2 +e2 )+a e +a e 1 1 2 2 1 1 2 2 1 2 1 2 1 2 1 2 g(z) = 56 z e1 −ie2 so it belongs to the case 1.2 with b = d = 0. 5. Relations among the families in Theorem 4.4 For some special values of the parameters the surfaces from Theorem 4.4 obviously coincide. Namely if d = −b2 c in r 12 [a, b, c, d](u, v), the surface is of type 3. On the other hand even when this equality is not satisfied, the corresponding surfaces may look very similar, as Fig. 5.1 and Fig. 5.2 show. Fig. 5.1: r 12 [1, 0, 1, 1](u, v) Fig. 5.2: r 3 [1, 1, 0](u, v) We will see that despite the resemblance these two surfaces are different as well as that in general the families r 11 , r12 and r 3 give different surfaces. Suppose that a surface r 12 [a, b, c, d](u, v) generated via the Weierstrass formula by the functions cz 2 + d (5.1) f12 (z) = a(z + b)2 g12 (z) = z+b coincides (up to position in space) with r 3 [A, B, C](u, v) generated by (5.2) f3 (z) = Az 2 + B g3 (z) = z + C . POLYNOMIAL MINIMAL SURFACES OF DEGREE FIVE 9 Denote z12 (w), z3 (w) solutions of the respective equations (2.2), so that (according to Theorem 2.3) the generating functions in canonical principal parameters (5.3) g̃12 (w) = g12 (z12 (w)) g̃3 (w) = g3 (z3 (w)) are related by α + g̃12 (w) eiϕ or g̃3 (w) = . 1 − ᾱ g̃12 (w) g̃12 (w) We will consider only the first possibility. The second can be considered analogously. Note that according to the equation (2.2) the functions z12 (w) and z3 (w) are related by g̃3 (w) = eiϕ (5.4) ′ ′ 2 f12 (z12 (w))g12 (z12 (w))(z12 ) = f3 (z3 (w))g3′ (z3 (w))(z3′ )2 . (5.5) From the last equality, using (5.1)–(5.4) and comparing the coefficients of z12 (w) (note that z12 (w) may not be constant) we may derive α=0 a = Ac3 e4iϕ C + 2 b c eiϕ = 0 b2 c + d = 0. B=0 So the surfaces can coincide only if b2 c + d = 0. In particular the surfaces defined by r 12 [1, 0, 1, 1](u, v) and r3 [1, 1, 0](u, v) are different despite the resemblance in Figures 5.1 and 5.2. Actually in a smaller neighborhood of (u, v) = (0, 0) (with the same viewpoint as for Figures 5.1 and 5.2) the difference is clear, see Figures 5.3 and 5.4. Fig. 5.3: r 12 [1, 0, 1, 1](u, v), |u|, |v| ≤ 1 Fig. 5.4: r 3 [1, 1, 0](u, v), |u|, |v| ≤ 1.2 We use a similar idea to investigate a possible coincidence of surfaces from cases 1.1 and 1.2. Let the surface S11 : r 11 [A, B](u, v) be generated via the Weierstrass formula by (5.6) f11 (z) = A g11 (z) = z 2 + B . and suppose that it coincides with S12 : r 12 [a, b, c, d](u, v). Then some functions g̃11 and g̃12 that generate them in canonical parameters are related by (5.7) g̃11 (w) = eiϕ α + g̃12 (w) 1 − ᾱ g̃12 (w) or g̃11 (w) = eiϕ . g̃12 (w) As before, we consider only the first possibility. Denote z11 (w), z12 (w) respective solutions of the equation (2.2). Then (5.8) g̃11 (w) = (z11 (w))2 + B and hence (5.9) ′ ′ 2z11 (w)z11 = g̃11 (w) . On the other hand analogously to (5.5)
2 ′ 4
2 ′ 4 ′ ′ f11 (z11 (w))g11 (z11 (w)) (z11 ) = f12 (z12 (w))g12 (z12 (w)) (z12 ) 10 OGNIAN KASSABOV holds. Applying (5.6)–(5.9) we can find the left hand side of the above equality as a function of z12 (w). Then looking at the coefficients of z12 (w) we conclude that this equality implies a contradiction. So a surface S11 can not coincide with a surface S12 . Applying the same arguments we may prove that a surface S11 can not coincide with a surface S3 . So we have Theorem 5.1. The families from Theorem 4.4 contain different surfaces except if b2 c + d = 0 in case 2.1 and then the surface belongs also to the case 3. Remark 5.1. The surfaces generated via the Weierstrass formula by the pairs of functions (f (z), g(z)) and (Cf (z), g(z)) are homothetic for any positive real number C. On the other hand, if C 6= 0 is not real, these surfaces are different in general. More precisely let C = |C|eiϕ for a real number ϕ. The pairs (f (z), g(z)) and (|C|f (z), g(z)) generate two homothetic surfaces, but the surface generated by (eiϕ |C|f (z), g(z)) belongs to the associated family of the surface generated by (|C|f (z), g(z)). Thus we see that if a surface belongs to a family from Theorem 4.4 then its homothetic surfaces, as well as their associated surfaces, belong to the same family. References [1] Cosı́n, C., Monterde, J.: Bézier surfaces of minimal area. Proc. Int. Workshop of Computer Graphics and Geom. Modelig. Lecture Notes in Comput. Sci. 2330, 2002, 72–81. [2] Ganchev, G.: Canonical Weierstrass representation of minimal surfaces in Euclidean space. To appear. Available as arXiv:0802.2374. [3] Gray, A., Abbena, E., Salomon, S.: Modern Differential Geometry of Curves and Surfaces with MATHEMATICA. Boca Raton, FL: CRC Press, 2006. [4] Kassabov O.: Transition to Canonical Principal Parameters On Minimal Surfaces. Comput. Aided Geom. Design, 31(2014), 441-450. [5] Kassabov O., Vlachkova, K: On polynomial bi-quartic minimal surfaces. AIP Conf. Proc. 1684, 110002 (2015); doi: 10.1063/1.4934345. [6] Xu, G., Wang, G.: Parametric polynomial minimal surfaces of degree six with isothermal parameter. Lecture Notes in Comput. Sci. 4975, 2008, 329–343. [7] Xu G., Wang G.: Quintic parametric polynomial minimal surfaces and their properties. Differential Geometry and its Applications, 28(2010), 697-704. [8] Xu G., Zhu Y., Wang G., Galligo A., Zhang L., Hui K.: Explicit form of parametric polynomial minimal surfaces with arbitrary degree. Appl. Math. Comput., 259(2015), 124-131. University of Transport, Sofia, Bulgaria E-mail address: [email protected]