Volume 2, Issue 2, June 2017, Page: 38-42
On the Intersection of a Hyperboloid and a Plane
Sebahattin Bektas, Faculty of Engineering, Geomatics Engineering, Ondokuz Mayis University, Samsun, Turkey
Received: Jan. 19, 2017;       Accepted: Feb. 7, 2017;       Published: Mar. 1, 2017
DOI: 10.11648/j.dmath.20170202.12      View  1623      Downloads  82
The intersection topic is quite popular at an interdisciplinary level. It can be the friends of geometry, geodesy and others. The curves of intersection resulting in this case are not only ellipses but rather all types of conics: ellipses, hyperbolas and parabolas. In text books of mathematics usually only cases are treated, where the planes of intersection are parallel to the coordinate planes. Here the general case is illustrated with intersecting planes which are not necessarily parallel to the coordinate planes. We have developed an algorithm for intersection of a hyperboloid and a plane with a closed form solution. To do this, we rotate the hyperboloid and the plane until inclined plane moves parallel to the XY plane. In this situation, the intersection ellipse and its projection will be the same. This study aims to show how to obtain the center, the semi-axis and orientation of the intersection curve.
Hyperboloid, Intersection, 3D Reverse Transformation, Plane
To cite this article
Sebahattin Bektas, On the Intersection of a Hyperboloid and a Plane, International Journal of Discrete Mathematics. Vol. 2, No. 2, 2017, pp. 38-42. doi: 10.11648/j.dmath.20170202.12
Copyright © 2017 Authors retain the copyright of this article.
This article is an open access article distributed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0/) which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Bektas, S, (2014) Orthogonal Distance From An Ellipsoid, Boletim de Ciencias Geodesicas, Vol. 20, No. 4 ISSN 1982-2170 http://dx.doi.org/10.1590/S1982-217020140004000400053, 2014
Bektas, S, (2015a), Least squares fitting of ellipsoid using orthogonal distances, Boletim de Ciencias Geodesicas, Vol. 21, No. 2 ISSN 1982-2170, http://dx.doi.org/10.1590/S1982-21702015000200019.
Bektas, S, (2015b), Geodetic Computations on Triaxial Ellipsoid, International Journal of Mining Science (IJMS) Volume 1, Issue 1, June 2015, PP 25-34 www.arcjournals.org ©ARC Page | 25.
Coxeter, H. S. M. (1961) Introduction to Geometry, p. 130, John Wiley & Sons.
David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999) Geometry, pp. 39–41 Cambridge University Press.
C. C. Ferguson, (1979), Intersections of Ellipsoids and Planes of Arbitrary Orientation and Position, Mathematical Geology, Vol. 11, No. 3, pp. 329-336. doi:10.1007/BF01034997.
Hawkins T. (2000) Emergence of the Theory of Lie Groups: an essay in the history of mathematics, 1869—1926, §9.3 "The Mathematization of Physics at Göttingen", see page 340, Springer ISBN 0-387-98963-3.
Heinrich, L (1905). Nichteuklidische geometrie. Vol. 49. GJ Göschen, 1905.
Klein, P. (2012), On the Hyperboloid and Plane Intersection Equation, Applied Mathematics, Vol. 3 No. 11, pp. 1634-1640. doi: 10.4236/am.2012.311226.
Porteous, R (1995) Clifford Algebras and the Classical Groups, pages 22, 24 & 106, Cambridge University Press ISBN 0-521-55177-3.
Shifrin, Ted; Adams, Malcolm (2010), Linear Algebra: A Geometric Approach (http://books.google.com/books?id=QwHcZ7cegD4C&pg=PA32) (2nd ed.), Macmillan, p. 32, ISBN 9781429215213.
Story, William E. (1882) “On non-Euclidean properties of conics”. American Journal of Mathematics 5, no. 1 (1882): 358–381.
Wilhelm, B. (1948) Analytische Geometrie, Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt.
H. Liebmann, Nichteuklidische Geometrie, B. G. Teubner, Leipzig, 1905.
URL 1 The Math Forum, “Ellipsoid and Plane Intersection Equation,” 2000. http://mathforum.org/library/drmath/view/51781.html, Accessed 26 March 2016.
URL 2 The Math Forum, “Intersection of Hyperplane and an Ellipsoid,” 2007. http://mathforum.org/library/drmath/view/72315.html, Accessed 26 March 2016.
Browse journals by subject