Finding the equation of an ellipse for modeling a physical system

Question

I'm trying to model a physical system in parameter space and I got the following equations: $$ x=P[a\cos(wt)+b\sin(wt)] $$ $$ y=Q\sin(wt) $$ all constants are known parameters. I'm having extreme difficulty attempting to get this into a form of an ellipse either in $$(x^2/\alpha^2)+(y^2/\beta^2)=1$$ Any advice?

Answer 1

Yes; you cannot write your curve as $\frac{x^2}{\alpha^2}+\frac{y^2}{\beta^2}=1$, because your ellipse is not parallel to the $x$- and $y$-axes.

To obtain a Cartesian description of your curve, first note that $$\sin{\omega t}=\frac{y}{Q}$$ Thus $$\cos{\omega t}=\frac{\frac{x}{P}-\frac{by}{Q}}{a}$$ Squaring, $$1=\sin^2{(\omega t)}+\cos^2{(\omega t)}=\left(\frac{y}{Q}\right)^2+\left(\frac{\frac{x}{P}-\frac{by}{Q}}{a}\right)^2$$