The Composition of Bennett’s Hyperboloids From the Loop Itself

Abstract

Despite the tidiness with which the Bennett linkage can be related geometrically to its associated quadric surfaces, the corresponding algebraic determinations have generally turned out to be cumbersome, and so practically noninvertible. In particular, although earlier work has delineated the linkage in terms of parameters peculiar to its quadric surfaces, the inverse dependences could not be given. We here use the loop strictly as a basis for derivation of the surfaces and employ cartesian co-ordinates throughout, thereby effectively completing the conversion process.