Journal for Geometry and Graphics Volume 10 (2006), No. 2, 207–213.
A Cylinder of Revolution on Five Points
Paul Zsombor-Murray, Sawsan El Fashny
McGill Centre for Intelligent Machines, Dept. of Mechanical Eng’g, McGill University 817 Sherbrooke Street West, Montr eal (Qu ebec), H3A 2K6, Canada email: {paul,sfashn}@cim.mcgill.ca
Abstract. Although a general quadric surface is uniquely de ned on nine linearly independent given points, special and possibly degenerate quadrics can be gener- ated on fewer if certain constraints, implied or explicit, apply. E.g., coe cients of the implicit equation of a unique sphere may be generated with four given points and ve constraint equations. The sphere is special but not degenerate. This article addresses a speci c degenerate case, an arbitrary disposition of ve given points so as to unambiguously de ne up to six cylinders of revolution upon them. An approach based on geometric constraints concerning the distance between any two points on the surface yields a sestic univariate in one of the cylinder axis direction numbers. Three linear variables are eliminated from the ve initially formulated second order constraints. A cubic and quadratic intermediate pair of equations is produced. These contain the three homogeneous axial direction num- bers. Projection of the given points onto any normal plane reveals that the ve projected images lie on a circle. Key Words: Algebraic Geometry, cylinder of revolution, degenerate quadric MSC: 51N20, 51N05
1. Introduction
Recently representatives of FARO Technologies, Inc. in Florida exhibited their six revolute jointed serial robot arm equipped with end e ectors like a simple miniature touch-sphere or a raster scanning laser range- nder that can capture thousands of points on a spatial surface and resolve their location to within a sphere of 25 radius as the user manually moves the end e ector to scan the part. One example was to measure a round cylinder by touching many points on its surface. The device is capable of incorporating many more points than the minimum number required and statistically establishing a consensus that improves the accuracy of the solid model obtained. “How many represent a minimum number?” was asked. “Six” was the reply. This established a reason for this article. What are the principles underlying the minimal point de nition of simple surfaces ? Using analytical and descriptive geometry a robust and e cient algorithm to unambiguously identify cylinders of revolution on
ISSN 1433-8157/$ 2.50 c 2006 Heldermann Verlag 208 P. Zsombor-Murray, S. El Fashny: A Cylinder of Revolution on Five Points