An ellipsoid is a closed 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
where a 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, an arbitrarily oriented 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:
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.
Any planar cross section passing through the center of an ellipsoid forms an ellipse on its surface, with the possible special case of a circle if the three radii are the same (i.e, the ellipsoid is a sphere) or if the plane is parallel to two radii that are equal.
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