True anomaly of Hyperbola

In the section above it is shown that using the coordinate system in which the equation of the hyperbola takes its canonical form

$\frac{{x}^{2}}{a^{2}} - \frac{{y}^{2}}{b^{2}} = 1$

the distance $r$ from a point $(x\ ,\ y)$ on the left branch of the hyperbola to the left focal point $( -e a\ ,\ 0)$ is

$r = -e x - a\,\!$ .

Introducing polar coordinates $( r\ ,\ \theta)$ with origin at the left focal point the coordinates relative the canonical coordinate system are

$x\ =\ -ae+r \cos \theta$

$y\ =r \sin \theta$

and the equation above takes the form

$r = -e (-ae+r \cos \theta) - a\,\!$

from which follows that

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

This is the representation of the near branch of a hyperbola in polar coordinates with respect to a focal point.

The polar angle $\theta$ of a point on a hyperbola relative the near focal point as described above is called the true anomaly of the point.

Pages