If one applies an invertible linear transformation to a sphere, one obtains an ellipsoid; it can be brought into the above standard form by a suitable rotation, a consequence of the spectral theorem. If the linear transformation is represented by a symmetric 3-by-3 matrix, then the eigenvectors of the matrix are orthogonal (due to the spectral theorem) and represent the directions of the axes of the ellipsoid: the lengths of the semiaxes are given by the eigenvalues.
The intersection of an ellipsoid ...
The surface area of an ellipsoid is given by:
where
and F(θ,m) and E(θ,m) are the incomplete elliptic integrals of the first and second kind.
Exact formulae are:
If flat:
If prolate:
If oblate:
Approximate fo ...
