Eccentricity
A conic has foci and directrices : ellipse for , parabola for , hyperbola for , with polar form around a focus.
Definition 4.27 (Eccentricity)
The eccentricity is a non-negative parameter. The eccentricity and scale properties of a conic section satisfy
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the foci of a conic are ;
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the directrices are the vertical lines .
Focus-directrix property
A conic is the set of points whose distance from the focus equals times the distance from the closest directrix (for the other directrix is taken):
- : an ellipse, , equivalent to with semi-major axis and semi-minor axis ; for it is a circle of radius .
- : a hyperbola, , equivalent to with .
- : a parabola, , equivalent to .
Polar form about a focus
Let satisfy equal to the distance from focus to directrix, so . Polar coordinates centred on a focus turn the focus-directrix condition into
Hence an ellipse () has , a hyperbola () has , and a parabola () has .
Related
Stated in
- Definition 4.27 (Eccentricity)§4.7.4 Focus-Directrix Property of Conics