According to mathematics, the eccentricity of a conic section is a non-negative real number that uniquely characterizes shape. The two conic sections are said to be similar if they have the same eccentricity.

Eccentricity is a measure of how a conic section deviates from being circular. For example,

  • The eccentricity of a circle is zero
  • In the case of an ellipse that is not a circle, the eccentricity is greater than zero but less than 1.
  • For parabola, eccentricity is 1
  • For hyperbola, eccentricity is greater than 1
  • For a pair of lines, eccentricity is infinity.

It can also be defined in the terms of the intersection of a plane and a double-napped cone that is associated with a conic section.

If the cone is oriented with its vertical angle

Eccentricity (e) = sin β / sin α  

Where, 0 < α < 90o , 0 ≤ β ≤ 90o

Here, β is the angle between the plane and the horizontal and α is the angle between the cone’s slant generator and the horizontal.

 

Eccentricities of different conical sections;

  1.  ELLIPSE:

 The eccentricity of an ellipse is less than 1. For instance, if we consider a circle also which has an eccentricity of 0. then the value is either 0 or greater than 0. But if circles are given a special category and are excluded then the value is strictly greater than 0.

 

Name 

Symbol

Terms of a and b

In terms of e

First eccentricity

e

1-b2a2

e

Second eccentricity

e,

a2b2-1

e1-e2

Third eccentricity

e,,=√m

a2-b2a2+b2

e2-e2

Angular eccentricity

 

cos-1ab

sin-1e

 

The formula for eccentricity of an eclipse is represented;

 The eccentricity of an ellipse is, the ratio of the distance c between the centre of the ellipse and each focus to the length of the semimajor axis a.

                                                             Hence, e=ca

In case of an ellipse the eccentricity can grow, it grows and becomes skinnier. The formula for ellipse also shows that every ellipse can be produced by taking a circle in a plane, lifting it up and out, tilting it and projecting it back into the plane 

HYPERBOLA 

In case of a hyperbola the eccentricity can be any real number greater than 1, with no upper bound. The eccentricity of a rectangular is √2.

Three notational conventions are in common use:

  1. e for the eccentricity and c for the linear eccentricity.
  2. ε for the eccentricity and e for the linear eccentricity.
  3. e or ϵ< for the eccentricity and f for the linear eccentricity (mnemonic for half-focal separation).

This article uses the first notation.

Some important values of all the conical sections;