Erzincan Üniversitesi Erzincan University Fen Bilimleri Enstitüsü Dergisi Journal of Science and Technology 2020, 13(2), 544-549 2020, 13(2), 544-549 ISSN: 1307-9085, e-ISSN: 2149-4584 DOI: 10.18185/erzifbed.672100 Araştırma Makalesi Research Article
A Study On Hyperbolic Lifted Developable Surfaces
İlkay Arslan Güven1* , Mustafa Dede2 , Cumali Ekici3
1Gaziantep Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Gaziantep 2Kilis 7 Aralık Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Kilis 3Eskişehir Osmangazi Üniversitesi, Fen Edebiyat Fakültesi, Matematik Bölümü, Eskişehir
Geliş / Received: 14/01/2020, Kabul / Accepted: 01/06/2020
Abstract In this paper, we build an envelope ruled surface by means of hyperbolic lifting transformation which is performed to a plane curve,. We demonstrate that it is a developable surface in three dimensional Euclidean space and we impute the statements to be a minimal surface of this surface and constant mean curvature surface. Finally, we construct some examples and graph the curves and surfaces given in examples. Keywords: Ruled surface, developable surface, envelope surface, curve lifting
Hiperbolik Yükseltme İle Açılabilir Yüzeyler Üzerine Bir Çalışma
Öz Bu makalede, bir düzlemsel eğriye hiperbolik yükseltme dönüşümü uygulayarak, regle zarf yüzeyi inşa ettik. Daha sonra bu yüzeyin üç boyutlu Öklid uzayında açılabilir yüzey olduğunu gösterdik ve bu yüzeyin minimal yüzey ve sabit ortalama eğrilikli yüzey olabilmesi için koşulları verdik. Son olarak, bazı örnekler verdik ve örneklerdeki eğrilerin ve yüzeylerin grafiklerini çizdik.
Anahtar Kelimeler: Regle yüzey, açılabilir yüzey, zarf yüzeyi, eğri yükseltme
*Corresponding Author: [email protected] 544
A Study On Hyperbolic Lifted Developable Surfaces
1. Introduction Bertrand offsets, yet a linear equation should be provided among the torsion and curvature Surfaces are called developable, when they of a developable ruled surface’s edge of can be created through ordinary bending of a regression for it to possess a developable Bertrand offset, in (Ravani and Ku, 1991). planar surface without stretching, cutting or 3 wrinkling the material. These surfaces are A ruled surface in R is surface which can be characterized by only bending in one described as the set of points swept out by direction at a time, like the cylinder or cone. moving a straight line in surface. It therefore Developable surfaces are useful because they has a parametrization of the form allow round forms to be made out of .at (s, v)(s)v(s), (1) materials like plywood, sheet metal or cloth. where and are curves lying on the This is why they are used for shiping surface called base curve and ruling, building, tent sewing and fabrication of respectively. By using the equation of ruled ventiliation ducts ect. surface we assume that is never zero and Developable surfaces have natural is not identically zero. The ruled surface’s applications in many areas of engineering rulings are asymptotic curves. In addition to and manufacturing. For example; they are this, the ruled surface’s Gaussian curvature is used to design the airplane wings by aircraft everywhere non-positive. A ruled surface is designer or to connect two tubes of different doubly ruled if through every one of its shapes with planar segments of metal sheets points there are two distinct lines that lie on by tinsmith. the surface. Cylinder, cone, helicoid, Mobius So developable surfaces are more efficient strip, right conoid are some examples of then the general surfaces. There are many ruled surfaces and hyperbolic paraboloid and works about developable surfaces in hyperboloid of one sheet are doubly ruled literature. Choi, Kim and Elber constructed a surfaces (Gray, 1993). developable surface as the envelope of a one Minimal surface is a shape whose surface has parameter family of tangent planes along the the least amount of area needed to occupy lifted curve on the paraboloid z x22 y , space. Since the material which is needed to by lifting a rational planar curve in (Choi et build shapes is least, minimal surfaces are al., 1997). Izumiya and Takeuchi gave a preferable in architecture, industrial design classification of special developable surfaces and mathematics. So mathematicians was under the condition of the existence of such a investigated this kind of surfaces for a long special curve as a