Re: Finding all tangents for two ellipses

anonymous_at_mathforum.org
Date: 08/16/04 

Date: Mon, 16 Aug 2004 17:24:59 +0000 (UTC)

I would like to point out that your problem can be solved in the same
way as that of intersecting two circles.
Indeed the dual of an ellipse, seen as a locus of points, is the envelope of the ellipse ie, the assemblage of its (tangent) lines.

If the ellipse has point-equation
     (x,y,1)^T * M * (x,y,1) = 0,
          [ A B/2 D/2 ]
where M = [ B/2 C E/2 ] is the 3-by-3 matrix
          [ D/2 E/2 F ]
of the ellipse coefficients, and (x,y,1)^T is the (homogeneous) coordinate vector of any point on the ellipse, then the dual line-equation of its envelope is
     (a,b,c)^T * inv(M) * (a,b,c) = 0,
where inv(M) is the matrix of the envelope of the ellipse and (a,b,c)^T denotes the coordinate vector of any tangent line.

It suffices to solve your problem for the two inverse matrices of the ellipses with a standard algorithm and consider the obtained "intersection points" as "common tangent lines".

I hope it helps!

Pierre

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