On the Ellipsoid and Plane Intersection Equation
Author(s)
Peter Paul Klein
ABSTRACT
It is well known that the line of intersection of an ellipsoid and a plane is an ellipse. In this note simple formulas for the semi-axes and the center of the ellipse are given, involving only the semi-axes of the ellipsoid, the componentes of the unit normal vector of the plane and the distance of the plane from the center of coordinates. This topic is relatively common to study, but, as indicated in [1], a closed form solution to the general problem is actually very difficult to derive. This is attemped here. As applications problems are treated, which were posed in the internet [1,2], pertaining to satellite orbits in space and to planning radio-therapy treatment of eyes.
It is well known that the line of intersection of an ellipsoid and a plane is an ellipse. In this note simple formulas for the semi-axes and the center of the ellipse are given, involving only the semi-axes of the ellipsoid, the componentes of the unit normal vector of the plane and the distance of the plane from the center of coordinates. This topic is relatively common to study, but, as indicated in [1], a closed form solution to the general problem is actually very difficult to derive. This is attemped here. As applications problems are treated, which were posed in the internet [1,2], pertaining to satellite orbits in space and to planning radio-therapy treatment of eyes.
Cite this paper
P. Klein, "On the Ellipsoid and Plane Intersection Equation," Applied Mathematics, Vol. 3 No. 11, 2012, pp. 1634-1640. doi: 10.4236/am.2012.311226.
P. Klein, "On the Ellipsoid and Plane Intersection Equation," Applied Mathematics, Vol. 3 No. 11, 2012, pp. 1634-1640. doi: 10.4236/am.2012.311226.
References
[1] The Math Forum, “Intersection of Ellipsoid and Plane,” 2007. http://mathforum.org/library/drmath/view/71275.html [2] The Math Forum, “Ellipsoid and Plane Intersection Equation,” 2000. http://mathforum.org/library/drmath/view/51781.html [3] The Math Forum, “Intersection of Hyperplane and an Ellipsoid,” 2007. http://mathforum.org/library/drmath/view/72315.html [4] A. Korn and M. Korn, “Mathematical Handbook for Scientists and Engineers,” Mc Graw-Hill Book Company, Inc., New York, Toronto, London, 1961. [5] C. C. Ferguson, “Intersections of Ellipsoids and Planes of Arbitrary Orientation and Position,” Mathematical Geology, Vol. 11, No. 3, 1979, pp. 329-336. doi:10.1007/BF01034997 [6] M. P. Verma and J. C. Upadhyaya, “On the Electron-Ion Interaction in Hexagonal and Tetragonal Metals,” Journal of Physics F: Metal Physics, Vol. 1, No. 5, 1971, pp. 618620. doi:10.1088/0305-4608/1/5/315 [7] The Math Forum, “Equation of an Ellipse in 3-Space,” 2003. http://mathforum.org/library/drmath/view/63373.html
[1] The Math Forum, “Intersection of Ellipsoid and Plane,” 2007. http://mathforum.org/library/drmath/view/71275.html [2] The Math Forum, “Ellipsoid and Plane Intersection Equation,” 2000. http://mathforum.org/library/drmath/view/51781.html [3] The Math Forum, “Intersection of Hyperplane and an Ellipsoid,” 2007. http://mathforum.org/library/drmath/view/72315.html [4] A. Korn and M. Korn, “Mathematical Handbook for Scientists and Engineers,” Mc Graw-Hill Book Company, Inc., New York, Toronto, London, 1961. [5] C. C. Ferguson, “Intersections of Ellipsoids and Planes of Arbitrary Orientation and Position,” Mathematical Geology, Vol. 11, No. 3, 1979, pp. 329-336. doi:10.1007/BF01034997 [6] M. P. Verma and J. C. Upadhyaya, “On the Electron-Ion Interaction in Hexagonal and Tetragonal Metals,” Journal of Physics F: Metal Physics, Vol. 1, No. 5, 1971, pp. 618620. doi:10.1088/0305-4608/1/5/315 [7] The Math Forum, “Equation of an Ellipse in 3-Space,” 2003. http://mathforum.org/library/drmath/view/63373.html