Where does this equation for an ellipse come from?

[davidbenari]

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:

$\frac{x}{x_o}^2+\frac{y}{y_o}^2-\frac{2xy}{x_oy_o}\cos\delta=\sin^2\delta$

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.

Thanks.

 

[andrewkirk]

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.

 

[davidbenari]

Nice, i'll try it in a while. How would time dependence vanish though?

 

[davidbenari]

Oh I think I see it... The arguments would be subtracted, right?

 

[andrewkirk]

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 ) that a parametric equation matching the given one will suggest itself once an equation for the path has been obtained.