Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections

Abstract

A method is presented for the determination of a triaxial ellipsoid (such as a strain ellipsoid)from three nonparallel plane sections of the ellipsoid. The sections need be neither orthogonal nor central sections of the ellipsoid. Measurement errors are used to adjust the observed plane ellipses so that they are exact sections of the nearest true ellipsoid, whose dimensions and orientation are then found by solution of a system of six linear equations. A solution of the inverse problem is also presented: given a triaxial ellipsoid with known orientation, to determine the shape and orientation of the ellipse on a plane section. The problem is solved by expanding the equation of an ellipsoid with rotated coordinates, then setting one dimension to zero. Also, a method is presented for the projection of a triaxial ellipsoid onto a plane surface. This is solved by taking the derivative of the ellipsoid equation in the direction of the normal to the plane surface.