Abstract. Keywords. 1. Introduction. Gülnur ŞAFFAK ATALAY 1,*

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1 Journal of Alied Mathematics and Comutation (JAMC), 218, 2(4), htt:// ISSN Online: ISSN Print: Surfaces family with a common Mannheim geodesic cure Gülnur ŞAFFAK ATALAY 1,* 1 Education Faculty, Deartment of Mathematics and Science Education, Ondokuz Mayis Uniersity, Samsun, Turkey How to cite this aer: ATALAY, G.Ş. (218) Surfaces family with a common Mannheim geodesic cure. Journal of Alied Mathematics and Comutation, 2(4), htt://dx.doi.org/ / jamc *Corresonding author: Gülnur ŞAFFAK ATALAY, Education Faculty, Deartment of Mathematics and Science Education, Ondokuz Mayis Uniersity, Samsun, Turkey gulnur.saffak@omu.edu.tr Abstract In this aer, we analyzed surfaces family ossessing a Mannheim artner cure of a gien cure as a geodesic. Using the Frenet frame of the cure in Euclidean -sace, we exress the family of surfaces as a linear combination of the comonents of this frame and derie the necessary and sufficient conditions for coefficients to satisfy both the geodesic and isoarametric requirements. The extension to ruled surfaces is also outlined. Finally, examles are gien to show the family of surfaces with common Mannheim geodesic cure. Keywords Geodesic cure; Mannheim artner; Frenet Frame; Ruled Surface. 1. Introduction At the corresonding oints of associated cures, one of the Frenet ectors of a cure coincides with one of the Frenet ectors of other cure. This has attracted the attention of many mathematicians. One of the well-known cures is the Mannheim cure, where the rincial normal line of a cure coincides with the binormal line of another cure at the corresonding oints of these cures. The first study of Mannheim cures has been resented by Mannheim in 1878 and has a secial osition in the theory of cures (Blum, 1966). Other studies hae been reealed, which introduce some characterized roerties in the Euclidean and Minkowski sace (Lee, 211; Liu & Wang, 28; Orbay &Kasa, 29; Öztekin & Ergüt, 211). Liu and Wang called these new cures as Mannheim artner cures: Let x and x 1 be two cures in the three dimensional Euclidean E. If there exists a corresonding relationshi between the sace cures x and x 1 such that, at the corresonding oints of the cures, the rincial normal lines of x coincides with the binormal lines of x 1, then x is called a Mannheim cure, and x 1 is called a Mannheim artner cure of x. The air {x, x 1 } is said to be a Mannheim air. They showed that the cure: x 1 (s 1 ) is the Mannheim artner cure of the cure x(s) if and only if the curature and the torsion of x 1 1 (s 1 ) satisfy following equation 1 d ' 1 1 ds 1 for some non-zero constant. They also study the Mannheim cures in Minkowski -sace. The generalizations of the Mannheim cures in the 4-dimensional saces hae been gien (Matsuda & Yorozu, 29; Akyiğit, et al. 211). Later, Mannheim offset the ruled surfaces and dual Mannheim cures hae been defined in Orbay et al.29; Özkaldı et al. 29; Güngör & Tosun, 21). Aart from these, some roerties of Mannheim cures hae been analyzed according to different frames such as the weakened Mannheim cures, quaternionic Mannheim cures and quaternionic Mannheim cures of Aw(k) - tye (Karacan, 211; Okuyucu, 21; Önder & Kızıltuğ, 212; Kızıltuğ & Yaylı, 215). In differential geometry, there are many imortant consequences and roerties of cures ( O Neill, 1966; do Carmo, 1976). Researches follow labours about the cures. One of most significant cure on a surface is geodesic cure. Geodesics are imortant in the relatiistic descrition of graity. Einstein s rincile of equialence tells us that geodesics reresent the DOI: /jamc Journal of Alied Mathematics and Comutation(JAMC)

