Elliptic polar coordinate

Question

Does anybody know why the elliptic polar coordinate of the ellipse is : $$ \begin{cases} x=a \ r\cos\theta \\ \\ y=b \ r\sin\theta \end{cases} $$

Answer 1

The general equation of an ellipse centerd at the origin and focus on the same Axis is $$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$$ If you make the substitutio $X=x/a$, $Y=y/b$, you'll get $$X^2+Y^2=1$$ that is, the unit circumpherence. Using standard polar coordinates here, $$X=r\cos\theta,\ Y=r\sin\theta.$$ Finally, undo the substitution and you'll get your Elliptic Polar Coordinates.