geodesic in (Izumiya and time, see (Osserman, 1986). Some well Takeuchi, 2004). known minimal surfaces are plane, helicoid In the surface theory of geometry, ruled and catenoid. surfaces were found by French In this study, we construct a developable mathematician Gaspard Monge who was a surface as the envelope of one parameter founder of constructive geometry. Recently, family of tangent planes along the lifted many mathematicians have studied the ruled curve which is lifted from the planar curve to surfaces on Euclidean space and Minkowski the hyperbolic paraboloid zxy22 . Then space for a long time. For the information we give the conditions of being minimal about these topic, see, e.g., Ali et al. (2013); surface and constant mean curvature surface Çöken et al. (2008); Izumiya and Takeuchi of this developable surface. Also we illustrate (2003); Yu et al. (2014) for a systematic some examples. work. Also Kühnel studied a type of ruled surfaces which was called ruled W-surface and the properties of being a ruled W-surface 2. Preliminaries were given in (Kühnel, 1994). Ravani and Ku showed that, predominantly, entire ruled In Choi et al. (1997), the envelope surface of surface could have a double infinity of the one parameter family of tangent planes
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A Study On Hyperbolic Lifted Developable Surfaces
along a curve of the paraboloid was given by A surface’s mean curvature vanishes the following procedure. necessary and sufficient condition it Let A(t)x(t),y(t) be a rational planar is minimal. curve which is regular (i.e A (t) 0 ). Then A surface’s mean curvature is a curve is constructed as constant necessary and sufficient condition it is constant mean ˆ 22 (2) A(t)x(t), y(t), x(t)y(t), curvature surfaces. This comprises by lifting the curve A(t) to the paraboloid Q; minimal surfaces as a subset , but z x y . 22 Let the unnormalized normal they are dealed by special case. vector field of the surface Q be n(t). Then For a surface given with the parametrization n(t) can be count up by the gradient function (u, v) and the surface’s normal vector field of the surface as; n, the Gaussian and mean curvatures are n(t)2x(t), 2y(t), 1. (3) attained respectively by egf 2 The derivative of the unnormalized normal K, vector field and vector product of n(t) and its EGF 2 (5) derivative are easily attained as; eG2fFgE H 2 n (t)2x (t), 2y (t), 0, 2(EGF) (6)
2y (t),2x (t), where n(t)n (t). 4x(t)y (t)4x (t)y(t) E , , e= ,n , By bearing in mind one parameter family of u u uu (7) F u , v , f= uv ,n , tangent planes T (Q) of the surface Q; A(t)ˆ G v , v , g= vv ,n . zxy22 along the curve Aˆ (t) , the envelope surface is constructed by taking 3. Envelope Surface Of The n(t)n(t) as a ruling direction and Aˆ (t) as Hyperbolic Paraboloid Surface a base curve. Therefore the envelope surface of one-parameter family of tangent planes Before giving the envelope surface of the T(Q) is defined by; hyperbolic paraboloid zxy22, lets A(t)ˆ ˆ express the following theorems and their D(t,A(t) v)A(t)v(n(t)n (t)). (4) proofs about the envelope surface D(t,v) This surface is rational when the curve A(t) A(t) is rational and it is developable because its of the paraboloid given above. Remark 1. The planar curve Gaussian curvature K(t, v) is zero. Since the A(t)x(t),y(t) can be lifted to all of the envelope surface D(t,v)A(t) in equation (4) is quadratics and the curve A(t)ˆ can be created tangent to paraboloid Q; zxy22 along by the equation of the quadratic surface. Also ˆ the curve A(t) , the unnormalized normal the envelope surface of the one parameter vector field of the paraboloid Q; z x22 y family of tangent planes of the quadratic and the envelope surface in (4) surface is given with the equation (4). are same (Choi et al., 1997). Now lets give some basic conception which Theorem 1. The envelope surface can be encountered in (Do Carmo, 1976; D(t,v)A(t) given with the equation (4) of the Gray, 1993; Kühnel, 2006). one parameter family of tangent planes of An envelope surface is a surface that any quadratic surface, is minimal if and only is tangent to each member of a family if the sequent equation is provided; of surfaces. Aˆ ,n n n ,n n A developable surface possesses vn,n n n n,nn 0, negative Gaussian curvature every where n is the unnormalized normal vector place. field of the quadratic surface.
