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Linked References

1. J. A. Adam, “The mathematical physics of rainbows and glories,” Physics Reports, vol. 356, pp. 229–365, 2002.
2. S. Bara, Z. Jaroszewicz, A. Kolodziejczyk, and M. Sypek, “Method for scaling the output focal curves formed by computer generated zone plates,” Optics & Laser Technology, vol. 23, pp. 303–307, 1991.
3. I. A. Popov, N. V. Sidorovsky, I. L. Veselov, and S. G. Hanson, “Statistical properties of focal plane speckle,” Optics Communications, vol. 156, pp. 16–21, 1998.
4. J. Weiss, “Bäcklund transformations, focal surfaces and the two-dimensional Toda lattice,” Physics Letters A, vol. 137, no. 7-8, pp. 365–368, 1989. View at Publisher · View at Google Scholar
5. S. Izumiya, D. Pei, and M. T. Akahashi, “Curves and surfaces in hyperbolic space,” Banach Center Publications, vol. 65, pp. 107–123, 2004.
6. S. Izumiya, D. Pei, and T. Sano, “Horospherical surfaces of curves in hyperbolic space,” Publicationes Mathematicae Debrecen, vol. 64, no. 1-2, pp. 1–13, 2004.
7. S. Izumiya, D. Pei, and T. Sano, “The lightcone Gauss map and the lightcone developable of a spacelike curve in Minkowski 3-space,” Glasgow Mathematical Journal, vol. 42, no. 1, pp. 75–89, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
8. S. Izumiya, D. Pei, and T. Sano, “Singularities of hyperbolic Gauss maps,” Proceedings of the London Mathematical Society, vol. 86, no. 2, pp. 485–512, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
9. S. Izumiya, D. H. Pei, T. Sano, and E. Torii, “Evolutes of hyperbolic plane curves,” Acta Mathematica Sinica, vol. 20, no. 3, pp. 543–550, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
10. D. Pei and T. Sano, “The focal developable and the binormal indicatrix of a nonlightlike curve in Minkowski 3-space,” Tokyo Journal of Mathematics, vol. 23, no. 1, pp. 211–225, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
11. K. L. Duggal, “Constant scalar curvature and warped product globally null manifolds,” Journal of Geometry and Physics, vol. 43, no. 4, pp. 327–340, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
12. A. Ferrández, A. Giménez, and P. Lucas, “Geometrical particle models on 3D null curves,” Physics Letters B, vol. 543, no. 3-4, pp. 311–317, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
13. A. Ferrández, A. Giménez, and P. Lucas, “Null helices in Lorentzian space forms,” International Journal of Modern Physics A, vol. 16, no. 30, pp. 4845–4863, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
14. L. P. Hughston and W. T. Shaw, “Classical strings in ten dimensions,” Proceedings of the Royal Society, vol. 414, no. 1847, pp. 423–431, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
15. L. P. Hughston and W. T. Shaw, “Constraint-free analysis of relativistic strings,” Classical and Quantum Gravity, vol. 5, no. 3, pp. L69–L72, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
16. J. Samuel and R. Nityananda, “Transport along null curves,” Journal of Physics A, vol. 33, no. 14, pp. 2895–2905, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
17. G. Tian and Z. Zheng, “The second variation of a null geodesic,” Journal of Mathematical Physics, vol. 44, no. 12, pp. 5681–5687, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
18. H. Urbantke, “Local differential geometry of null curves in conformally flat space-time,” Journal of Mathematical Physics, vol. 30, no. 10, pp. 2238–2245, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
19. A. Nersessian and E. Ramos, “Massive spinning particles and the geometry of null curves,” Physics Letters B, vol. 445, no. 1-2, pp. 123–128, 1998. View at Publisher · View at Google Scholar
20. A. Nersessian and E. Ramos, “A geometrical particle model for anyons,” Modern Physics Letters A, vol. 14, no. 29, pp. 2033–2037, 1999. View at Publisher · View at Google Scholar
21. K. L. Duggal, “A report on canonical null curves and screen distributions for lightlike geometry,” Acta Applicandae Mathematicae, vol. 95, no. 2, pp. 135–149, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
22. K. L. Duggal, “On scalar curvature in lightlike geometry,” Journal of Geometry and Physics, vol. 57, no. 2, pp. 473–481, 2007. View at Publisher · View at Google Scholar
23. W. B. Bonnor, “Null curves in a Minkowski spacetime,” Tensor (N.S.), vol. 20, pp. 229–242, 1969.
24. Z. Wang and D. Pei, “Singularities of ruled null surfaces of the principal normal indicatrix to a null Cartan curve in de Sitter 3-space,” Physics Letters B, vol. 689, no. 2-3, pp. 101–106, 2010. View at Publisher · View at Google Scholar
25. Z. Wang and D. Pei, “Null Darboux developable and pseudo-spherical Darboux image of null Cartan curve in Minkowski 3-space,” Hokkaido Mathematical Journal, vol. 40, no. 2, pp. 219–240, 2011.
26. J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, UK, 2nd edition, 1992.
27. J. Martinet, Singularities of Smooth Functions and Maps, vol. 58, Cambridge University Press, Cambridge, UK, 1982.
28. G. Wassermann, “Stability of caustics,” Mathematische Annalen, vol. 216, pp. 43–50, 1975. View at Publisher · View at Google Scholar · View at Zentralblatt MATH