# 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 (for a circle ), for a hyperbola , and for a parabola . 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 , , and , the eccentricity is equal to, respectively, (if ), 1 and . A focus and a corresponding directrix for the three cases are given by , (if ); , ; , . 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. A.B. Ivanov (originator),

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Eccentricity&oldid=15306