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A new version of Bishop frame and an application to spherical images. (English) Zbl 1207.53003

A classical Bishop frame is a moving frame of a curve in Euclidean three space [R. L. Bishop, Am. Math. Mon. 82, 246–251 (1975; Zbl 0298.53001)] with the following properties. Considering a curve parametrized by arclength s, the first vector is the unit tangent vector T(s), the second and third vector M 1 (s),M 2 (s) are selected arbitrarily so that (T(s),M 1 (s),M 2 (s)) is an orthonormal frame and the (skew symmetric) frame equations read T ' =k 1 M 1 +k 2 M 2 ,M 1 ' =-k 1 T,M 2 ' =-k 2 T.

In the present paper the authors introduce an other version of a Bishop frame using the binormal vector shared with the Frenet frame. The relations to the Frenet frame and the classical Bishop frame are discussed as well as the spherical images. As applications, general helices and slant helices (regular curves the principal normals of which form a constant angle with a fixed direction) are discussed.

MSC:
53A04Curves in Euclidean space
References:
[1]Ali, A. T.: Position vectors of slant helices in Euclidean 3-space, (2009)
[2]Ali, A. T.; López, R.: Slant helices in Minkowski space E13, (2008)
[3]Ali, A. T.; López, R.: Timelike B2-slant helices in Minkowski space E14, Arch. math. (Brno) 46, 39-46 (2010)
[4]Ali, A. T.; Turgut, M.: Position vector of a time-like slant helix in Minkowski 3-space, J. math. Anal. appl. 365, 559-569 (2010) · Zbl 1185.53004 · doi:10.1016/j.jmaa.2009.11.026
[5]Bükcü, B.; Karacan, M. K.: The Bishop Darboux rotation axis of the spacelike curve in Minkowski 3-space, J. fac. Sci. 3, No. 1, 1-5 (2007)
[6]Bükcü, B.; Karacan, M. K.: On the slant helices according to Bishop frame of the timelike curve in Lorentzian space, Tamkang J. Math. 39, No. 3, 255-262 (2008) · Zbl 1168.53306
[7]Bükcü, B.; Karacan, M. K.: Special Bishop motion and Bishop Darboux rotation axis of the space curve, J. dyn. Syst. geom. Theor. 6, No. 1, 27-34 (2008) · Zbl 1172.53001
[8]Bükcü, B.; Karacan, M. K.: The slant helices according to Bishop frame, Int. J. Math. comput. Sci. 3, No. 2, 67-70 (2009)
[9]Bükcü, B.; Karacan, M. K.: Bishop motion and Bishop Darboux rotation axis of the timelike curve in Minkowski 3-space, Kochi J. Math. 4, 109-117 (2009) · Zbl 1181.53005
[10]Bishop, L. R.: There is more than one way to frame a curve, Amer. math. Monthly 82, No. 3, 246-251 (1975) · Zbl 0298.53001 · doi:10.2307/2319846
[11]Do Carmo, M. P.: Differential geometry of curves and surfaces, (1976) · Zbl 0326.53001
[12]Izumiya, S.; Takeuchi, N.: New special curves and developable surfaces, Turkish J. Math. 28, No. 2, 531-537 (2004) · Zbl 1081.53003
[13]Karacan, M. K.; Bükcü, B.: An alternative moving frame for tubular surfaces around timelike curves in the Minkowski 3-space, Balkan J. Geom. appl. 12, No. 2, 73-80 (2007) · Zbl 1138.53018
[14]Karacan, M. K.; Bükcü, B.: An alternative moving frame for tubular surface around the spacelike curve with a spacelike binormal in Minkowski 3-space, Math. morav. 11, 47-54 (2007) · Zbl 1164.53001
[15]Karacan, M. K.; Bükcü, B.: An alternative moving frame for a tubular surface around a spacelike curve with a spacelike normal in Minkowski 3-space, Rend. circ. Mat. Palermo 57, No. 2, 193-201 (2008) · Zbl 1165.53004 · doi:10.1007/s12215-008-0013-8
[16]Karacan, M. K.; Bükcü, B.: Bishop frame of the timelike curve in Minkowski 3-space, Fen derg. 3, No. 1, 80-90 (2008) · Zbl 1179.53002
[17]Karacan, M. K.; Bükcü, B.; Yuksel, N.: On the dual Bishop Darboux rotation axis of the dual space curve, Appl. sci. 10, 115-120 (2008) · Zbl 1171.53014 · doi:http://vectron.mathem.pub.ro/apps/v10/a10.htm
[18]Kula, L.; Yayli, Y.: On slant helix and its spherical indicatrix, Appl. math. Comput. 169, No. 1, 600-607 (2005) · Zbl 1083.53006 · doi:10.1016/j.amc.2004.09.078
[19]Kula, L.; Ekmekçi, N.; Yaylı, Y.; Ilarslan, K.: Characterizations of slant helices in Euclidean 3-space, Turkish J. Math. 34, 261-274 (2010) · Zbl 1204.53003 · doi:http://mistug.tubitak.gov.tr/bdyim/abs.php?dergi=mat&rak=0809-17
[20]Scofield, P. D.: Curves of constant precession, Amer. math. Monthly 102, 531-537 (1995) · Zbl 0881.53002 · doi:10.2307/2974768
[21]Yılmaz, S.: Position vectors of some special space-like curves according to Bishop frame in Minkowski space E13, Sci. magna 5, No. 1, 48-50 (2009)