Ellipsoid

An ellipsoid is a quadric surface such that every planar cross section is either an ellipse, or is empty, or is reduced to a single point. This explains the name, which means "ellipse like". Equivalently, an ellipsoid is a quadric surface that is bounded, which means that it may be enclosed in a sufficiently large sphere.

An ellipsoid has three pairwise perpendicular axes of symmetry which intersect at a center of symmetry, called the center of the ellipsoid. The line segments that are delimited on the axes of symmetry by the ellipsoid are called the principal axes, or simply axes of the ellipsoid. If the three axes have different lengths, the ellipsoid is said to be tri-axial or (rarely) scalene, and the axes are uniquely defined.

If two of the axes have the same length, then the ellipsoid is an "ellipsoid of revolution", also called a spheroid. In this case, the ellipsoid is invariant under a rotation around the third axis, and there are thus infinitely many ways of choosing the two perpendicular axes of the same length. If the third axis is shorter, the ellipsoid is an oblate spheroid, if it is longer, it is prolate spheroid. If the three axes have the same length, the ellipsoid is a sphere.

When choosing a Cartesian coordinate system, such that the origin is the center of the ellipsoid, and the coordinate axes are axes of the ellipsoid, the implicit equation of the ellipsoid has the standard form

where a, b, c are positive real numbers.

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