Editing Hyperbola (section)

==In polar coordinates==
[[File:Hyperbel-pold-f-s.svg|thumb|Hyperbola: Polar coordinates with pole = focus]]
[[File:Hyperbel-pold-m-s.svg|thumb|Hyperbola: Polar coordinates with pole = center]]
[[File:Hyperbola polar animation.gif|thumb|Animated plot of Hyperbola by using <math>r = \frac{p}{1 - e \cos \theta}</math>]]

===Origin at the focus===
The polar coordinates used most commonly for the hyperbola are defined relative to the Cartesian coordinate system that has its ''origin in a focus'' and its x-axis pointing toward the origin of the "canonical coordinate system" as illustrated in the first diagram.

In this case the angle <math>\varphi</math> is called '''true anomaly'''.

Relative to this coordinate system one has that

<math display="block">r = \frac{p}{1 \mp e \cos \varphi}, \quad p = \frac{b^2}{a}</math>

and

<math display="block">-\arccos \left(-\frac 1 e\right) < \varphi < \arccos \left(-\frac 1 e\right). </math>

===Origin at the center===
With polar coordinates relative to the "canonical coordinate system" (see second diagram)
one has that

<math display="block">r =\frac{b}{\sqrt{e^2 \cos^2 \varphi -1}} .\,</math>

For the right branch of the hyperbola the range of <math> \varphi </math> is
<math display="block">-\arccos \left(\frac 1 e\right) < \varphi < \arccos \left(\frac 1 e\right).</math>

===Eccentricity===
{{Anchor|Polar coordinate eccentricity}}
When using polar coordinates, the eccentricity of the hyperbola can be expressed as <math>\sec\varphi_\text{max}</math> where <math>\varphi_\text{max}</math> is the limit of the angular coordinate. As <math>\varphi</math> approaches this limit, ''r'' approaches infinity and the denominator in either of the equations noted above approaches zero, hence:<ref name=Casey1885>Casey, John, (1885) [https://archive.org/details/cu31924001520455/page/n219/mode/2up "A treatise on the analytical geometry of the point, line, circle, and conic sections, containing an account of its most recent extensions, with numerous examples"]</ref>{{rp|219}}

<math display="block">e^2 \cos^2 \varphi_\text{max} - 1 = 0</math>

<math display="block">1 \pm e \cos \varphi_\text{max} = 0</math>

<math display="block">\implies e = \sec\varphi_\text{max}</math>