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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;
- 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:
- e for the eccentricity and c for the linear eccentricity.
- ε for the eccentricity and e for the linear eccentricity.
- 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;
