#### What is eccentricity in terms of ellipses and hyperbolae?

That eccentricity is a ratio of some sort...

Anonymous

0

That eccentricity is a ratio of some sort...

Krishna

0

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 and 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.

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.

Read more: https://www3.ul.ie/~rynnet/swconics/eccentricity1.htm

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