Distance from Point to EllipseDate: 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/ |
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