the second-order elliptical gear

Question

I've across a term "second-order elliptical gear" in an article. The polar radius of gear, $r$, as a function of its polar angle, $θ_1$, is given by:

$$r = \frac{2ab}{(a+b)-(a-b)\cos(2θ_1)}$$

I have also searched on google, but I can't find what I need. This elliptical-gear curve is a kind of the normal ellipse? And what is its equation in Cartesian coordinates?

If you has any information about this elliptical curve, please tell me. Thank you a lot!

Answer 1

This geometric function is not a normal ellipse:

$x = \frac{4 \cos (\theta )}{\cos (2 \theta )+3}$

$y = \frac{4 \sin (\theta )}{\cos (2 \theta )+3}$

Answer 2

If you Google "elliptical gears" you will find a font of information on elliptical and other non-circular gears. As for the Cartesian coordinates, since the curve can be expresses in the complex plane as

$$z=re^{i\theta},\quad \theta\in[0,2\pi]$$

then

$$ x=r\cos\theta\\ y=r\sin\theta $$

as usual.