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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.

mathreadler
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    If you want to simulate orbital motion then you also need to take into account that the orbital speed varies (as described by Kepler and observed before him). If you want equal distance between consecutive points then you are asking a lot. Because you run into elliptic integrals. There is no formula (in terms of elementary functions) for the arc length of (a fraction of) an ellipse. – Jyrki Lahtonen Jul 04 '18 at 08:11
  • @JyrkiLahtonen I don't want to simulate it right now (but maybe later!), just try to plot it with evenly spaced points along the contour. What, elliptic integral from this seemingly easy question? That makes me even more curious. – mathreadler Jul 04 '18 at 08:13
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    You might look at this article – Robert Israel Jul 04 '18 at 08:21
  • Oh wow, thank you prof. Robert. – mathreadler Jul 04 '18 at 08:21

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