Hyperbola in Polar coordinates | Conic Section

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Friday 5 October 2012

Hyperbola in Polar coordinates

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 towards the origin of the "canonical coordinate system" as illustrated in the figure of the section "True anomaly".

Relative to this coordinate system one has that

$r = \frac{a(e^2-1)}{1+e\cos \theta}$

and the range of the true anomaly $\theta$ is:

$-\arccos {\left(-\frac{1}{e}\right)} < \theta < \arccos {\left(-\frac{1}{e}\right)}$

With polar coordinate relative to the "canonical coordinate system"

$x = R\, \cos t$

$y = R\, \sin t$

one has that

$R^2 =\frac{b^2}{e^2 \cos^2 t -1} \,$

For the right branch of the hyperbola the range of $t$ is:

$-\arccos {\left(\frac{1}{e}\right)} < t < \arccos {\left(\frac{1}{e}\right)}$

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