Circular Cylinders through Four or Five Points in Space

The focus of this paper is the analysis of circular cylinders through sets of points in three dimensions. This investigation has a number of motivations. Clearly, if a cylinder of radius R and direction t passes through a set of points P , it means also that there is a line of direction t tangent to the sphere of radius R and centered at the points of P . Another interpretation is that an observer looking at P from infinity in direction t sees the points of P as cocircular. So, this kind of operations arises in surface reconstruction, visibility and metrology. In Section 2 we describe precisely the set of cylinders through the vertices of a regular tetrahedron. These preliminary observations give insights into the problem and provide lower bounds on the number of solutions. In Section 3, given four points in space, we prove a degree three condition on a direction of a cylinder through these four points. We exploit this result to prove that there are at most six cylinders through five points (Section 4), at most 186 cylindrical shells through six points (Section 5), and at most nine pairs of parallel cylinders through two sets of four points (Section 6). These bounds are tight as illustrated by explicit examples. Such primitives are used for smallest enclosing cylinders [1], [17] or for Delaunay triangulation of projected points [4]. The number of cylindrical shells appear in metrology problems [2], [8]. The first problem of cylinders through five points has already been considered in the literature. The problem of cylinders through four and five points is discussed by Bottema