A method to determine the two parameter set of circular cylinders, whose
surfaces contain three given points, is presented in the context of an efficient
algorithm, based on the set of two parameter projections of the points onto
planar sections, to compute radius and a point where the axes intersect the
plane of the given points. The geometry of the surface of points, whose position
vectors represent cylinder radius, r, and axial orientation, is
revealed and described in terms of symmetry and singularity inherent in the
triangle with vertices on the given points. This strongly suggests that, given
one constraint on the axial orientation of the cylinder, there are up to six
cylinders of identical radius on the three given points. A bivariate function,
in two of the three line direction Plücker coordinates, is derived to prove
this. By specifying r and an axis direction, say, perpendicular to a
given direction, one obtains a sixth order univariate polynomial in one of the
line coordinates which yields six axis directions. These ideas are needed in the
design of parallel manipulators