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Encyclopedia > Spheroid

In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes. If the ellipse is rotated about its major axis, the surface is called a prolate spheroid (similar to the shape of a rugby ball or cigar). If the minor axis is chosen, the surface is called an oblate spheroid (similar to the shape of the planet Earth or a pancake). Euclid, detail from The School of Athens by Raphael. ... Ellipsoid Elliptic Paraboloid Hyperbolic Paraboloid Hyperboloid of One Sheet Hyperboloid of Two Sheets Cone Elliptic Cylinder Hyperbolic Cylinder Parabolic Cylinder In mathematics a quadric, or quadric surface, is any D-dimensional (hyper-)surface represented by a second-order equation in spatial variables (coordinates). ... An open surface with X-, Y-, and Z-contours shown. ... The ellipse and some of its mathematical properties. ... A Rugby player Rugby football refers to sports descended from a common form of football developed at Rugby School. ... Four cigars of different brands (from top: H. Upmann, Montecristo, Macanudo, Romeo y Julieta) An airtight cigar storage tube and a guillotine-style cutter A cigar is a tightly rolled bundle of dried and fermented tobacco, one end of which is ignited so that its smoke may be drawn into... Oblate also refers to a member of the Roman Catholic religious order of the Missionary Oblates of Mary Immaculate, or in some cases to a lay or religious person who has officially associated himself (or herself) with a monastic community such as the Benedictines for reasons of personal enrichment without... The expression figure of the Earth has various meanings in geodesy according to the way it is used and the precision with which the Earths size and shape is to be defined. ... Pancakes with strawberries A pancake is a batter cake fried in a pan or on a griddle with oil or butter. ...


A spheroid can also be characterised as an ellipsoid having two equal semi-axes (b = c), as represented by the equation 3D rendering of an ellipsoid In mathematics, an ellipsoid is a type of quadric that is a higher dimensional analogue of an ellipse. ... A coordinate axis is one of a set of vectors that defines a coordinate system. ...

frac{x^2}{a^2}+frac{y^2}{b^2}+frac{z^2}{b^2}=1

A prolate spheroid has a semi-minor axis shorter than the semi-major axis (a > b); an oblate spheroid has a semi-minor axis longer than the semi-major axis (a < b) and can resemble a disk. In geometry, a disk is the region in a plane contained inside of a circle. ...

Image:ProlateSpheroid.PNG
Prolate spheroid.
Image:OblateSpheroid.PNG
Oblate spheroid.


The sphere is a special case of the spheroid in which the generating ellipse is a circle. Prolate Spheroid. ... Oblate Spheroid This image was made by AugPi using Mathematica. ... A sphere (< Greek σφαίρα) is a perfectly symmetrical geometrical object. ...


Volume

Prolate spheroid:

  • volume is frac{4}{3}pi a b^2

Oblate spheroid:

  • volume is frac{4}{3}pi a^2 b

where

  • a is the semi-major axis length
  • b is the semi-minor axis length

Surface area

A prolate spheroid has surface area

2pi bleft(b + a frac{arcsin{e}}{e}right).

An oblate spheroid has surface area This article explains the meaning of area as a Physical quantity. ...

pileft(2 a^2 + frac{b^2}{e} logleft(frac{1+e}{1-e}right) right).

Here e is the eccentricity of the ellipse, defined as (This page refers to eccentricity in mathematics. ...

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


where

  • a is the semi-major axis length
  • b is the semi-minor axis length

See also


  Results from FactBites:
 
Spheroid - Wikipedia, the free encyclopedia (191 words)
In mathematics, a spheroid is a quadric surface in three dimensions obtained by rotating an ellipse about one of its principal axes.
A prolate spheroid has a semi-minor axis shorter than the semi-major axis (a > b); an oblate spheroid has a semi-minor axis longer than the semi-major axis (a < b) and can resemble a disk.
The sphere is a special case of the spheroid in which the generating ellipse is a circle.
  More results at FactBites »

 
 

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