Tangent Common to Two EllipsesDate: 10/24/2002 at 05:53:10 From: Amyn Poonawala Subject: Common tangent to 2 ellipses Hello Dr. Math, I have two general ellipses in space and I want to find the equation of the tangent common to these ellipses. The two ellipses do not intersect each other and are not enclosed within each other. Hence, I believe there will be four such lines that will be common tangents. The question is, how do I find the equations of these lines? Thanks, Amyn Date: 10/24/2002 at 08:29:17 From: Doctor Mitteldorf Subject: Re: Common tangent to 2 ellipses Dear Amyn, The straightforward approach is to solve 4 equations in 4 unknowns to find the two points (x1,y1) and (x2,y2) (on the two ellipses) that are linked. Here is how you can get your 4 equations: 1) Two of the equations are the known equations for the ellipses, and they connect x1 to y1, and connect x2 to y2. 2) From the equations for the ellipses, derive the slope at any point. You can do this with "implicit differentiation." For example, if x^2/a^2 + y^2/b^2 = 1, then 2x dx 2y dy ---------- + ---------- = 0 a^2 b^2 -x*b^2 so that dy/dx = --------- y*a^2 Two more equations come from (a) setting these two slopes equal to each other, and (b) setting them equal to the slope of the line connecting the two points y2-y1 ------- x2-x1 You still have a mess of algebra to do in combining these 4 equations. If you make progress simplifying the equations, will you write back and let me know what you find? - Doctor Mitteldorf, The Math Forum http://mathforum.org/dr.math/ |
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