Ellipsoider

Ellipsoider, often rendered in English as ellipsoids, are closed quadric surfaces in three-dimensional Euclidean space. They can be defined as the set of points (x, y, z) that satisfy x^2/a^2 + y^2/b^2 + z^2/c^2 = 1 after a suitable translation and rotation, or equivalently as the image of a sphere under a linear transformation that scales along three principal axes. The principal axes align with the eigenvectors of the associated quadratic form.

An ellipsoid is determined by its three semi-axes a, b, c > 0. If two axes are equal, the surface is an ellipsoid of revolution (a spheroid); if a = b = c, it is a sphere. When all three axes differ, it is a triaxial ellipsoid. The orientation of the axes is given by the corresponding eigenvectors, and the center is at the translation used to place the ellipsoid.

Key measurements include its volume V = 4/3 π a b c. The surface area generally has no simple closed form; closed expressions exist only for special cases such as spheroids, while the general case involves elliptic integrals.

Cross-sections by planes through the center are ellipses, with circles arising in special orientations. Ellipsoids serve as useful geometric and bounding shapes in various fields, and their projections and sections yield a range of ellipses.

Applications span geodesy, astronomy and planetary science (Earth is commonly modeled as an oblate spheroid), computer graphics (as bounding volumes and collision shapes), and statistics (ellipsoidal confidence regions). Ellipsoids also form a fundamental example of quadratic surfaces in analytic geometry.