Congruence of circularcylinders on three given points

Abstract

A method to determine the two parameter set of circular cylinders, whosesurfaces contain three given points, is presented in the context of an efficientalgorithm, based on the set of two parameter projections of the points ontoplanar sections, to compute radius and a point where the axes intersect theplane of the given points. The geometry of the surface of points, whose positionvectors represent cylinder radius, r, and axial orientation, isrevealed and described in terms of symmetry and singularity inherent in thetriangle with vertices on the given points. This strongly suggests that, givenone constraint on the axial orientation of the cylinder, there are up to sixcylinders of identical radius on the three given points. A bivariate function,in two of the three line direction Plücker coordinates, is derived to provethis. By specifying r and an axis direction, say, perpendicular to agiven direction, one obtains a sixth order univariate polynomial in one of theline coordinates which yields six axis directions. These ideas are needed in thedesign of parallel manipulators