Directional Tubular Surfaces

Transcription

1 Inernaional Journal of Algebra, Vol. 9, 015, no. 1, HIKARI Ld, hp://dx.doi.org/ /ija Direcional Tubular Surfaces Musafa Dede Deparmen of Mahemaics, Faculy of Ars Sciences Kilis 7 Aralık Universiy, Kilis, Turkey Cumali Ekici Deparmen of Mahemaics-Compuer Eskişehir Osmangazi Universiy, 6480, Eskişehir, Turkey Haice Tozak Deparmen of Mahemaics-Compuer Eskişehir Osmangazi Universiy, 6480, Eskişehir, Turkey Copyrigh c 015 Musafa Dede, Cumali Ekici Haice Tozak. This aricle is disribued under he Creaive Commons Aribuion License, which permis unresriced use, disribuion, reproducion in any medium, provided he original work is properly cied. Absrac In his paper, we inroduce a new version of ubular surfaces. We firs define a new adaped frame along a space curve, denoe his he q-frame. We hen reveal he relaionship beween he Frene frame he q-frame. We give a parameric represenaion of a direcional ubular surface using he q-frame. Finally, some comparaive examples are shown o confirm he effeciveness of he proposed mehod. Mahemaics Subjec Classificaion: 53A04, 53A05 Keywords: Frene frame, pipe surface, ube, adaped frame 1 Inroducion I is well known ha he ubular (pipe) surface is defined as he envelope of he se of spheres wih radius r which are cenered a a spine curve α()

2 58 Musafa Dede, Cumali Ekici Haice Tozak [9]. The imporance of he ubular surface lies in he fac ha i is used in many pracical applicaions in compuer aided geomeric design. The ubular surface can be parameerized using he Frene frame, however, his frame is undefined wherever he curvaure vanishes, such as a poins of inflecion or along sraigh secions of he curve [13]. Thus, various alernaive mehods have been proposed for compuing he ubular surfaces [11]. Klok [] defined he sweep surfaces using roaion-minimizing frames. A robus compuaion of he roaion minimizing frame for sweep surfaces was inroduced by Wang [3]. There are a number of differen adaped frames along a space curve, such as he parallel ranspor frame [1,1] he Frene frame [5]. Alhough he Frene frame is he mos well-known frame along a space curve, is roaion abou he angen of a general spine curve ofen leads o undesirable wiss in he ubular surface modelling. Owing o is minimal wis, he Bishop frame is widely used in compuer graphics, however, i is no easy o compue [3]. Le α() be a space curve wih a non-vanishing second derivaive. The Frene frame is defined as follows, = α α, b = α α, n = b. (1) α α The curvaure κ he orsion τ are given by κ = α α α 3, τ = de(α, α, α ) α α. () The well-known Frene formulas are given by 0 κ 0 n = v κ 0 τ n b 0 τ 0 b where, (3) v = α (). (4) In order o consruc he 3D curve offse, Coquillar [6] inroduced he quasi-normal vecor of a space curve. The quasi-normal vecor is defined for each poin of he curve, lies in he plane perpendicular o he angen of he curve a his poin [7]. q-frame Along a Space Curve In his secion, as an alernaive o he Frene frame we define a new adaped frame along a space curve, he q-frame. Given a space curve α() he q-frame

3 D-ubular surfaces 59 consiss of hree orhonormal vecors, hese being he uni angen vecor, he quasi-normal n q he quasi-binormal vecor b q. The q-frame {, n q, b q, k} is given by = α α, n q = k k, b q = n q (5) where k is he projecion vecor. For simpliciy, we have chosen he projecion vecor k = (0, 0, 1) in his paper. However, he q-frame is singular in all cases where k are parallel. Thus, in hose cases where k are parallel he projecion vecor k can be chosen as k = (0, 1, 0) or k = (1, 0, 0). We can define he Euclidean angle θ beween he principal normal n quasi-normal n q vecors. Then, as one can see immediaely, he relaion marix may be expressed as Thus, n q b q n b = = cos θ sin θ 0 sin θ cos θ cos θ sin θ 0 sin θ cos θ n b n q b q. (6). (7) Le α(s) be a curve ha is parameerized by arc lengh s. Differeniaing (6) wih respec o s, hen subsiuing (3) (7) ino he resuls gives he variaion equaions of he q-frame in he following form n q b q where he q-curvaures are = 0 k 1 k k 1 0 k 3 k k 3 0 n q b q, (8) k 1 = κ cos θ, k = κ sin θ, k 3 = dθ + τ. (9) Fig. 1a 1b show he q-frame he roaion minimizing frame(rmf) of he curve r() = (,, 3 /3). Noe ha he behavior of he q-frame is similar o ha of he RMF. I is well known ha compuing he roaion minimizing frame along he curve is difficul, alhough boh of he frames have similar accuracy.

