About the Algebraic Solutions of Smallest Enclosing Cylinders Problems

TL;DR

This paper introduces an algebraic algorithm for computing the smallest enclosing cylinder of a set of points in Euclidean space, with explicit solutions for specific cases like tetrahedra and bipyramids.

Contribution

It develops an algebraic method to analytically determine the minimal enclosing cylinder, including the axis, radius, and position, especially for three-dimensional cases with four or five points.

Findings

Analytical solutions for minimal enclosing cylinders of tetrahedra and bipyramids.

Algorithm reduces to solving polynomial equations of degree up to 6.

Explicit formulas for the cylinder parameters in symmetric configurations.

Abstract

Given n points in Euclidean space E^d, we propose an algebraic algorithm to compute the best fitting (d-1)-cylinder. This algorithm computes the unknown direction of the axis of the cylinder. The location of the axis and the radius of the cylinder are deduced analytically from this direction. Special attention is paid to the case d=3 when n=4 and n=5. For the former, the minimal radius enclosing cylinder is computed algebrically from constrained minimization of a quartic form of the unknown direction of the axis. For the latter, an analytical condition of existence of the circumscribed cylinder is given, and the algorithm reduces to find the zeroes of an one unknown polynomial of degree at most 6. In both cases, the other parameters of the cylinder are deduced analytically. The minimal radius enclosing cylinder is computed analytically for the regular tetrahedron and for a trigonal bipyramids family with a symmetry axis of order 3.