# 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) |

**How to Cite This Entry:**

Eccentricity.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Eccentricity&oldid=46789