
Journal for Geometry and Graphics 13 (2009), No. 2, 131144 Copyright Heldermann Verlag 2009 Quadrics of Revolution on Given Points Anton Gfrerrer Institut für Geometrie, Technische Universität, Kopernikusgasse 24, 8010 Graz, Austria gfrerrer@tugraz.at Paul J. ZsomborMurray Dept. of Mechanical Engineering, McGill University, 817 Sherbrook Street w., Montreal H3A 2K6, Canada paul@cim.mcgill.ca In general, 4 points define a 3 parameter set of axisymmetric quadrics while 5 and 6 given points reduce these 3 degrees of freedom to 2 and 1, respectively. Similarly, 7 supporting points confine members of the set to a finite number. By imposing 2 constraints on the quadric coefficient matrix the 5 points are sufficient to find the axis direction of up to 6 right cylinders. Imposing only 1 constraint allows 6 points to support up to 12 right cones. Without either constraint, that implies a singular coefficient matrix or singular conic submatrix, up to 4 quadrics of revolution, possibly of mixed species, can contain 7 points. Formal arguments and proofs are presented to substantiate these observations. Algorithms are developed and applied to exhibit cases with 6 right cylinders, 12 right cones and 4 quadrics of revolution, at least 3 of which are of different type. Spheres, being uniquely defined on 4 points, are specifically excluded from consideration. The cases of 12 cones and 4 quadrics of revolution are believed to be original revelations. Methods to fit quadrics of revolution to more than 7 points are suggested. Keywords: Quadric of revolution, cone of revolution, cylinder of revolution, special quadric, finite given point set, repeated eigenvalues. MSC: 51N20; 51N35 [ Fulltextpdf (309 KB)] for subscribers only. 