Editing Ellipsoid (section)

==Standard equation==
The general ellipsoid, also known as triaxial ellipsoid, is a quadratic surface which is defined in [[Cartesian coordinate system|Cartesian coordinates]] as:

:<math>\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1,</math>
where <math>a</math>, <math>b</math> and <math>c</math> are the length of the semi-axes.  

The points <math>(a, 0, 0)</math>, <math>(0, b, 0)</math> and <math>(0, 0, c)</math> lie on the surface. The line segments from the origin to these points are called the principal semi-axes of the ellipsoid, because {{math|''a'', ''b'', ''c''}} are half the length of the principal axes. They correspond to the [[semi-major axis]] and [[semi-minor axis]] of an [[ellipse]].

In [[spherical coordinate system]] for which <math>(x,y,z)=(r\sin\theta\cos\varphi, r\sin\theta\sin\varphi,r\cos\theta)</math>, the general ellipsoid is defined as:

:<math>{r^2\sin^2\theta\cos^2\varphi\over a^2}+{r^2\sin^2\theta\sin^2\varphi \over b^2}+{r^2\cos^2\theta \over c^2}=1,</math>

where <math>\theta</math> is the polar angle and <math>\varphi</math> is the azimuthal angle.

When <math>a=b=c</math>, the ellipsoid is a sphere.

When <math>a=b\neq c</math>, the ellipsoid is a spheroid or ellipsoid of revolution.  In particular, if <math>a = b > c</math>, it is an [[oblate spheroid]]; if <math>a = b < c</math>, it is a [[prolate spheroid]].