1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Where does this equation for an ellipse come from?

  1. Oct 25, 2015 #1
    I'm reading the professors notes and he gives this general equation for the ellipse. The professor has already been mistaken in some of his notes so I wanted you to help me validate what he's saying, as I can't prove the equation.

    Suppose we have the vector ##\mathbf{r}=\big(x_o \cos(-\omega t + \phi_x), y_o \cos(-\omega t +\phi_y)\big)##

    Then, he says that the general equation for the path is:


    where ##\delta=\phi_y-\phi_x##.

    So my question was where does this equation come from? How can I derive it? I know these type of equations are tedious to prove so it's okay if you give me a rough outline, or point me towards a source which does go through it. Or at least tell me you attest to its validity. I can't find the equation elsewhere and I haven't been able to prove it myself.

  2. jcsd
  3. Oct 25, 2015 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    If ##\phi_x-\phi_y## is not an odd multiple of ##\frac{\pi}{2}## the ##xy## term will be nonzero, so the ellipse axes will be rotated relative to the x and y axes.

    Start with an x-y coordinate system and then consider a second coordinate system X-Y whose origin is at (u,v) in the x,-y coordinates and whose X and Y axes are angle ##\theta## to the anticlockwise direction from the x and y axes.

    Consider an ellipse centred at (u,v) with semi-major and semi-minor axes of a and b aligned with the X and Y axes. The equation of that ellipse is ##\frac{X^2}{a^2}+\frac{Y^2}{b^2}=1##.

    With a bit of messy trigonometry you should be able to convert that into an equation in x-y coordinates. Then match terms to the given equation and work out what ##\phi_x,\phi_y,x_0,y_0## represent in the diagram.
  4. Oct 25, 2015 #3
    Nice, i'll try it in a while. How would time dependence vanish though?
  5. Oct 25, 2015 #4
    Oh I think I see it... The arguments would be subtracted, right?
  6. Oct 25, 2015 #5


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    The equation without time in it is an equation of a 'path', which differs from a 'curve' in that it is not parameterised. It simply specifies the set of points that are traced out by a particle following the curve. The equation with t in it is a parametrised equation, which shows how the path is traversed over time. A simple analog is the circle, whose parametric equation is ##x=r\cos t,\ y=\sin t## and the equation for the path traced out is ##x^2+y^2=1##.

    I expect (but do not guarantee :wink:) that a parametric equation matching the given one will suggest itself once an equation for the path has been obtained.
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Similar Discussions: Where does this equation for an ellipse come from?