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I've got a function, $r(\theta)$, of the radius of an ellipse relative to one focus of the ellipse: $$ r(\theta) = \frac{l}{1 - e\cos \theta} $$ where $e$ is the eccentricity and $l$ is the semi-latus rectum.

I've also found an equation for the arc length from $\theta = 0$ as a function of time, $L(t)$. ($L(t)$ is a bit of a mess, so I'll leave it out unless someone wants to see it).

I'm trying to find $\theta$ as a function of $t$, so that I can use that in conjunction with my ellipse equation to find a parametric form of the ellipse, $\bigl(r(t),\,\theta(t)\bigr)$. The only way that I can think to do this is to solve the equation $$ L(t) = \int_0^t \!\sqrt{r^2 + \left(\frac{\mathrm{d}r}{\mathrm{d}\theta}\right)^2}\mathrm\;{d}\theta $$ for $\theta$. However, this is proving to be beyond both my and Mathematica's abilities. Is there a better way to get $\theta(t)$?

Rose Kunkel
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  • The question's still a bit on the vague side. Do you know the usual polar form for an ellipse? – dfeuer Oct 24 '13 at 16:19
  • What can I add to make the question more specific? – Rose Kunkel Oct 24 '13 at 19:35
  • Well, time only makes sense if something physical is going on. Is it? What exactly is your goal? Yes, revealing $L$ would probably be a good idea, if you say where it comes from and what it means. – dfeuer Oct 24 '13 at 22:38
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    There is a slight difference to the question @MvG linked to. Here the radius is given as measured relative to one focal point. There the radius is given as measure relative to the center. – Willie Wong Jan 28 '15 at 17:37
  • @WillieWong: I agree that this is not exactly the same problem. However, once that problem is solved, one can convert the solution to the focus-centered system. This post of mine for example has a line about the relation between eccentric anomaly $\varphi$ (for circle from center, which is easier to find than actual angle, since $E(\varphi,|,m)$ uses it) and true anomaly $\theta$ (from focus): $$\tan\frac \varphi2=\sqrt{\frac{1-e}{1+e}}\tan\frac\theta2$$ – MvG Jan 28 '15 at 17:47

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