<p>The problem can be expressed as the solution of a cubic equation which
gives 1, 2, or 3 real roots.  For the derivation and closed form
solution see Appendix B of <a href="http://arxiv.org/abs/1102.1215" rel="nofollow">Geodesics on an ellipsoid of revolution</a>.  The boundary between 1 and 3 solutions is an astroid.</p>
Find all 4 possible normals to an ellipse
<p><strong>Given a point <em>p</em> exterior to an axially aligned, origin centered ellipse <em>E</em>, find the (upto) four unique normals to <em>E</em> passing through <em>p</em>.</strong></p>

<p>This is not a Mathematica question. Direct computation is too slow; I am willing to sacrifice precision and accuracy for speed.</p>

<p>I have searched the web, but all I found involved overly complex calculations which if implemented directly appear to lack the performance I need. Is there a more "programmatical" way to do this, like using matrices or scaling the ellipse into a circle?</p>