Decomposition and approximation of three-dimensional solids

Abstract In order to determine the physical properties of a rock sample represented digitally as a set of serial cross sections it is necessary first to decompose the sample into discrete objects and then to approximate each of those objects by another with well-defined mathematical properties. For decomposition, the convex enclosure is defined and shown to be a good approximation of the three-dimensional convex hull yet less complex to calculate. The convex enclosure approximates the convex hull of simple objects with error no more than twenty %, and the enclosure deficiency can be used as a measure of the compactness of the object. Using this measure, it is possible to determine whether an object is sufficienty complex to require its decomposition into a set of subobjects. The decomposition procedure continues recursively until each subobject is sufficiently compact. The subobjects are then approximated by ellipsoids. For each subobject, the axes of the approximating ellipsoid are given by the eigenvalues and eigenvectors of the matrix used in the computation of its convex enclosure.