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Intersection between rotated & translated ellipse and line

  1. Sep 5, 2014 #1
    I have a rotated ellipse, not centered at the origin, defined by x,y,a,b and angle.
    Then I have a segment defined by two points x1,y1 and x2,y2
    Is there a quick way to find the intersection points?

    I used wolfram alpha equation solver, I tried to insert the equation of a line into the one of a standard non rotated, non translated ellipse,

    10how40.gif

    and resolving for x this is the result

    15qw704.png

    which is nice.

    Then I took the equation of a rotated and translated ellipse

    vngnwi.png

    and this is the result

    2rnzx1h.png
    6p5bat.png

    Which is obviously impractical, can anyone suggest a different method?
     
  2. jcsd
  3. Sep 5, 2014 #2

    Simon Bridge

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    Do it by hand rather than use the calculator.
    You can either -
    1. Solve for one variable in the line equation and sub into the ellipse equation - solve for the other coordinate ... you know: as usual.

    2. transform the ellipse into standard form, apply the same transformation to the line, find solutions for the transformed system, apply the inverse transformation to the solution.
     
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