Editing True anomaly (section)

===From the eccentric anomaly===
The relation between the true anomaly {{mvar|ν}} and the [[eccentric anomaly]] <math>E</math> is:

:<math>\cos{\nu} = {{\cos{E} - e} \over {1 - e \cos{E}}}</math>

or using the [[sine]]<ref>Fundamentals of Astrodynamics and Applications by David A. Vallado</ref> and [[tangent]]:

:<math>\begin{align}
                                    \sin{\nu} &= {{\sqrt{1 - e^2\,} \sin{E}} \over {1 - e \cos{E}}} \\[4pt]
  \tan{\nu} = {{\sin{\nu}} \over {\cos{\nu}}} &= {{\sqrt{1 - e^2\,} \sin{E}} \over {\cos{E} -e}}
\end{align}</math>

or equivalently:

:<math>\tan{\nu \over 2} = \sqrt{{{1 + e\,} \over {1-e\,}}} \tan{E \over 2}</math>

so

:<math>\nu = 2 \, \operatorname{arctan}\left(\, \sqrt{{{1 + e\,} \over {1 - e\,}}} \tan{E \over 2} \, \right)</math>

Alternatively, a form of this equation was derived by <ref>{{cite journal | last1=Broucke | first1=R. | last2=Cefola | first2=P. | title=A Note on the Relations between True and Eccentric Anomalies in the Two-Body Problem | journal=Celestial Mechanics | year=1973 | volume=7 | issue=3 | issn=0008-8714 | doi=10.1007/BF01227859 | pages=388–389 | url=https://ui.adsabs.harvard.edu/abs/1973CeMec...7..388B/abstract | bibcode=1973CeMec...7..388B| s2cid=122878026 }}</ref> that avoids numerical issues when the arguments are near <math>\pm\pi</math>, as the two tangents become infinite. Additionally, since <math>\frac{E}{2}</math> and <math>\frac{\nu}{2}</math> are always in the same quadrant, there will not be any sign problems. 

:<math>\tan{\frac{1}{2}(\nu - E)} = \frac{\beta\sin{E}}{1 - \beta\cos{E}}</math> where <math> \beta = \frac{e}{1 + \sqrt{1 - e^2}} </math>

so
:<math>\nu = E + 2\operatorname{arctan}\left(\,\frac{\beta\sin{E}}{1 - \beta\cos{E}}\,\right)</math>