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Spacelike Developable Surfaces Through a Common Line of Curvature in Minkowski 3-Space

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Spacelike Developable Surfaces Through a Common Line of Curvature in Minkowski 3-Space Bulletin of the JSME Vol.9, No.4, 2015 Journal of Advanced Mechanical Design, Systems, and Manufacturing Spacelike developable surfaces through a common line of curvature in Minkowski 3-space Cai-Yun LI∗, Chun-Gang ZHU∗∗ and Ren-Hong WANG∗∗ ∗ School of Science, Dalian University of Technology No.2 Dagong Road, Liaodong Bay District, Panjin, 124221, China. ∗∗ School of Mathematical Sciences, Dalian University of Technology No.2 Linggong Road, Ganjingzi District, Dalian, 116024, China. E-mail: [email protected] Received 23 December 2014 Abstract This paper studies the problem to construct a spacelike developable surface possessing a given non-null parametric curve as the line of curvature in Minkowski 3-space. By using Frenet frame to express the surface, we derive the necessary and sufficient conditions when the resulting spacelike developable surface is cylinder, cone or tangent surface. Finally, we give some representative examples. Key words : Spacelike developable surface, Line of curvature, Frenet frame 1. Introduction In E3 (Euclidean 3-space), developable surfaces are a subset of ruled surfaces. They can be developed onto a plane without streching and tearing. Therefore, they have important application in many product design which use leather, paper, sheet metal as materials. There are three types of developable surfaces: cylinders, cones and tangent surfaces. There are mainly two methods for the design of developable surfaces. One approach is to prove characterizing equations for the free form surfaces to be developable. The readers can refer to papers (Nolan, 1971, Aumann, 1991, Chalfant and Maekawa, 1998). However, the complex system of coupled equations is very difficult for the design of developable Bezier´ surfaces in a CAD system. Another approach to design developable surfaces is a direct surface represented in terms of geometric duality between points and planes in 3D projective space by Chu and Sequin´ (Chu and Sequin,´ 2002). In paper (Bodduluri and Ravani, 1993), the concept of duality between points and planes in 3D projective space was used to develop a new representation for developable surfaces in terms of plane geometry. Line of curvature is one of the most important characteristic curve on a surface and it plays an important role in differential geometry, please refer to the book by Carmo (1976) and Willmore (1959). It is a useful tool in surface analysis for exhibiting variations of the principal direction. There are a number of papers studied the lines of curvature. Martin (Martin, 1983) introduced the principal patches whose sides are lines of curvature. He showed that the existence of such patches was contingent upon certain position matching and frame matching conditions along the patch boundary curves. Alourdas et al. (Alourdas, et al., 1990) studied how to construct a net of lines of curvature on a B-spline surface. In paper (Li, et al., 2011), the authors studied the parametric surface pencils which possess the given curve as the line of curvature. Moreover, the authors further gave an approach to design the developable surface through the given curve as line of curvature in paper (Li, et al., 2013). As we all know, Minkowski geometry provides the theoretical model in mathematics for Einstein’s relativity theory. 4 4 3 Especially, E1 (Minkowski 4-space) has the very strong physical background. As the sub-space of E1, E1 (Minkowski 3-space) has many properties which are different from E3. Some basic concepts, such as vector, frame, and the motion 3 of point, have qualitative changes. Since the metric in E1 is Lorentzian metric which is not positive definite metric. 