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