4 530 Musafa Dede, Cumali Ekici Haice Tozak Figure 1a: The roaion minimizing frame along he curve. The normal(red) he binormal(black) vecors are shown. 3 D-Tubular Surfaces Figure 1b: The q-frame along he curve. The normal(red) he binormal(black) vecors are shown. In his secion, we inroduce a new form of ubular surface, call his surface a direcional ubular surface, or D-ubular surface for shor. The D-ubular surface, a a disance r from he spine curve α(s), may be represened as I is easy o see ha ψ r (s, v) = α(s) + r(cos vn q + sin vb q ). (10) ψ r α(s) = r, (11) where denoes he Euclidean norm. The parial derivaives of ψ r (s, v), wih respec o s v, are deermined by ψ r s = (1 r(k 1 cos v + k sin v)) rk 3 sin vn q + rk 3 cos vb q (1) ψ r v = r(cos vb q sin vn q ). (13) By aking he cross produc of (1) (13) we ge Thus, ψ r s ψ r v = r(1 r(k 1 cos v + k sin v))(cos vn q + sin vb q ). (14) ψ r s ψ r v = ±r(1 r(k 1 cos v + k sin v)). (15) From (14) (15), we obain he uni normal vecor of he D-ubular surface U = (cos vn q + sin vb q ). (16) Corollary 3.1. I is well know ha he poins where ψ r s ψ r v = 0 are singular. I follows ha he singular poins of he D-ubular surface can be obained by 1 r = k 1 cos v + k sin v. (17)

5 D-ubular surfaces 531 We can hen sae he following heorem: Theorem 3.1. The Gaussian mean curvaures of he D-ubular surface are given by (k 1 cos v + k sin v) K = (18) r(1 r(k 1 cos v + k sin v)) H = r(k 1 cos v + k sin v) 1 r(1 r(k 1 cos v + k sin v)). (19) Proof: From (1) (13), he componens E = ψ r s, ψ r s, F = ψ r s, ψ r v G = ψ r v, ψ r v of he firs fundamenal form are obained by E = (1 r(k 1 cos v + k sin v)) + r k 3 (0) F = r k 3, G = r. (1) Similarly we can derive he componens L = ψ r ss, U, M = ψ r sv, U N = ψ r vv, U of he second fundamenal form as L = [ (k 1 cos v + k sin v)(1 r(k 1 cos v + k sin v)) rk 3 ] () M = ±rk 3, N = ±r. (3) I is well known ha he Gaussian mean curvaures of a surface are given by K = LN M LG MF + NE, H =. (4) EG F EG F By subsiuing (0)-(3) ino (4), he Gaussian mean curvaures of he D-ubular surface are obained by K = (k 1 cos v + k sin v) r(1 r(k 1 cos v + k sin v)) (5) H = 1 r(cos xk 1 + sin xk ) r(1 r(k 1 cos x + k sin x)), (6) respecively.

