Normal Lines Drawn to Ellipses and Elliptic Integrals

Consider the ellipse in the standard position, and P, an arbitrary point on the plane. Finding the minimum distance from P to the ellipse is a well-known problem. One can easily show that the minimum distance path lies along a normal line to the ellipse, passing through P. This paper deals with a study of all the normal lines drawn from point P to the ellipse. We have used Mathematica to illustrate and discover several aspects of these normal lines. Mathematica can be used in contemporary mathematical research and education as a computational, visualization, experimentation and conjecture-forming tool. The paper well illustrates such usage via a study of the normal lines to the ellipse. We used Mathematica 3.0 on a Windows 95 platform.