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Points on two ellipses with identical tangent lines

  1. Aug 7, 2010 #1
    Hi, I'm trying to get this working for a program I'm making. I've been working on this for a while, but I can't seem to figure it out.

    I have multiple rotated ellipses. Imagine you took a rubber band and stretched it around the ellipses. The rubber band would follow the curve of the outside of an ellipse until it reached a point on the ellipse whose tangent line was the same as the tangent line of another point on the next ellipse. What I need to figure out is where those points are.

    For each ellipse, I know: center x, center y, semimajor axis, semiminor axis, and the amount by which it has been rotated. So any ellipse E has known variables x, y, a, b, and theta.

    I don't have equations for the ellipses, all I have are those variables.

    How do I go about solving this?

    Thanks in advance!
    -Zippy Dee
  2. jcsd
  3. Aug 8, 2010 #2
    I assume you can write the cartesian equation of the ellipse (if you don't, I'll explain it in the next post). This is of the form

    [tex]A_1x_1^2+B_1y_1^2+C_1x_1y_1+D_1x_1+E_1y_1+F_1=0[/tex] (1)

    and the same for the other ellipse:

    [tex]A_2x_2^2+B_2y_2^2+C_2x_2y_2+D_2x_2+E_2y_2+F_2=0[/tex] (2)

    where all the A,B,... are known numbers.

    Now write another two equations:


    and the same for the second one:


    If you collect terms, you can write the last two equations as


    and for the second one


    Now put

    [tex]G_1=G_2[/tex] (4)


    [tex]H_1=H_2[/tex] (3)

    You have to solve the system composed of equation (1), (2), (3), (4). You will get the four unknowns x_1, y_1, x_2 and y_2. Typically (but not always) you will find four 4-uples of solutions.
  4. Aug 8, 2010 #3
    Thank you. That makes a lot of sense! However, I am not sure how to write the Cartesian equations for the ellipses. Could you explain how to do that?
  5. Aug 8, 2010 #4
    Start vriting the canonical equation of an ellipse centered at the origin and "unrotated":


    Then rotate it (clockwise) by the angle theta. This means you have to do the transformation

    [tex]x\rightarrow x\cos\theta-y\sin\theta[/tex]
    [tex]y\rightarrow x\sin\theta+y\cos\theta[/tex]



    And finally, put the center of the ellipse in the right place:

    [tex]x\rightarrow x+x_C[/tex]
    [tex]y\rightarrow y+y_C[/tex]

    that is,


    (I won't expand this for you! :mad:)

    Other questions? o:)
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