6 53 Musafa Dede, Cumali Ekici Haice Tozak 4 Examples In his secion, we presen several examples o highligh he advanages of his new approach. Example 4.1. I is well known ha he Frene frame (n b) is no defined a poins where he curvaure of he curve is zero. Hence, he analyical expression of he ubular surface around a sraigh line is no obainable. In his example, we have obained he analyical expression of he D-ubular surface generaed by a line wih a q-frame, as shown in Fig.. Now, le us consider a spine curve(line) parameerized by From (5), I is easy o see ha, α() = (,, 0). (7) Figure : The D-ubular surface generaed by he line wih he q-frame. = (,, 0), n q = (,, 0) (8) b q = (0, 0, 1). (9) For r = 3, he D-ubular surface is parameerized by ψ r (, v) = ( + 3 cos v, 3 cos v, 3 sin v). (30) Noe ha, compared o ubular surfaces generaed by he RMF, he analyic expressions of D-ubular surfaces can be easily obained. Example 4.. Assume ha he curve is given by α() = (,, 9 ) (31) I is easy o see ha he Frene curvaure κ orsion τ of his curve is obained by κ = 7 7, τ = 0 (3) ( ) 3

7 D-ubular surfaces 533 respecively. Hence τ = 0, he angle beween he roaion minimizing frame(bishop frame) he Frene frame is consan, herefore he Bishop frame is also no suiable for his example. Figure 3a: The ubular surface. Figure 3b: The D-ubular surface. The q frame can be calculaed by = (1, 1, 9 8 ), n q = 1 (,, 0), b q = (9 8, 9 8, ) (33) For r = 1, he ubular he D-ubular surfaces are shown in Fig. 3a 3b, respecively. Example 4.3. In his example, le us consider a more complex spine curve α(), parameerized by α() = (cos( 3 10 )( + cos()), sin( 3 10 )( + cos()), sin( )). (34) 15 For r = 0., he ubular he D-ubular surfaces are illusraed in Fig. 4a 4b, respecively. Figure 4a: The ubular surface. Figure 4b: The D-ubular surface.

8 534 Musafa Dede, Cumali Ekici Haice Tozak References [1] R.L. Bishop, There is more han one way o frame a curve, Am. Mah. Mon., 8 (1975), no. 3, hp://dx.doi.org/10.307/ [] F. Klok, Two moving coordinae frames for sweeping along a 3D rajecory, Compu. Aided Geom. Des., 3 (1986), hp://dx.doi.org/ / (86) [3] W. Wang, B. Jüler, D. Zheng, Y. Liu, Compuaion of roaion minimizing frames, ACM Trans. Graph., 7 (008), hp://dx.doi.org/ / [4] H. Guggenheimer, Compuing frames along a rajecory, Compu. Aided Geom. Des, 6 (1989), hp://dx.doi.org/ / (89) [5] P.M. do Carmo, Differenial Geomery of Curves Surfaces, Prenice- Hall, Englewood Cliffs, New Jersey, [6] S. Coquillar, Compuing offses of B-spline curves, Compuer-Aided Design, 19 (1987), hp://dx.doi.org/ / (87) [7] H. Shin, S. K. Yoo, S. K. Cho, W. H. Chung, Direcional Offse of a Spaial Curve for Pracical Engineering Design, Compuaional Science is Applicaions-ICCSA, 003, hp://dx.doi.org/ / x 7 [8] Z. Xu, R. Feng, J. Sun, Analyic Algebraic Properies of Canal Surfaces, Journal of Compuaional Applied Mahemaics, 195 (006), 0-8. hp://dx.doi.org/ /j.cam [9] T. Maekawa, N.M. Parikalakis, T. Sakkalis, G. Yu, Analysis applicaions of pipe surfaces, Compu. Aided Geom. Design, 15 (1998), hp://dx.doi.org/ /s (97) [10] M. Dede, Tubular surfaces in Galilean space, Mah. Commun., 18 (013), no. 1, [11] F. Dogan, Y. Yayli, Tubes wih Darboux Frame, In. J. Conemp. Mah. Sciences, 7 (01), no , [1] S. Yilmaz M. Turgu, A new version of Bishop frame an applicaion o spherical images, J. Mah. Anal. Appl., 371 (010), hp://dx.doi.org/ /j.jmaa

9 D-ubular surfaces 535 [13] J. Bloomenhal, Calculaion of Reference Frames Along a Space Curve, Graphics Gems, Academic Press Professional, Inc., San Diego, CA Received: December 15, 015; Published: December 30, 015