3 3 The distance function in E1 can be positive, negative or zero. Therefore, the vectors in E1 can be classified into three types: spacelike vectors, timelike vectors and lightlike vectors according to the sign of the distance function. Especially, Paper No.14-00543 [DOI: 10.1299/jamdsm.2015jamdsm0050] © 2015 The Japan Society of Mechanical Engineers 1 0123456789 Li, Zhu and Wang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.4 (2015) the emergence of lightlike vectors makes the results of some problems are surprising. As the spatio-temporal model of relativity theory, Minkowski space obtains much attention in the field of mathematics and physics. The theory of sub- manifold in Minkowski space has been an important topic in the field of geometry. In this paper, we continue the research of the paper (Li, et al., 2013). We give an approach to design the spacelike developable surface which possess the given curve as line of curvature, and we also derive the necessary and sufficient conditions for the spacelike developable surface to be cylinder, cone or tangent surface respectively. The paper is organized as follows. In Section 2, we briefly recall the relevant definitions and results about curve and ff 3 surface of Di erential Geometry in E1. In Section 3, for a given non-null parametric curve r(s), where s is arc-length parameter, we construct a spacelike developable surface which possesses r(s) as the line of curvature and give the concrete expression of the resulting surface. Moreover, we derive the conditions when the developable surface is cylinder, cone or tangent surface respectively. Some representative examples are given in Section 4. Finally, we conclude the paper in Section 5. 2. Backgrounds 2.1. Curves and surfaces in Minkowski 3-space 3 = 3; <; > <; > Definition 1 (Lopez, 2008)The Lorentz-Minkowski space is the metric space E1 (R ), where the metric is given by < u; v >= u1v1 + u2v2 − u3v3; u = (u1; u2; u3); v = (v1; v2; v3): (1) The metric <; > is called the Lorentzian metric. 2 3 Definition 2 (Lopez, 2008)A vector v E1 is called ( 1 ) spacelike if < v; v >> 0 or v = 0, ( 2 ) timelike if < v; v >< 0 and ( 3 ) lightlike if < v; v >= 0 and v , 0. We point out that the null vector v = 0 is considered of spacelike type although it satisfies < v; v >= 0. The definition of vector product is the same as the given one in the Euclidean ambient. ; 2 3 Definition 3 (Lopez, 2008)If u v E1, we call the Lorentzian vector product of u and v to the unique vector denoted by u × v that satisfies < u × v; w >= det(u; v; w); where det(u; v; w) is the determinant of the matrix obtained by putting by columns the coordinates of the three vectors u; v and w. For any two vectors u = (u1; u2; u3) and v = (v1; v2; v3), the vector product is defined by u × v = (u2v3 − u3v2; u3v1 − u1v3; −(u1v2 − u2v1)). 3 Let r(s) be a curve in E1, where s is the arc-length parameter. We say that r(s) is spacelike (resp. timelike, lightlike) at s if r˙(s) is a spacelike (resp. timelike, lightlike) vector. In this paper, r˙(s) denote the derivatives of r with respect to arc length parameter s. The curve r(s) is called spacelike (resp. timelike, lightlike) if it is for any s 2 I. 3 τ We suppose that r(s) be a unit speed timelike curve in E1. Let k(s) and (s) be the curvature and torsion. Assume r¨(s) , 0 for all s 2 [0; L], since this would give us a straight line. For each point of r(s), the corresponding Serret-Frenet formulae is given by (Walfare 1995): T˙ (s) = k(s)N(s); (2) N˙ (s) = k(s)T(s) + τ(s)B(s); (3) B˙ (s) = −τ(s)N(s); (4) and T(s) × N(s) = B(s); T(s) × B(s) = −N(s); N(s) × B(s) = −T(s): (5) If r(s) is a unit speed spacelike curve, the Serret-Frenet formulaes are as follows: [DOI: 10.1299/jamdsm.2015jamdsm0050] © 2015 The Japan Society of Mechanical Engineers 2 Li, Zhu and Wang, Journal of Advanced Mechanical Design, Systems, and Manufacturing, Vol.9, No.4 (2015) ( 1 ) N(s) is spacelike and B(s) is timelike T˙ (s) = k(s)N(s); (6) N˙ (s) = −k(s)T(s) + τ(s)B(s); (7) B˙ (s) = τ(s)N(s); (8) and T(s) × N(s) = −B(s); T(s) × B(s) = −N(s); N(s) × B(s) = T(s): (9) ( 2 ) N(s) is timelike and B(s) is spacelike T˙ (s) = k(s)N(s); (10) N˙ (s) = k(s)T(s) + τ(s)B(s); (11) B˙ (s) = τ(s)N(s); (12) and T(s) × N(s) = B(s); T(s) × B(s) = N(s); N(s) × B(s) = T(s): (13) Here N(s) and B(s) are called the normal vector and the binormal vector respectively. For a parametric surface P(s; t), it is called a spacelike (timelike) if the induced metric on P(s; t) is a Riemannian (Lorentzian) metric on each tangent plane. That is to say, the unit normal vector n(s; t) of the surface P(s; t) is timelike (spacelike) on each point of P(s; t), where P (s; t) × P (s; t) n(s; t) = s t : (14) kPs(s; t) × Pt(s; t)k 2.2. Ruled surface 3 3 Ruled surface is the well-known geometric object. Extending the research of ruled surface from E to E1, and studying its geometric properties is an important issue. De Woestijne (de Woestijne, 1990) classified all ruled surfaces in 3 3 E1. Kim and Yoon (Kim and Yoon, 2004) classified the ruled surfaces satisfying Gauss map conditions in E1. 3 α = α β = β Now, we define a ruled surface M in E1. Let (s) be a curve defined on an open interval I1 of R and (s) be a transversal vector field along α.
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