A New Approach about the Determination of a Developable Spherical Orthotomic Ruled Surface in R(1,3)

Year 2016, Volume: 9 Issue: 1, 62 - 69, 30.04.2016

Ö. Gökmen Yıldız , Sıddıka Ö. Karakuş , H. Hilmi Hacısalihoğlu

https://doi.org/10.36890/iejg.591892

Abstract

Keywords

Ruled Surface, Developable Surface, Dual Lorentzian Unit Sphere, Minkowski Space

References

• [1] Alamo, N. and Criado, C., Generalized Antiorthotomics and their Singularities, Inverse Problems, 18(3) (2002), 881-889.
• [2] Blaschke, W., Vorlesungen Über Differential Geometry I., Verlag von Julieus Springer, Berlin 1930.
• [3] Bruce, J. W., On Singularities, Envelopes and Elementary Differential Geometry, Math. Proc. Cambridge Philos. Soc., 89 (1) (1981), 43-48.
• [4] Bruce, J. W. and Giblin, P. J., Curves and Singularities. A Geometrical Introduction to Singularity Theory, Second Edition, University Press, Cambridge, (1992).
• [5] Bruce, J. W. and Giblin, P. J., One-parameter Families of Caustics by Reflexion in the Plane, Quart. J. Math. Oxford Ser., (2), 35 (139) (1984), 243-251.
• [6] Georgiou, C., Hasanis, T. and Koutroufiotis, D., On the Caustic of a Convex Mirror, Geom. Dedicata, 28 (2) (1988), 153-169.
• [7] Gibson, C. G., Elementary Geometry of Differentiable Curves, Cambridge University Press, 2011.
• [8] Hoschek, J., Smoothing of curves and surfaces, Computer Aided Geometric Design, Vol. 2, No. 1-3 (1985), special issue, 97-105.
• [9] Köse, Ö., A Method of the Determination of a Developable Ruled Surface, Mechanism and Machine Theory, 34 (1999), 1187-1193.
• [10] Li, C.Y., Wang, R.H. and Zhu, C.G., An approach for designing a developable surface through a given line of curvature, Computer Aided Design, 45 (2013), 621-627.
• [11] McCarthy, J.M., On The Scalar and Dual Formulations of the Curvature Theory of Line Trajectories, ASME Journal of Mechanisms, Transmissions and Automation in Design, (1987), 109-101.
• [12] Study, E., Geometrie der Dynamen, Leibzig, 1903.
• [13] Uğurlu, H. H. and Çaliskan, A., The Study mapping for directed spacelike and timelike lines in Minkowski 3-space, Mathematical and Computational Applications, Vol. 1, No:2 (1996), 142-148.
• [14] Veldkamp, G. R., On the use of dual numbers, vectors and matrices in instantaneous, spatial kinematics, Mech. Mach. Theory, 11 (1976), no. 2, 141-156.
• [15] Xiong, J.F., Spherical Orthotomic and Spherical Antiorthotomic, Acta Mathematica Sinica, 23 (2007), Issue 9, 1673-1682.
• [16] Yaylı, Y., Çalışkan, A., and Uğurlu, H.H., The E. Study Maps of Circles on Dual Hyperbolic and Lorentzian Unit Spheres H^2_0 and S^2_1 Mathematical Proceedings of the Royal Irish Academy, 102A (2002), 1, 37-47.
• [17] Yıldız, Ö. G., Hacısalihogğlu, H.H., Study Map of Spherical Orthotomic of a Circle, International J. Math. Combin, Vol.4, (2014), 07-17.
• [18] Yıldız, Ö. G., Karakus¸, S. Ö., Hacısalihoğlu, H.H., On the determination of a developable spherical orthotomic ruled surface, Bull. Math. Sci., (2014) 5:137-146.
• [19] Yıldız, Ö. G., Karakus¸, S. Ö., Hacısalihogğlu, H.H., On the Determination of a Timelike Developable Spherical Orthotomic RuledSurface,Konuralp Journal of Mathematic, Volume 3 No. 1 (2015) 75-83.