Distance from Point to Ellipse

Date: 05/19/97 at 18:28:41
From: Chuck Ingrum
Subject: Distance from point to an ellipse

I desire a method to find the (minimum) distance from a point to an 
ellipse (point and ellipse both in the same plane). The point may be 
inside or outside the ellipse. We have been trying to find the tangent 
line to the ellipse that is perpendicular to the line drawn from the 
point to the same tangent point. So far we have failed. Any thoughts 
you have on the subject would be appreciated.

Chuck Ingrum

Date: 05/20/97 at 08:40:03
From: Doctor Jerry
Subject: Re: Distance from point to an ellipse

Hi Chuck,

The slope of the ellipse x^2/a^2+y^2/b^2 = 1 at the pt (x,y) is 
-b^2*x/(a^2*y).

If (X,Y) is a point outside, then the slope of line from (X,Y) to a 
point (x,y) on the ellipse is (Y-y0)/(X-x0).

As you suggested, you want to choose (x0,y0) such that the slope of 
the segment from (x0,y0) to (X,Y) is the negative reciprocal of the 
slope of the ellipse at (x0,y0).  So:

  (Y-y0)/(X-x0) = a^2*y0/(b^2*x0)

Regarding (X,Y) as given, another equation is:

  x0^2/a^2+y0^2/b^2 = 1

You must solve these two equations for x0 and y0 in terms of a, b, X, 
and Y. However, this is not so easy. You can solve the first equation 
for x0 in terms of y0. Substituting into the second equation and 
simplifying gives, after some effort, a fourth degree equation to 
solve for y0.  

One could apply Ferrari's method, which solves quartics exactly, but I 
think it would be a giant mess and probably would not simplify very 
much.

So, my opinion is that though in specific cases a quick solution could 
be found, there is no convenient formula for the general case.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/