Ellipse equation

  1. Hi, does exist an easy way to change the center of circle or a ellipse in polar coordinates?

  2. jcsd
  3. HallsofIvy

    HallsofIvy 41,264
    Staff Emeritus
    Science Advisor

    Yes, and it was given in any text book dealing with the conic sections that I have ever seen:
    If the equation of an ellipse centered at (0, 0) is
    [tex]\frac{x^2}{a^2}+ \frac{y^2}{b^2}= 1[/tex]
    then the same ellipse, centered at (a, b) has equation
    [tex]\frac{(x- a)^2}{a^2}+ \frac{(y-b)^2}{b^2}= 1[/tex]
  4. dextercioby

    dextercioby 12,327
    Science Advisor
    Homework Helper

    Now you have to pass from the cartesian eqns that Halls wrote to polar coordinates and you're done.
  5. Yes thanks...but I need it in polar coordinates
  6. arildno

    arildno 11,265
    Science Advisor
    Homework Helper
    Gold Member

    Insert polar representations for "x" and "y", multiply out parentheses and simplify and redefine variables/constants.
    In particular, remember simplifying trig identities, such as, for example:
    1 person likes this.
  7. arildno

    arildno 11,265
    Science Advisor
    Homework Helper
    Gold Member

    Now, HallsofIvy made out a special case, with the center with the same values as the lengths of the semi-axes.

    You shouldn't make that restriction here (call one of the (a,b)-pairs (c,d)-for example).

    To give you the first step on your way, multiplying up and out, we get (with (c,d) centre coordinates):
    [tex]b^{2}r^{2} \cos^{2}\theta+a^{2}r^{2} \sin^{2}\theta-2b^{2}cr \cos\theta-2a^{2} dr\sin\theta=a^{2}b^{2}-c^{2}-d^{2}[/tex]
    There would be various ways to simplify this expression further, and redefing independent constants.

    One very compact way of doing so would be to transform your equation into the following form:
    where the angle "phi" is a phase-shifted version of "gamma"/2 with a fourth constant D to be determined along with A, B and C (gamma being twice the value of "theta")
    Last edited: Mar 29, 2014
Know someone interested in this topic? Share this thead via email, Google+, Twitter, or Facebook

Have something to add?

Draft saved Draft deleted