2 aths of freely falling articles in a gien sace. (Freely falling in this context means moing only under the influence of graity, with no other forces inoled). The geodesics rincile states that the free trajectories are the geodesics of sace. It lays a ery imortant role in a geometric-relatiity theory, since it means that the fundamental equation of dynamics is comletely determined by the geometry of sace, and therefore has not to be set as an indeendent equation. In architecture, some secial cures hae nice roerties in terms of structural functionality and manufacturing cost. One examle is lanar cures in ertical lanes, whichcan be used as suort elements. Another examle is geodesic cures, Deng described methods to create atterns of secial cures on surfaces, which find alications in design and realization of freeform architecture. (Deng, B. 211). He resented an eolution aroach to generate a series of cures which are either geodesic or iecewise geodesic, starting from a gien source cure on a surface. The concet of family of surfaces haing a gien characteristic cure was first introduced by Wang et.al. in Euclidean -sace. Kasa et.al. generalized the work of Wang by introducing new tyes of marching-scale functions, coefficients of the Frenet frame aearing in the arametric reresentation of surfaces. Atalay and Kasa, studied the roblem: gien a cure (with Bisho frame), how to characterize those surfaces that osess this cure as a common isogeodesic and Smarandache cure in Euclidean -sace. Also they studied the roblem: gien a cure (with Frenet frame), how to characterize those surfaces that osess this cure as a common isogeodesic and Smarandache cure in Euclidean -sace. As is well-known, a surface is said to be ruled if it is generated by moing a straight line continuously in Euclidean sace E (O'Neill, 1997). Ruled surfaces are one of the simlest objects in geometric modeling. One imortant fact about ruled surfaces is that they can be generated by straight lines. A ractical alication of this tye surfaces is that they are used in ciil engineering and hysics (Guan et al.,1997). Since building materials such as wood are straight, they can be considered as straight lines. The result is that if engineers are lanning to construct something with curature, they can use a ruled surface since all the lines are straight (Orbay et al., 29). In this aer, we analyzed surfaces family ossessing an Mannheim artner of a gien cure as a geodesic. Using the Frenet frame of the cure in Euclidean -sace, we exress the family of surfaces as a linear combination of the comonents of this frame, and derie the necessary and sufficient conditions for coefficents to satisfy both the geodesic and isoarametric requirements. The extension to ruled surfaces is also outlined. Finally, examles are gien to show the family of surfaces with common Mannheim geodesic cure. 2. Preliminaries Let E be a -dimensional Euclidean sace roided with the metric gien by where( x1, x2, x) is a rectangular coordinate system of, dx dx dx E. Recall that, the norm of a arbitrary ector X E is gien by X X, X. Let ( s) : I IR E is an arbitrary cure of arc-length arameter s. The cure is called a unit seed cure if elocity ector ' of a satisfies ' 1. LetT(s), N(s),B(s) be the moing Frenet frame along, Frenet formulas is gien by where the function T s s T s d N s s s N s, ds B s s B s s and s are called the curature and torsion of the cure s, resectiely. Let C: (s) be the Mannheim cure in E arameterized by its arc length s and C*: *(s *) is the Mannheim artner cure of C with an arc length arameter s *. Denote by {T(s), N(s), B(s)} the Frenet frame field along C: (s), that is; T(s) is the tangent ector field, N(s) is the normal ector field, and B(s) is the binormal ector field of the cure C resectiely also denote by {T * (s), N * (s), B * (s)} the Frenet frame field along C * : * (s), that is; T * (s) is the tangent ector field, N * (s) is the normal ector field, and B * (s) is the binormal ector field of the cure C * resectiely. If there DOI: /jamc Journal of Alied Mathematics and Comutation

3 exists one to one corresondence between the oints of the sace cures C and C* such that the binormal ector of C is in the direction of the rincial normal ector of the cure C*, then the (C, C* ) cure coule is called Mannheim airs (Liu & Wang, 28). Figure 1. The Mannheim artner cures. From the figure1, we can write (s) *(s*) (s*)b*(s*). A cure on a surface is geodesic if and only if the normal ector to the cure is eerywhere arallel to the local normal ector of the surface. Another criterion for a cure in a surface M to be geodesic is that its geodesic curature anishes. An isoarametric cure α(s) is a cure on a surface Ψ=Ψ(s,t) is that has a constant s or t-arameter alue. In other words, there exist a arameter or such that α(s)= Ψ(s, ) or α(t)= Ψ(, ). Gien a arametric cure α(s), we call α(s) an isogeodesic of a surface Ψ if it is both a geodesic and an isoarametric cure on Ψ.. Surfaces with common Mannheim geodesic cure Suose we are gien a -dimensional arametric cure () s, L1 s L2, in which s is the arc length and ''( s). Let Surface family that interolates () s, L1 s L2, be the Mannheim artner of the gien cure () s. () s as a common cure is gien in the arametric form as (s, ) (s) [x(s, ) T(s) y(s, ) N(s) z(s, ) B(s)], L1 s L2, T1 T2, (.1) 1 where x( s, ), y( s, ) and z( s, ) are C functions. The alues of the marching-scale functions x( s, ), y( s, ) and z( s, ) indicate, resectiely; the extension-like, flexion-like and retortion-like effects, by the oint unit through the time, starting from () s and T(s), N(s),B(s) is the Frenet frame associated with the cure () s. Our goal is to find the necessary and sufficient conditions for which the Mannheim artner cure of the unit sace cure () s s,. is an arametric cure and a geodesic cure on the surface Firstly, since Mannheim artner cure of the cure () s is an arametric cure on the surface s, arameter T1, T2 such that xs, ys, z s,, L1 s L2, T1 T2 Secondly, since Mannheim artner cure of () s is a geodesic cure on the surface s, T, T such that 1 2, there exists a. (.2), there exist a arameter DOI: /jamc Journal of Alied Mathematics and Comutation

