Hyperbola - True Anomaly

True Anomaly

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 from a point on the left branch of the hyperbola to the left focal point is

.

Introducing polar coordinates with origin at the left focal point the coordinates relative the canonical coordinate system are

and the equation above takes the form

from which follows that

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

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

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