Results 1 to 3 of 3

Math Help - Intersection of two ellipses

  1. #1
    Newbie
    Joined
    Oct 2008
    Posts
    1

    Intersection of two ellipses

    Hello all,

    I have two ellipses, both share the same origin, one is rotated relative to the other which is aligned with the coordinate system. For the unrotated ellipse, it is in the form Ax^2+By^2+F=0. The rotated ellipse has a similar form, but includes a rotational component of course: Ax^2+By^2+Exy+F=0. I am looking for the intersection points (there should be four) of these two ellipses. It is where the two equations are equal to one another. Setting these two equations equal to one another and simplifying yields something like: (B1-B0)y^2=-E1xy-(A1-A0)x^2. My question is how can I solve for both x and y to find these intersections? Is there a simpler manner? Thank you for any input.
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by againstthebes View Post
    Hello all,

    I have two ellipses, both share the same origin, one is rotated relative to the other which is aligned with the coordinate system. For the unrotated ellipse, it is in the form Ax^2+By^2+F=0. The rotated ellipse has a similar form, but includes a rotational component of course: Ax^2+By^2+Exy+F=0. I am looking for the intersection points (there should be four) of these two ellipses. It is where the two equations are equal to one another. Setting these two equations equal to one another and simplifying yields something like: (B1-B0)y^2=-E1xy-(A1-A0)x^2. My question is how can I solve for both x and y to find these intersections? Is there a simpler manner? Thank you for any input.
    Solve both as quadratics in x (so the solutions are in terms of y and the coefficients), and equate the solutions and solve for y.

    CB
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    Joined
    Mar 2008
    Posts
    934
    Thanks
    33
    Awards
    1
    I think you will end up having to solve a 4th degree equation (a.k.a. a biquadratic). This can be done analytically but is not very pleasant.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. ellipses
    Posted in the Pre-Calculus Forum
    Replies: 7
    Last Post: December 17th 2010, 06:39 PM
  2. Help with Ellipses
    Posted in the Pre-Calculus Forum
    Replies: 4
    Last Post: December 7th 2010, 06:15 PM
  3. Ellipses
    Posted in the Algebra Forum
    Replies: 0
    Last Post: March 13th 2008, 08:54 AM
  4. Ellipses
    Posted in the Pre-Calculus Forum
    Replies: 8
    Last Post: November 24th 2007, 03:40 PM
  5. intersection of the two ellipses
    Posted in the Pre-Calculus Forum
    Replies: 1
    Last Post: February 3rd 2007, 05:06 AM

Search Tags


/mathhelpforum @mathhelpforum