4 where n is a normal ector of ( s, ) and n( s, ) // N(s). (.) N is a normal ector of () s. Theorem.1: Let () s, L1 s L2, be a unit seed cure with nonanishing curature and Mannheim artner cure. is a geodesic cure on the surface (.1) if and only if x(s, ) y(s, ) z(s, ), y(s, ) z(s, ) and () s, L1 s L2, be a Proof: Let () s be a Mannheim artner of the cure () s. From (.1), ( s, ) arametric surface is defined by as follows: Let () s Mannheim artner cure of the cure () s arameter T, T such that, 1 2 Secondly, since Mannheim artner cure of () s 1 2 (s, ) (s) [x(s, ) T(s) y(s, ) N(s) z(s, ) B(s)]. is an arametric cure on the surface s, L1 s L2, T1 T2 is an geodesic cure on the surface s,, there exists a x s, y s, z s,, ( fixed ) (.4) T, T such that n( s, ) // N(s) where n is a normal ector of ( s, ) and, there exist a arameter N is a normal ector of () s. The normal ector can be exressed as z( s, ) y( s, ) y( s, ) z( s, ) n( s, ) ( s) x( s, ) ( s) z( s, ) ( s) y( s, ) T ( s) s s x( s, ) z( s, ) z( s, ) x( s, ) ( s) y( s, ) 1 ( s) y( s, ) N( s) s s y(, ) (, ) (, ) (, ) s 1 x s ( ) (, ) x s y s s y s ( s) x( s, ) ( s) z( s, ) B( s) s s (.5) Thus, if we let z( s, ) y( s, ) y( s, ) z( s, ) 1 s, ( s) x( s, ) ( s) z( s, ) ( s) y( s, ), s s x( s, ) z( s, ) z( s, ) x( s, ) 2 s, ( s) y( s, ) 1 ( s) y( s, ), s s (, ) (, ) (, ) (, ) s, y s 1 x s ( ) (, ) x s s y s y s ( s) x( s, ) ( s) z( s, ). s s DOI: /jamc Journal of Alied Mathematics and Comutation

5 We obtain n( s, ) s, T( s) s, N( s) s, B( s) (.6) 1 2 We know that From (.4), () s is a geodesic cure if and only if s s s,,,, (.7) s, y( s, ) z( s, ) s,, we hae 2 s,, we hae (.8) Combining the conditions (.4) and (.8), we hae found the necessary and sufficient conditions for the s, the Mannheim artner cure of the cure () s is an isogeodesic. to hae Now let us consider other tyes of the marching-scale functions. In the Eqn. (.1) marching-scale functions x s,, y s, and z s, can be choosen in two different forms: 1) If we choose k k xs, a1 kl s x, k 1 k k y s, a2km( s) y, k 1 k k z s, akn s z, k 1 then we can simly exress the sufficient condition for which the cure s, as x y z, dy a21 or ms or, d dz a1 ns and const d, () s is a geodesic cure on the surface (.9) DOI: /jamc Journal of Alied Mathematics and Comutation

