Ellipse::their    Surface::spheroid    Oblate::which    Qquad::mathrm    Prolate::volume    Where::''c''


Tri-axial ellipsoid with distinct semi-axis lengths <math>c>b>a</math>
Tri-axial ellipsoid with distinct semi-axes a, b and c
Ellipsoids of revolution (spheroid) with a pair of equal semi-axes (a) and a distinct third semi-axis (c) which is an axis of symmetry. The ellipsoid is prolate (top) or oblate (bottom) as c is greater than or less than a.
Circular sections of an ellipsoid

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

<math>{x^2 \over a^2}+{y^2 \over b^2}+{z^2 \over c^2}=1,</math>

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.

There are four distinct cases of which one is degenerate:

  • <math>a>b>c</math> — tri-axial or (rarely) scalene ellipsoid;
  • <math>a=b>c</math> — oblate ellipsoid of revolution (oblate spheroid);
  • <math>a=b<c</math> — prolate ellipsoid of revolution (prolate spheroid);
  • <math>a=b=c</math> — the degenerate case of a sphere;

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. Older literature uses 'spheroid' in place of 'ellipsoid of revolution'.

Any planar cross section passing through the center 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.)

Ellipsoid sections
Intro  Generalised equations   Parameterization   Volume and surface area  Dynamical properties   Fluid properties    Equations in specific coordinate systems    See also    References    External links   

PREVIOUS: IntroNEXT: Generalised equations