Mathematics - What is eccentricity in terms of ellipses and hyperbolae?

Krishna (last edited 4 years ago) (Expert, Educator @Qalaxia)

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.

FORMULA OF ELLIPSE:

For any ellipse, let a be the length of its semi-major axis and b be the length of its semi-minor axis.

We define a number of related additional concepts (only for ellipses):

The eccentricity of an ellipse is, most simply, the ratio of the distance f between the center of the ellipse and each focus to the length of the semi-major axis a.

{\displaystyle e={\frac {f}{a}}.}

The eccentricity is also the ratio of the semi-major axis a to the distance d from the center to the directrix:

{\displaystyle e={\frac {a}{d}}.}

The eccentricity can be expressed in terms of the flattening g (defined as g = 1 – \frac{b}{a} for semi-major axis a and semi-minor axis b):

{\displaystyle e={\sqrt {g(2-g)}}.}

(Flattening is denoted by f in some subject areas, particularly geodesy.)

Define the maximum and minimum radii $r_\text{max}$ and $r_\text{min}$ as the maximum and minimum distances from either focus to the ellipse (that is, the distances from either focus to the two ends of the major axis). Then with semi-major axis a, the eccentricity is given by

{\displaystyle e={\frac {r_{\text{max}}-r_{\text{min}}}{r_{\text{max}}+r_{\text{min}}}}={\frac {r_{\text{max}}-r_{\text{min}}}{2a}},}

which is the distance between the foci divided by the length of the major axis.

Hyperbola:

Any branch of a hyperbola can also be defined as a curve where the distances of any point from:

a fixed point (the focus), and
a fixed straight line (the directrix) are always in the same ratio.

This ratio is called the eccentricity, and for a hyperbola it is always greater than 1.

The eccentricity (usually shown as the letter e) shows how "uncurvy" (varying from being a circle) the hyperbola is.

On this diagram:

P is a point on the curve,
F is the focus and
N is the point on the directrix so that PN is perpendicular to the directrix.

The eccentricity is the ratio \frac{PF}{PN}, and has the formula:

e = \frac{\sqrt{a^2\ +\ b^2}}{a}

Using "a" and "b" from the diagram above.

https://www.mathopenref.com/ellipseeccentricity.html

https://www.mathsisfun.com/geometry/eccentricity.html