6 where l s, ms, ns, x, y and z are 1 C functions, 2) If we choose k k xs, f a1 kl s x, k 1 k k y s, g a2kms y, k 1 k k z s, h akn s z, k 1 then we can write the sufficient condition for which the cure () s aij IR, i 1,2,, j 1,2,...,. is a geodesic cure on the surface s, x y z f g h, dy a21 or ms or or g ', d dz a1 ns h and const d 1 l s, m s, n s, x, y, z, f, g and h are C functions. where,, '. Also conditions for different tyes of marching-scale functions can be obtained by using the Eqn. (.4) and (.8). 4. Ruled surfaces with common Mannheim geodesic cure as (.1) Ruled surfaces are one of the simlest objects in geometric modelling as they are generated basically by moing a line in sace. A surface is a called a ruled surface in Euclidean sace, if it is a surface swet out by a straight line l moing alone a cure. The generating line l and the cure are called the rulings and the base cure of the surface, resectiely. We show how to derie the formulations of a ruled surfaces family such that the common Mannheim geodesic is also the base cure of ruled surfaces. Let ( s, ) be a ruled surface with the Mannheim isogeodesic base cure. From the definition of ruled surface, there is a ector R R s such that; and where.1 is used, we get ( s, ) ( s, ) ( ) Rs ( ) ( ) R( s) x( s, ) T ( s) y( s, ) N( s) z( s, ) B( s) For the solutions of three unknows x( s, ), y( s, ) and z( s, ) we hae, DOI: /jamc Journal of Alied Mathematics and Comutation

7 x( s, ) ( ) T ( s), R( s) y( s, ) ( ) N( s), R( s) (4.1) z( s, ) ( ) B( s), R( s). From (.8) and (4.1), we hae N( s), R( s) and B( s), R( s) (4.2) Including, R( s) x( s) T( s) y( s) N( s) z( s) B( s) using (4.2) we obtain, So, the ruled surfaces family with common Mannheim isoasymtotic gien by; ys ( ) and zs ( ) (4.) (s, ) (s) [x(s) T(s) z(s) B(s)]; z(s) (4.4) 5. Examles of generating simle surfaces with common Mannheim geodesic cure s s 4 Examle 5.1. Let s cos, sin, s be a unit seed cure. Then it is easy to show that s s 4 T s sin, cos,, s s Ns cos,sin,, s 4 s Bs sin, cos, and the curatures of this cure is and The arametric reresentation of the Mannheim artner (with resect to Frenet frame) cure (s) is obtained as (where cons tan t, for 1) (s) s N(s) s s 4 4 cos, 4sin, s The Frenet ectors of the cure are found as of the cure s DOI: /jamc Journal of Alied Mathematics and Comutation

8 2 s 2 s 7 2 Ts sin, cos,, s 7 2 s 2 Ns sin, cos,, s s Bs cos,sin,. 5 5 x s, y s,, z s, and then the Eqn. (.4) and (.8) are satisfied. Thus, we obtain a If we take member of the surface with common Mannheim geodesic cure (s) as s s s s 4s ( s, ) 4cos cos, 4sin sin, x s, y s,, z s, and, we obtain a member of the surface with common geodesic Also, for cure (s) as s 4 s s 4 s 4s ( s, ) cos cos, sin cos, where s, -1 1 (Fig.2). ( s, ) ( s, ) Figure 2. s, surface with cure () s and s, Mannheim offset surface with Mannheim artner cure ( () s ) of the cure () s DOI: /jamc Journal of Alied Mathematics and Comutation

9 x s, cos( s / 5), y s,, z s, e 1 and then the Eqn. (.4) and (.8) are satisfied. If we take Thus, we obtain a member of the surface with common Mannheim geodesic cure as s 2 s s s s 2 2 s 4 cos cos sin ( 1) cos, 4sin cos e ( s, ) s 4s 7 2 s ( e 1) cos, cos x s, cos( s / 5), y s,, z s, e 1 and, we obtain a member of the surface with Also, for common geodesic cure as s 4 s s 4 s 4 s cos cos sin ( e 1) sin, ( e 1) cos, ( s, ) 4s 4 s cos ( e 1) where s 2, -1 1 (Fig. ). ( s, ) s, Figure. s, surface with cure () s and s, Mannheim offset surface with Mannheim artner cure ( () s ) of the cure () s DOI: /jamc Journal of Alied Mathematics and Comutation

