Although the Fogel-Huang algorithm, utilised in ellipsoid parameter estimation, uses the unique minimum-volume (Behrend-Löwner/John - BLJ) ellipsoid containing the intersection of an ellipsoid and a strip, the ellipsoid resulting from an iteration of the FH algorithm to find an ellipsoid bounding the intersection of an ellipsoid and several strips is not the BLJ ellipsoid for that intersection. After a literature survey, FH algorithm performance is examined, in simulation and theoretically. Comparisons are made with the BLJ ellipsoid containing the intersection of the strips. The result of recycling the FH algorithm is also examined. Modifications of the FH algorithm are proposed. These involve the examination of families of ellipsoids containing the intersection of several strips and an ellipsoid, and then choosing the family-optimal ellipsoid (in general, not the BLJ ellipsoid for the intersection of the strips and the ellipsoid). The resulting ellipsoids are often smaller than the ellipsoid found by iterating the FH algorithm, and lead to closer approximations to the BLJ ellipsoid. As the first of these modifications leading to smaller ellipsoids than the iterated FH algorithm finds a family each member of which contains the intersection of an ellipsoid and two strips, the investigation of the use of the true BLJ ellipsoid for the intersection of an ellipsoid and two strips is motivated.