Journal for Geometry and Graphics 10 (2006), No. 2, 207--213
Copyright Heldermann Verlag 2006
A Cylinder of Revolution on Five Points
Paul Zsombor-Murray
McGill Centre for Intelligent Machines, Dept. of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montréal H3A 2K6, Canada
Sawsan El Fashny
McGill Centre for Intelligent Machines, Dept. of Mechanical Engineering, McGill University, 817 Sherbrooke Street West, Montréal H3A 2K6, Canada
sfashn@cim.mcgill.ca
Although a general quadric surface is uniquely defined on nine linearly independent given points, special and possibly degenerate quadrics can be generated on fewer if certain constraints, implied or explicit, apply. E.g., coefficients of the implicit equation of a unique sphere may be generated with four given points and five constraint equations. The sphere is special but not degenerate. This article addresses a specific degenerate case, an arbitrary disposition of five given points so as to unambiguously define 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 five initially formulated second order constraints. A cubic and quadratic intermediate pair of equations is produced. These contain the three homogeneous axial direction numbers. Projection of the given points onto any normal plane reveals that the five projected images lie on a circle.
Keywords: Algebraic geometry, cylinder of revolution, degenerate quadric.
MSC: 51N20; 51N05
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