CONGRUENCE OF CIRCULAR-CYLINDERS ON 3 GIVEN POINTS

A method to determine the two parameter set of circular cylinders, who se 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 whe re the axes intersect the plane of the given points. The geometry of t he surface of points, whose position vectors represent cylinder radius , r, and axial orientation, is revealed and described in terms of symm etry and singularity inherent in the triangle with vertices on the giv en points. This strongly suggests that, given one constraint on the ax ial orientation of the cylinder, there are up to six cylinders of iden tical radius on the three given points. A bivariate function, in two o f the three line direction Plucker coordinates, is derived to prove th is. By specifying r and an axis direction, say, perpendicular to a giv en direction, one obtains a sixth order univariate polynomial in one o f the line coordinates which yields six axis directions. These ideas a re needed in the design of parallel manipulators.