One can derive a parametrization for ellipse in polar coordinates (origo at one of the focal points)
$$\varphi(t) = ct$$ $$r(\varphi) = \frac {k+1}{k+\cos(\varphi)} $$ where for width of ellipse $w$: $$k = 1+\frac{2}{w-2} = \frac w {w-2}$$ However, when I plot it, sampling from $t$ linearly, I get very unevenly spaced sampling points.
How can I modify my parametrization to achieve same "speed" of linear sampling along the ellipse boundary for small $\varphi$ as for "large" $\varphi$ ($\approx \pi$)?
Answers which don't use calculus/analysis are of extra interest (as I think the pre-Newtonian astronomers did not have access to such advanced tools), but I won't forbid anyone to use it.