Chemistry Reference and Research
           
 
Periodic Table
- standard table
- large table
 
Chemical Elements
- by name
- by symbol
- by atomic number
 
Chemical Properties
 
Chemical Reactions
 
Organic Chemistry
 
Branches of Chemistry
Analytical chemistry
Biochemistry
Computational Chemistry
Electrochemistry
Environmental chemistry
Geochemistry
Inorganic chemistry
Materials science
Medicinal chemistry
Nuclear chemistry
Organic chemistry
Pharmacology
Physical chemistry
Polymer chemistry
Supramolecular Chemistry
Thermochemistry

Spheroid

(Redirected from Prolate)

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).

A spheroid can also be characterised as an ellipsoid having two equal semi-axes , as represented by the equation

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

A prolate spheroid has one semiaxis longer than the other two, (a > b); an oblate spheroid has two equal semiaxes that are longer than the third one(a < b) and can resembles a disk.

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.

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 major axis length
  • b is the minor axis length

Surface area

A prolate spheroid has surface area

\pi\left(2 a^2 + \frac{b^2}{e} \ln\left(\frac{1+e}{1-e}\right) \right).

An oblate spheroid has surface area

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

Here e is the eccentricity of the ellipse, defined as

\left(1-(b^2/a^2)\right)^{1/2}.

01-04-2007 01:16:19
The contents of this article are licensed from Wikipedia.org under the GNU Free Documentation License. How to see transparent copy