  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.
