Editing Eccentric anomaly (section)

===From the true anomaly===
The ''[[true anomaly]]'' is the angle labeled <math>\theta</math> in the figure, located at the focus of the ellipse. It is sometimes represented by {{mvar|f}} or {{mvar|v}}. The true anomaly and the eccentric anomaly are related as follows.<ref name=Tsui>{{cite book |first=James Bao-yen |last=Tsui |year=2000 |title=Fundamentals of Global Positioning System receivers: A software approach |edition=3rd |page=48 |publisher=[[John Wiley & Sons]] |isbn=0-471-38154-3 |url=https://books.google.com/books?id=jPRCxNDZqDQC&pg=PA48}}</ref>

Using the formula for {{mvar|r}} above, the sine and cosine of {{mvar|E}} are found in terms of {{mvar|f}}&nbsp;:
:<math>\begin{align}
              \cos E &= \frac{\,x\,}{a} = \frac{\, a e + r \cos f \,}{a} = e + (1 - e \cos E) \cos f \\
  \Rightarrow \cos E &= \frac{\, e + \cos f \,}{1 + e \cos f} \\
              \sin E &= \sqrt{\, 1 - \cos^2 E \;} = \frac{\, \sqrt{\, 1 - e^2 \;} \, \sin f \,}{ 1 + e\cos f } ~.
\end{align}</math>

Hence,
:<math>\tan E = \frac{\, \sin E \,}{\cos E} = \frac{\, \sqrt{\, 1 - e^2 \;} \, \sin f \,}{e + \cos f} ~.</math>

where the correct quadrant for {{mvar|E}} is given by the signs of numerator and denominator, so that {{mvar|E}} can be most easily found using an [[atan2]] function.

Angle {{mvar|E}} is therefore the adjacent angle of a right triangle with hypotenuse <math>\; 1 + e \cos f \;,</math> adjacent side <math>\; e + \cos f \;,</math> and opposite side <math>\;\sqrt{ \, 1 - e^2 \; } \, \sin f \;.</math>

Also,
:<math>\tan\frac{\, f \,}{2} = \sqrt{\frac{\, 1 + e \,}{1 - e}\,}  \,\tan\frac{\, E \,}{2}</math>

Substituting {{math|cos}}&nbsp;{{mvar|E}} as found above into the expression for {{mvar|r}}, the radial distance from the focal point to the point {{math|P}}, can be found in terms of the true anomaly as well:<ref name=Tsui/>
:<math>r = \frac{a \left(\, 1 - e^2 \,\right)}{\,  1 + e \cos f \, } =  \frac{p}{\,  1 + e \cos f \, }\,</math>
where
:<math>\, p \equiv a \left(\, 1 - e^2 \,\right) </math>
is called ''"the semi-latus rectum"'' in classical geometry.