Eccentricity (mathematics)

Encyclopedia : E : EC : ECC : Eccentricity (mathematics)

---

In mathematics, eccentricity 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 is greater than zero and less than 1.
• The eccentricity of a parabola is 1.
• The eccentricity of a hyperbola is greater than 1 and less than infinity.
• The eccentricity of a straight line is infinity.

It is given by:

[e = \sqrt}]

Where a is the length of the semimajor axis of the section, b the length of the semiminor axis, and k is equal to +1 for an ellipse, 0 for a parabola, and -1 for a hyperbola.

It is also called the first eccentricity when necessary to distinguish it from the second eccentricity, e', which is sometimes used for algebraic convenience. The second eccentricity is defined as:

[e' = \sqrt - 1}]

And is related to the first eccentricity by the equation:

[1 = (1 - e^2)(1 + e'^2)\,\!]

Ellipse

For any ellipse, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, the eccentricity is given by:

[e = \sqrt}]

The eccentricity is the ratio of the distance between the foci ([F_1] and [F_2]) to the major axis; i.e. [\left ( \frac}} \right )].

The term linear eccentricity is used for [].

Straight Line

A straight line or line segment can be shown as an ellipse with a minor axis of length 0, causing b to be 0. Entering this value of b into the equation of eccentricity for an ellipse gives a value of 1.

Hyperbola

For any hyperbola, where the length of the semi-major axis is a, and where the same of the semi-minor axis is b, eccentricity is given by:

[e = \sqrt}]

Surfaces

The eccentricity of a surface is the eccentricity of a designated section of the surface. For example, on a triaxial ellipsoid, the meridional eccentricity is that of the ellipse formed by a section containing both the longest and the shortest axes (one of which will be the polar axis), and the equatorial eccentricity is the eccentricity of the ellipse formed by a section through the centre, perpendicular to the polar axis (i.e. in the equatorial plane).

External links

• [MathWorld: Eccentricity]

 

---

From Wikipedia, the Free Encyclopedia. Original article here. Support Wikipedia by contributing or donating.
All text is available under the terms of the GNU Free Documentation License See Wikipedia Copyrights for details.

Search Titles

0

1

2

3

4

5

6

7

8

9

A

B

C

D

E

F

G

H

I

J

K

L

M

N

O

P

Q

R

S

T

U

V

W

X

Y

Z

?