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Shaded wireframe rendering of an
ellipsoid with a = 3, b = 2, c = 1 (scalene
ellipsoid is a type of quadric surface that is a higher
dimensional analogue of an ellipse. The equation of a
standard axis-aligned ellipsoid body in an xyz-Cartesian
coordinate system is
and b are the equatorial radii (along the x and y axes)
and c is the polar radius (along the z-axis), all of
which are fixed positive real numbers determining the
shape of the ellipsoid.
More generally, a not-necessarily-axis-aligned 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
If all three radii are equal, the solid body is a
sphere; if two radii are equal, the ellipsoid is a
(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
semi-principal axes. These correspond to the semi-major
axis and semi-minor 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.
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.
The mass of an ellipsoid of uniform
is the density.