Surface Area and Curvature of the general Ellipsoid
  This paper gives a detailed derivation of the surface of a tri-axial ellipsoid. The general result is in terms of the elliptic integrals of the first and second kind. It is in checked for all special cases included and the corresponding simplified formulae are given. Next, expressions for the mean and Gaussian curvature are derived. These curvatures depend only on: a) the three axes, and from the point under consideration, b) its distance to the centre and c) the length of the perpendicular from centre to the tangent plane of that point. Finally, it is noticed that at the four umbilic points of an ellipsoid, where the cicular sections degenerate to a point, the two principal curvatures are equal and have the simple expression (a c / b^3)where a>b>c are the half-axes. 
Daniel Poelaert, Joachim Schniewind, Frank Janssens
math.CA