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Shaded wireframe rendering of an
ellipsoid with a = 3, b = 2, c = 1 (scalene
ellipsoid). 
An
ellipsoid is a type of quadric surface that is a higher
dimensional analogue of an ellipse. The equation of a
standard axisaligned ellipsoid body in an xyzCartesian
coordinate system is
where a
and b are the equatorial radii (along the x and y axes)
and c is the polar radius (along the zaxis), all of
which are fixed positive real numbers determining the
shape of the ellipsoid.
More generally, a notnecessarilyaxisaligned ellipsoid
is defined by the equation
where A is a symmetric positive definite matrix and x is
a vector. In that case, the eigenvectors of A define the
principal directions of the ellipsoid and the inverse of
the square root of the eigenvalues are the corresponding
equatorial radii.
If all three radii are equal, the solid body is a
sphere; if two radii are equal, the ellipsoid is a
spheroid:
Sphere;
Oblate spheroid
(diskshaped);
Prolate spheroid
(like a rugby ball);
Scalene ellipsoid ("three unequal sides").
The points (a,0,0),
(0,b,0) and (0,0,c) lie on the surface and the line
segments from the origin to these points are called the
semiprincipal axes. These correspond to the semimajor
axis and semiminor axis of the appropriate ellipses.
Scalene ellipsoids are frequently called "triaxial
ellipsoids", the implication being that all three axes
need to be specified to define the shape.
Volume
The volume of an ellipsoid is given by the formula
Note that this equation
reduces to that of the volume of a sphere when all three
elliptic radii are equal, and to that of an oblate or
prolate spheroid when two of them are equal.
Mass
properties
The mass of an ellipsoid of uniform
density is:
where
is the density.
