Eccentricity (mathematics)

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,

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

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 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

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.)

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