On the Intersection of a Hyperboloid and a Plane

Sebahattin Bektas

doi:doi:10.11648/j.dmath.20170202.12

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On the Intersection of a Hyperboloid and a Plane

Sebahattin Bektas

Published in International Journal of Discrete Mathematics (Volume 2, Issue 2)

Received: 19 January 2017 Accepted: 7 February 2017 Published: 1 March 2017

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Abstract

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.

Published in	International Journal of Discrete Mathematics (Volume 2, Issue 2)
DOI	10.11648/j.dmath.20170202.12
Page(s)	38-42
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2017. Published by Science Publishing Group

Keywords

Hyperboloid, Intersection, 3D Reverse Transformation, Plane

References

[1]	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
[2]	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.
[3]	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.
[4]	Coxeter, H. S. M. (1961) Introduction to Geometry, p. 130, John Wiley & Sons.
[5]	David A. Brannan, M. F. Esplen, & Jeremy J Gray (1999) Geometry, pp. 39–41 Cambridge University Press.
[6]	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.
[7]	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.
[8]	Heinrich, L (1905). Nichteuklidische geometrie. Vol. 49. GJ Göschen, 1905.
[9]	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.
[10]	Porteous, R (1995) Clifford Algebras and the Classical Groups, pages 22, 24 & 106, Cambridge University Press ISBN 0-521-55177-3.
[11]	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.
[12]	Story, William E. (1882) “On non-Euclidean properties of conics”. American Journal of Mathematics 5, no. 1 (1882): 358–381.
[13]	Wilhelm, B. (1948) Analytische Geometrie, Kapital V: "Quadriken", Wolfenbutteler Verlagsanstalt.
[14]	H. Liebmann, Nichteuklidische Geometrie, B. G. Teubner, Leipzig, 1905.
[15]	URL 1 The Math Forum, “Ellipsoid and Plane Intersection Equation,” 2000. http://mathforum.org/library/drmath/view/51781.html, Accessed 26 March 2016.
[16]	URL 2 The Math Forum, “Intersection of Hyperplane and an Ellipsoid,” 2007. http://mathforum.org/library/drmath/view/72315.html, Accessed 26 March 2016.

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Sebahattin Bektas. (2017). On the Intersection of a Hyperboloid and a Plane. International Journal of Discrete Mathematics, 2(2), 38-42. https://doi.org/10.11648/j.dmath.20170202.12

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Sebahattin Bektas. On the Intersection of a Hyperboloid and a Plane. Int. J. Discrete Math. 2017, 2(2), 38-42. doi: 10.11648/j.dmath.20170202.12

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Sebahattin Bektas. On the Intersection of a Hyperboloid and a Plane. Int J Discrete Math. 2017;2(2):38-42. doi: 10.11648/j.dmath.20170202.12

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@article{10.11648/j.dmath.20170202.12,
  author = {Sebahattin Bektas},
  title = {On the Intersection of a Hyperboloid and a Plane},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {2},
  pages = {38-42},
  doi = {10.11648/j.dmath.20170202.12},
  url = {https://doi.org/10.11648/j.dmath.20170202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170202.12},
  abstract = {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.},
 year = {2017}
}

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TY  - JOUR
T1  - On the Intersection of a Hyperboloid and a Plane
AU  - Sebahattin Bektas
Y1  - 2017/03/01
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170202.12
DO  - 10.11648/j.dmath.20170202.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 38
EP  - 42
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170202.12
AB  - 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.
VL  - 2
IS  - 2
ER  -

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Author Information

Sebahattin Bektas

Faculty of Engineering, Geomatics Engineering, Ondokuz Mayis University, Samsun, Turkey

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APA Style

Sebahattin Bektas. (2017). On the Intersection of a Hyperboloid and a Plane. International Journal of Discrete Mathematics, 2(2), 38-42. https://doi.org/10.11648/j.dmath.20170202.12

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ACS Style

Sebahattin Bektas. On the Intersection of a Hyperboloid and a Plane. Int. J. Discrete Math. 2017, 2(2), 38-42. doi: 10.11648/j.dmath.20170202.12

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AMA Style

Sebahattin Bektas. On the Intersection of a Hyperboloid and a Plane. Int J Discrete Math. 2017;2(2):38-42. doi: 10.11648/j.dmath.20170202.12

Copy | Download

@article{10.11648/j.dmath.20170202.12,
  author = {Sebahattin Bektas},
  title = {On the Intersection of a Hyperboloid and a Plane},
  journal = {International Journal of Discrete Mathematics},
  volume = {2},
  number = {2},
  pages = {38-42},
  doi = {10.11648/j.dmath.20170202.12},
  url = {https://doi.org/10.11648/j.dmath.20170202.12},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.dmath.20170202.12},
  abstract = {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.},
 year = {2017}
}

Copy | Download

TY  - JOUR
T1  - On the Intersection of a Hyperboloid and a Plane
AU  - Sebahattin Bektas
Y1  - 2017/03/01
PY  - 2017
N1  - https://doi.org/10.11648/j.dmath.20170202.12
DO  - 10.11648/j.dmath.20170202.12
T2  - International Journal of Discrete Mathematics
JF  - International Journal of Discrete Mathematics
JO  - International Journal of Discrete Mathematics
SP  - 38
EP  - 42
PB  - Science Publishing Group
SN  - 2578-9252
UR  - https://doi.org/10.11648/j.dmath.20170202.12
AB  - 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.
VL  - 2
IS  - 2
ER  -

Copy | Download