Eccentricity

The number equal to the ratio of the distance between any point of a conic section (cf. Conic sections) and a given point (the focus) to that between the same point and a given line (the directrix). Two conic sections having the same eccentricity are similar. For an ellipse the eccentricity is $ e < 1 $( for a circle $ e = 0 $), for a hyperbola $ e > 1 $, and for a parabola $ e = 1 $. For an ellipse and hyperbola the eccentricity can also be defined as the ratio of the distances between the foci and the length of the major axis.

Comments

The quantity defined in the article above is often called the numerical eccentricity. The linear eccentricity equals half the distance between the foci (cf. Focus). See [a1], Chapt. 17; [a2], p. 117.

For the "standard" ellipse, parabola and hyperbola, given, respectively, by the equations $ {x ^ {2} } / {a ^ {2} } + {y ^ {2} } / {b ^ {2} } = 1 $, $ y ^ {2} = 2px $, and $ {x ^ {2} } / {a ^ {2} } - {y ^ {2} } / {b ^ {2} } = 1 $, the eccentricity is equal to, respectively, $ a ^ {-} 1 \sqrt {a ^ {2} - b ^ {2} } $( if $ a > b $), 1 and $ a ^ {-} 1 \sqrt {a ^ {2} + b ^ {2} } $. A focus $ f $ and a corresponding directrix $ D $ for the three cases are given by $ f = ( \sqrt {a ^ {2} - b ^ {2} } , 0) $, $ D : x = a ^ {2} ( a ^ {2} - b ^ {2} ) ^ {-} 1/2 $( if $ a > b $); $ f = (- p/2 , 0) $, $ D : x = p/2 $; $ f = ( \sqrt {a ^ {2} + b ^ {2} } , 0) $, $ D : x = a ^ {2} ( a ^ {2} + b ^ {2} ) ^ {-} 1/2 $. There are two foci for ellipses and hyperbolas and there is one for parabolas.

References

[a1]	M. Berger, "Geometry" , 1–2 , Springer (1987) (Translated from French)
[a2]	H.S.M. Coxeter, "Introduction to geometry" , Wiley (1963)