Ellipsoids

An ellipsoid is a closed quadric surface that is a three-dimensional analogue of an ellipse. The standard equation of an ellipsoid centered at the origin of a Cartesian coordinate system and aligned with the axes is

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 of length a, b, c. They correspond to the semi-major axis and semi-minor axis of the appropriate ellipses.

One can assume without loss of generality that ${\displaystyle a\geq b\geq c}$. If all three semi-axis are different, then it is a generic tri-axial or (rarely) scalene ellipsoid. If two or three semi-axis are equal (parameter degeneracy), then the ellipsoid is an 'ellipsoid of revolution', which is also called a spheroid. More specifically:

Mathematical literature often uses 'ellipsoid' in place of 'tri-axial ellipsoid'. Scientific literature (particularly geodesy) often uses 'ellipsoid' in place of 'ellipsoid of revolution' and only applies the adjective 'tri-axial' when treating the general case.

Any planar cross section of an ellipsoid forms an ellipse on its surface: this degenerates to a circle for sections normal to the symmetry axis of an ellipsoid of revolution (or all sections when the ellipsoid degenerates to a sphere.)

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