[en] Several methods for determining the shape parameters, which in two dimensions are the Courant-Snyder parameters, and the volume of an ellipse or hyperellipse that represents a set of phase-space points in a two-or-more-dimensional hyperspace are presented. The ellipse parameters are useful for matching a beam to an accelerating or transport system and in studies of emittance growth. The fitting procedure minimizes the total volume of a hyperellipse by adjusting the ellipse shape parameters. The total volume is the sum of the individual particle volumes defined by the hyperellipse that passes through a particle's phase-space point. A two-dimensional space is considered first; then results are generalized to a higher number of dimensions. Computer programs using these techniques were written. 1 figure