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A Study On Hyperbolic Lifted Developable Surfaces
The envelope surface of the one parameter Proof: For the envelope surface D (t,v ) , family of tangent planes T (S) of the A (t) A(t)ˆ the following derivatives are obtained as; hyperbolic paraboloid surface S; z x y 22
along the curve A(t)ˆ which we call ˆ DAvnn,t (t,v ) same as the equation (1) is Dnn, A (t) v imputed by DAvnnnn,ˆ tt ˆ (8) A(t) (t, v)A(t)v(n(t)n(t)). (12) Dnn,tv Here A(t)ˆ is the base curve and D0.vv By using the equations (7), we have; n(t) n (t) is direction of ruling. EA ,A2vAˆˆˆ ,nn The unnormalized normal vector field n(t) of 2 the surface S; , its derivative and vnn,nn, (9) FA ,nnvnn,nn,ˆ the vectoral product of n(t) and its derivative Gnn,nn, can be calculated simply as; and n(t)2x(t), -2y(t), -1, (13) e Aˆ ,n v n,n n , n (t)2x (t), -2y (t), 0,
f n,n n 0, (10) 2y (t),2x (t), n(t)n (t). g 0. 4x (t)y(t)4x(t)y (t) From the equations (6), (9) and (10) , the mean curvature of the envelope surface Notation 1: The envelope surface of the one is obtained as; parameter family of tangent planes T (S) DA (t,v (t) ) A(t)ˆ eG of the hyperbolic paraboloid surface S; H, 2 2(EGF) can be also denoted with the which yields to parametric equation ˆ eGA ,nnn ,nn v nn ,nnn ,nn. x(t)2vy (t), y(t)- 2vx (t), Thus the result is clear. (t, v). A(t) 22x (t)y(t) x(t)y(t)4v x(t)y (t) Theorem 2. The envelope surface D(t,v) A(t) given with the equation (4) of the one Corollary 1: The envelope surface of the one parameter family of tangent planes of any parameter family of tangent planes T(S) A(t)ˆ quadratic surface is developable. of the hyperbolic paraboloid surface S; Proof: By the equations in (9), (10) and is developable. using the Gaussian curvature in (5), we have directly; Proof: By means of the Theorem 3, it could K0 be simply revealed. which means the surface is developable. Theorem 3. The envelope surface of the one parameter family of tangent planes T(S) Now lets construct the envelope surface by A(t)ˆ using the planar curve A(t) x(t),y(t) and of the hyperbolic paraboloid surface S; lifting it to the hyperbolic paraboloid S; is minimal if the curve in plane zxy.22 We call this lifting as hyperbolic is a line parametrized by lifting and the curve on the surface S is A(t)x(t),x(t)k , imputed by where k is a constant. A(t)ˆ x(t), y(t), x(t)22 y(t) . (11)
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A Study On Hyperbolic Lifted Developable Surfaces
Proof: For the envelope surface given in z x y 22 is constant mean curvature (12), if we use the equations (7) and (13), we surface if the planar curve is a line get; parametrized by A(t) x(t), cx(t) k , G n n ,n n where c and k are constants. 2 =4x (t)22 +4y (t) 16 y(t)x (t) y (t)x(t) 2 Proof: It can be attained by a 2 2x(t) 4 =4 x (t) y (t) 16 y(t) . straightforward computation. y(t)
For ˆ , firstly we get; eA,nvn,nn Example 1. Lets conceive a line given by ˆ 22 A(t)x(t), y(t), x(t)y(t), A(t) (t, 3t), Aˆ (t)x (t), y (t), 2x (t)x(t)2y (t)y(t), then the lifted curve A(t)ˆ of A (t) to the x (t)y22 (t) hyperbolic paraboloid is; ˆ ˆ 2 A(t)x (t), y (t), 2x(t)x(t),A(t) (t, 3t, 8t ), y(t)y(t) and the developable surface is parametrized and by (t, v)(t6v, 3t2v, 8t). 2 A,n2yˆ (t)x (t), 22 A(t) It follows that the Gaussian and mean nn0,0,4y (t)x(t)x (t)y(t), curvates are calculated as; K0, H10240. y (t) 2 n,nn4x (t). x (t) In Figure 1, we illustrate the generator curve Since f=0 and g=0, the mean curvature is A (t) , the lifted curve A(t)ˆ and the constant obtained as; mean curvature surface A(t) . eG H. 2(EGF) 2 Here by the equation of G given above, we are conscious of G0 . Thereby if we have A,n0ˆ and n,n n 0. Then the surface is minimal. By solving the last equations, we acquire that; Figure 1. The generator curve A (t), the x (t)y22 (t)0 lifted curve Aˆ (t) , the constant mean y (t)x (t) curvature surface , respectively. y (t)x (t) A(t)
y(t)x(t)k, and Example 2. Let us consider the hyperbola y (t) parametrized by 0 x (t) A(t)sinh t, cosh t.
y (t) One can calculate the lifted curve of A(t) to c x (t) the hyperbolic paraboloid as y (t) cx (t) A(t)ˆ (sinht, cosht, 1). y(t) cx(t) k, Lastly the developable surface can be where k is a constant. By these solutions we parametrized as (in Figure 2); have e=0 which means H=0. sinh t 2vsinh t, (t,v) . A(t) cosh t - 2vcosh t, 1 4v Theorem 4. The envelope surface of the one parameter family of tangent planes T (S) A(t)ˆ of the hyperbolic paraboloid surface S;
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