Cylinders through Five Points: Computational Algebra and Geometry

We address the following question: Given five points in R 3 , determine a right circular cylinder containing those points. We obtain algebraic equations for the axial line and radius parameters and show that these give six solutions in the generic case. An even number (0, 2, 4, or 6) will be real valued and hence correspond to actual cylinders in R 3 . We will investigate computational and theoretical matters related to this problem. In particular we will show how exact and numeric Gr¨ obner bases, equation solving, and related symbolic-numeric methods may be used to advantage. We will also discuss some applications. obner bases, nonlinear systems, symbolic-numeric computation 1. Outline of the Problem and Related Work Given five points in R 3 , we are to determine all right circular cylinders containing those points. We do this by solving equations for the axial line and radius parameters. We will show that generically one obtains six solutions to these equations. Of these an even number are real valued, as the complex valued ones appear in conjugate pairs (an immediate consequence is that there is no "unique" real cylinder through five given points unless it a solution with multiplicity). Moreover there are open regions in the real configuration space that give each of these possibilities so none are disallowed. The basic problem of determining cylinders from five points may be recast in a computational geometry setting: Given five points in R 3 , find the smallest positive r and orientation parameters such that the cylinder of radius 2r with those parameters encloses tangentially the balls of radius r centered at the points. Here are some questions we will consider. The first three are classical; we address them here to illustrate the utility of symbolic computation in such investigations. The last ones are related to more recent work in computational and integral geometry.

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