# Eccentricity (mathematics)

All types of conic sections, arranged with increasing eccentricity. Note that curvature decreases with eccentricity, and that none of these curves intersect.

In mathematics, the eccentricity, denoted e or ${\displaystyle \varepsilon }$, is a parameter associated with every conic section. It can be thought of as a measure of how much the conic section deviates from being circular.

In particular,

• The eccentricity of a circle is zero.
• The eccentricity of an ellipse which is not a circle is greater than zero but less than 1.
• The eccentricity of a parabola is 1.
• The eccentricity of a hyperbola is greater than 1.

Furthermore, two conic sections are similar (identically shaped) if and only if they have the same eccentricity.

## Definitions

plane section of a cone

Any conic section can be defined as the locus of points whose distances to a point (the focus) and a line (the directrix) are in a constant ratio. That ratio is called the eccentricity, commonly denoted as e.

The eccentricity can also be defined in terms of the intersection of a plane and a double-napped cone associated with the conic section. If the cone is oriented with its axis vertical, the eccentricity is[1]

${\displaystyle e={\frac {\sin \beta }{\sin \alpha }},\ \ 0<\alpha <90^{\circ },\ 0\leq \beta \leq 90^{\circ }\ ,}$

where β is the angle between the plane and the horizontal and α is the angle between the cone's slant generator and the horizontal. For ${\displaystyle \beta =0}$ the plane section is a circle, for ${\displaystyle \beta =\alpha }$ a parabola. (The plane must not meet the vertex of the cone.)

The linear eccentricity of an ellipse or hyperbola, denoted c (or sometimes f or e), is the distance between its center and either of its two foci. The eccentricity can be defined as the ratio of the linear eccentricity to the semimajor axis a: that is, ${\displaystyle e={\frac {c}{a}}}$. (Lacking a center, the linear eccentricity for parabolas is not defined.)

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