Intersection Curves of Implicit and Parametric Surfaces in R³

Mohamed Abdel-Latif Soliman, Nassar Hassan Abdel-All, Soad Ali Hassan, Sayed Abdel-Naeim Badr
DOI: 10.4236/am.2011.28141                

Abstract

We present algorithms for computing the differential geometry properties of Frenet apparatus {t,n,b,κ,τ} and higher-order derivatives of intersection curves of implicit and parametric surfaces in R³ for transversal and tangential intersection. This work is considered as a continuation to Ye and Maekawa [1]. We obtain a classification of the singularities on the intersection curve. Some examples are given and plotted.

Keywords

Geometric Properties, Frenet Frame, Frenet Apparatus, Frenet-Serret Formulas, Surface-Surface Intersection, Transversal Intersection, Tangential Intersection, Dupin Indicatrices

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	X. Ye and T. Maekawa, “Differential Geometry of Intersection Curves of Two Surfaces,” Computer-Aided Geometric Design, Vol. 16, No. 8, September 1999, pp. 767-788. doi:10.1016/S0167-8396(99)00018-7
[2]	C. L. Bajaj, C. M. Hoffmann, J. E. Hopcroft and R. E. Lynch, “Tracing Surface Intersections,” Computer-Aided Geometric Design, Vol. 5, No. 4, November 1988, pp. 285-307. doi:10.1016/0167-8396(88)90010-6
[3]	N. M. Patrikalakis, “Surface-to-Surface Intersection,” IEEE Computer Graphics & Applications, Vol. 13, No. 1, January-February 1993, pp. 89-95. doi:10.1109/38.180122
[4]	T. J. Willmore, “An Introduction to Differential Geometry,” Clarendon Press, Oxford, 1959.
[5]	M. Düldül, “On the Intersection Curve of Three Parametric Hypersurfaces,” Computer-Aided Geometric Design, Vol. 27, No. 1, January 2010, pp. 118-127. doi:10.1016/j.cagd.2009.10.002
[6]	E. Kruppa, “Analytische und Konstruktive Differentialgeometrie,” Springer, Wien, 1957.
[7]	E. Hartmann, “G2 Interpolation and Blending on Surfaces,” The Visual Computer, Vol. 12, No. 4, 1996, pp. 181-192. doi: 10.1007/s003710050057
[8]	G. A. Kriezis, N. M. Patrikalakis and F.-E. Wolter, “Topological and Differential Equation Methods for Surface Intersections,” Computer-Aided Geometric Design, Vol. 24, No. 1, January 1992, pp. 41-55. doi:10.1016/0010-4485(92)90090-W
[9]	R. C. Luo, Y. Ma and D. F. McAllister, “Tracing Tangential Surface-Surface Intersections,” Proceedings Third ACM Solid Modeling Symposium, Salt Lake City, 1995, pp. 255-262. doi:10.1145/218013.218070
[10]	O. Aléssio, “Differential Geometry of Intersection Curves in of three Implicit Surfaces,” Computer-Aided Geometric Design, Vol. 26, No. 4, May 2009, pp. 455-471. doi:10.1016/j.cagd.2008.12.001
[11]	M. P. do Carmo, “Differential Geometry of Curves and Surface,” Prentice Hall, Englewood Cliffs, NJ, 1976.
[12]	J. J. Stoker, “Differential Geometry,” Wiley, New York, 1969.
[13]	D. J. Struik, “Lectures on Classical Differential Geometry,” Addison-Wesley, Reading, 1950.