Determining the angle degree of an arc in ellipse? Is it possible to determine the angle in degree of an arc in ellipse by knowing the arc length, ellipse semi-major and semi-minor axis ?
If I have an arc length at the first quarter of an ellipse and I want to know the angle of it, what is the data that I will need it to use it and what is the exact method to use it?
Please take a look at this picture:

Actually I know how to determine the arc length of a ellipse here. but I want to do the obverse.
 A: Parameterization of an ellipse by angle from the center is
$$
\gamma(\phi)=(\cos(\phi),\sin(\phi))\frac{ab}{\sqrt{a^2\sin^2(\phi)+b^2\cos^2(\phi)}}
$$
and
$$
\left|\gamma'(\phi)\right|=ab\sqrt{\frac{a^4\sin^2(\phi)+b^4\cos^2(\phi)}{\left(a^2\sin^2(\phi)+b^2\cos^2(\phi)\right)^3}}
$$
and integrating $\left|\gamma'(\phi)\right|$ gets extremely messy.
So instead, we use $\theta$, where
$$
b\tan(\theta)=a\tan(\phi)
$$
Then
$$
\gamma(\theta)=(a\cos(\theta),b\sin(\theta))
$$
and
$$
\left|\gamma'(\theta)\right|=\sqrt{a^2\sin^2(\theta)+b^2\cos^2(\theta)}
$$
Now, integrating $\left|\gamma'(\theta)\right|$ is a lot simpler.
$$
\int\left|\gamma'(\theta)\right|\,\mathrm{d}\theta
=b\,\mathrm{EllipticE}\left(\theta,\frac{b^2-a^2}{b^2}\right)
$$
However, to go from from arc length to angle, we still need to invert the Elliptic integral.

Solution of the problem given using Mathematica
Here we get the solution to 30 places.
In[]=  Phi[a_,b_,s_,opts:OptionsPattern[]] := Block[ {t}, t+ArcTan[(b-a)Tan[t]/(a+b Tan[t]^2)] /. FindRoot[b EllipticE[t,(b^2-a^2)/b^2] == s, {t, 0}, opts]]

In[]=  Phi[4, 2.9`32, 3.31`32, WorkingPrecision->30]

Out[]= 0.87052028743193111752524449959

Thus, $\phi\doteq0.87052028743193111752524449959\text{ radians}$.
Let me describe the algorithm a bit.
FindRoot[b EllipticE[t,(b^2-a^2)/b^2] == s, {t, 0}, opts]
inverts the elliptic integral to get $\theta$ from $a$, $b$, and $s$.
Now, we want to find $\phi$ so that $b\tan(\theta)=a\tan(\phi)$. However, simply using $\tan^{-1}\left(\frac ba\tan(\theta)\right)$ will only return a result in $\left(-\frac\pi2,\frac\pi2\right)$. To get the correct value, we use the relation
$$
\begin{align}
\tan(\phi-\theta)
&=\frac{\tan(\phi)-\tan(\theta)}{1+\tan(\phi)\tan(\theta)}\\
&=\frac{\frac ba\tan(\theta)-\tan(\theta)}{1+\frac ba\tan(\theta)\tan(\theta)}\\
&=\frac{b\tan(\theta)-a\tan(\theta)}{a+b\tan(\theta)\tan(\theta)}\\
\phi
&=\theta+\tan^{-1}\left(\frac{(b-a)\tan(\theta)}{a+b\tan^2(\theta)}\right)
\end{align}
$$
This is why we have
t+ArcTan[(b-a)Tan[t]/(a+b Tan[t]^2)]
A: The nice answer by robjohn demands a comment. His result for the integral of the arc-length of the elipse (with major axis $2a$ and minor axis $2b$) is $b\,E(\theta,1-a^2/b^2)$, but this is only a quarter of the complete elliptic circumference. Also, for those more interested readers, it is easy to show that $b\,E(\theta,1-a^2/b^2) = a\,E(\theta,1-b^2/a^2)$, as found at Wolfram World webpage http://mathworld.wolfram.com/Ellipse.html. This change rule is useful for avoiding negative values of parameters. F. M. S. Lima, University of Brasilia (fabio-at-fis.unb.br)