10 x( s) y( s), z( s) s and then the Eqn. (4.4) is satisfied. Thus, we obtain a member of the ruled surface with common Mannheim geodesic cure as s s s s 4s ( s, ) 4cos s cos, 4sin s sin, Also, for x( s) y( s), z( s) s and,we obtain a member of the ruled surface with common geodesic cure as s 4 s s 4 s 4s s ( s, ) cos s cos, sin s cos, where s, -1 1 (Fig.4). ( s, ) ( s, ) Figure 4. ( s, ) as a member of the ruled surface and its Mannheim geodesic cure. References Akyiğit, M., Ersoy, S., Özgür, İ. & Tosun, M. (211). Generalized timelike Mannheim cures in Minkowski Sace-Time, Mathematical Problems in Engineering, Article ID 5978: 19 ages, doi /211/5978. Atalay G.Ş, E. Kasa. (216). Surfaces Family with Common Smarandache Geodesic Cure According to Bisho Frame in Euclidean Sace, Mathematical Sciences and Alications E-Notes 4 (1) Atalay G.Ş, E. Kasa. (217). Surfaces Family with Common Smarandache Geodesic Cure According to Frenet Frame in Euclidean Sace, (in reiew: Journal of Science and Arts). Bisho, R.L. (1975). There is more than one way to Frame a cure. American Mathematical Monthly, 82(): Blum, R. (1966). A remarkable class of Mannheim-cures. Canadian Mathematical Bulletin, 9: Deng, B. (211). Secial Cure Patterns for Freeform Architecture Ph.D. thesis, submitted to the Vienna Uniersity of DOI: /jamc Journal of Alied Mathematics and Comutation

11 Technology, Faculty of Mathematics and Geoinformation of. Do Carmo, M.P. (1976). Differential Geometry of Cures and Surfaces, Prentice Hall, Inc., Englewood Cliffs, New Jersey. Güngör, M.A. & Tosun, M. (21). A study on dual Mannheim artner cures. International Mathematical Forum, 5(47): Karacan, M.K. (211). Weakened Mannheim cures. International Journal of the Physical Sciences, 6(2): Kızıltuğ, S. & Yaylı, Y. (215). On the quaternionic Mannheim cures of Aw (k) tye in Euclidean sace E. Kuwait Journal of Science, 42(2): Lee, J.W. (211). No null-helix Mannheim cures in the Minkowski Sace. International Journal of Mathematics and Mathematical Sciences, Article ID 5857: 7 ages, doi /211/5857. Liu, H. & Wang, F. (28). Mannheim artner cures in -sace. Journal of Geometry, 88: Matsuda, H. & Yorozu, S. (29). On generalized Mannheim cures in Euclidean 4-sace. Nihonkai Mathematical Journal, 2:-56. Okuyucu, O.Z. (21). Characterizations of the quaternionic Mannheim cures in Euclidean sace E4. International Journal of Mathematical Combinatorics, 2:44-5. Orbay, K. & Kasa, E. (29). On Mannheim artner cures in E. International Journal of Physical Sciences, 4(5): Orbay, K., Kasa, E. & Aydemir, İ. (29). Mannheim offsets of ruled surfaces. Mathematical Problems in Engineering, Article ID 16917: 9 Pages, doi:1.1155/29/ O Neill, B. (1966). Elementary Differential Geometry, Academic Press Inc., New York. Önder, M. & Kızıltuğ, S. (212). Bertrand and Mannheim artner D-cures on arallel surfaces in Minkowski -Sace. International Journal of Geometry, 1(2):4-45. Özkaldı, S., İlarslan, K. & Yaylı, Y. (29). On Mannheim artner cure in Dual sace. Ananele Stiintifice Ale Uniersitatii Oidius Constanta, 17(2): Öztekin, H.B. & Ergüt, M. (211). Null Mannheim cures in the Minkowski -sace. Turkish Journal of Mathematics, 5(1): Serret, J.A. (1851). Sur quelques formules relaties àla théorie des courbes àdouble courbure. Journal de mathématiques ures et aliquées. 16: Wang, F., Liu, H. (27). Mannheim artner cures in -Euclidean sace, Mathematics in Practice and Theory, ol. 7, no. 1, Wang, G. J., Tang, K. (24). C. L. Tai, Parametric reresentation of a surface encil with a common satial geodesic, Comut. Aided Des. 6 (5) Kasa, E., Akyildiz, F.T., Orbay, K. (28). A generalization of surfaces family with common satial geodesic, Alied Mathematics and Comutation, Guan Z, Ling J, Ping X, Rongxi T (1997). Study and Alication of Physics-Based Deformable Cures and Surfaces, Comuters and Grahics 21: 5-1. DOI: /jamc Journal of Alied Mathematics and Comutation