Cylinders Through Five Points: Computational Algebra and Geometry

We address the following question: Given five points in (mathbb{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 (mathbb{R}^3). We will investigate computational and theoretical matters related to this problem. In particular we will show how exact and numeric Gr{" o}bner bases, equation solving, and related symbolic-numeric methods may be used to advantage. We will also discuss some applications.