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Encyclopedia > Eccentricity (mathematics)

(This page refers to eccentricity in mathematics. For other uses, see the disambiguation page eccentricity.) Eccentric is from the Greek for out of the centre, as opposed to concentric, in the centre. ...


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, Mathematics is often defined as the study of topics such as quantity, structure, space, and change. ... In mathematics, a conic section (or just conic) is a curve formed by intersecting a cone (more precisely, a right circular conical surface) with a plane. ...

  • 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: A circle, in Euclidean geometry, is the set of all points at a fixed distance, called the radius, from a fixed point, the centre. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ... A parabola The parabola (from the Greek: παραβολή) is a conic section generated by the intersection of a a right circular conical surface and a plane parallel to a generating straight line of that surface. ... A graph of a hyperbola, where h = k = 0 and a = b = 2. ... A line, or straight line, is, roughly speaking, an (infinitely) thin, (infinitely) long, straight geometrical object, i. ...

e = sqrt{1 - kfrac{b^2}{a^2}}

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. In geometry, the semi-major axis (also semimajor axis) a applies to ellipses and hyperbolas. ... In geometry, the semi-minor axis (also semiminor axis) applies to ellipses and hyperbolas. ...


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{kfrac{a^2}{b^2} - 1}

And is related to the first eccentricity by the equation:

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

Contents


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: Image File history File links Ellipse. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ... In mathematics, an ellipse (from the Greek for absence) is a plane algebraic curve where the sum of the distances from any point on the curve to two fixed points is constant. ...

e = sqrt{1-frac{b^2}{a^2}}

The eccentricity is the ratio of the distance between the foci (F1 and F2) to the major axis; i.e. left ( frac{overline{F_1F_2}}{overline{AB}} right ).


The term linear eccentricity is used for ea.


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. In mathematics, a line segment is a part of a line that is bounded by two end points. ...


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: A graph of a hyperbola, where h = k = 0 and a = b = 2. ... A graph of a hyperbola, where h = k = 0 and a = b = 2. ...

e = sqrt{1+frac{b^2}{a^2}}

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). Section can be: A cross section (in the common sense or the physics sense) In mathematics: A conic section A section of a fiber bundle or sheaf A Caesarean section In UK law, Section 28 In the fictional Star Trek universe, Section 31 A military unit A section (land) is...


External links

  • MathWorld: Eccentricity

  Results from FactBites:
 
eccentricity - Search Results - MSN Encarta (190 words)
Eccentricity, in geometry, a property of a conic section (circle, ellipse, parabola, or hyperbola).
In mathematics, the eccentricity, denoted e or, is a parameter associated with every conic section.
A quantity defined for a conic section which can be given in terms of semimajor a and semiminor axes b.The eccentricity can also be interpreted as the fraction of the distance...
AllRefer.com - eccentricity Information (236 words)
Physics The distance between the center of an eccentric and its axis.
Mathematics The ratio of the distance of any point on a conic section from a focus to its distance from the corresponding directrix.
A circle has an eccentricity of zero; for an ellipse it is less than one; for a parabola it is equal to one; and for a hyperbola it is greater than one.
  More results at FactBites »

 
 

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