Finding the Coefficients for a Conic in a Rotated System

Key Questions

  • This is a good question, but a little bit vague. I'll try to guess:

    When you say, "a conic," do you (1) have an equation and you want to rotate the axes to get a simpler equation? Or do you (2) have a graph or some points that the curve goes through?

    The general conic equation is a relation between x and y:

    A x^2 + B x y +C y^2 + Dx + Ey + F = 0


    (1) • If you want to rotate this curve to "get rid of" the xy-term, you can substitute coordinates u and v, rotated by angle theta, where

    u = x cos theta + y sin theta
    v = -x sin theta + y cos theta

    or to substitute in the general conic equation,

    x = u cos theta - v sin theta
    y = u sin theta + v cos theta

    If you use the specific angle theta where

    cot (2 theta) = (A-C)/B

    then you will get an equation in u and v that has no xy-term.


    (2) • If you just have some points that the conic goes through, you need five points to determine a unique conic. You then have 5 equations in the 6 variables A, B, C, D, E, F.

    They are scalable so this determines a unique conic. For example x^2 + y^2 - 25 is the same circle as 3x^2 + 3y^2 - 75 = 0.

    Note: If you have six or more points, they may not all lie on the same conic.

    // dansmath strikes again! //

    p.s. Here's the story, and an example, from Stewart's Calculus:
    http://www.stewartcalculus.com/data/ESSENTIAL%20CALCULUS/upfiles/topics/ess_at_13_ra_stu.pdf

  • The general equation of a conic section is in the form

    Ax^2 + Bx + Cy^2 + Dy + E = 0

    When A or C is 0, the equation is that of a parabola.

    When A and C have the same value (including signs), the equation is that of a circle.

    When A and C have the same sign, the equation is that of an ellipse

    When A and C have different signs, the equation is that on a hyperbola

Questions