# How to compute uniformly distributed points on an ellipse

The ellipse can be parametrized in polar coordinates by

$$r(\theta)=\frac{1}{a+\cos\theta}$$

up to a scaling factor, and $a>1$. Suppose we measure $S$, the distance along the ellipse from the $\theta=0$ point, where $r$ is minimal. The metric gives small changes of S with respect to $\theta$:

$$dS^2=dr^2+r^2d\theta^2$$

$$dS=\sqrt{r^2+\left(\frac{dr}{d\theta}\right)^2}d\theta.$$

and finally to

$$dS=\frac{\sqrt{a^2+2 a\cos\theta+1}}{(a+\cos\theta)^2}d\theta.$$

Integrating we get

$$S(\theta)=\int_0^\theta \frac{\sqrt{a^2+2 a\cos{u}+1}}{(a+\cos{u})^2}du=F(\theta)$$

Now we have to generate a set $\{s_i\}$ of uniformly distributed numbers over $[0,1)$ and solve

$$Ls_i=F(\theta_i)$$

where $L$ is the total length aroung the ellipse. However, the above integral involves elliptic functions and the inversion is not possible analytically.

Is there any other way of doing this, perhaps geometrically before setting up some kind of root finding method?

Thanks a bunch!