On the geometry of geared 5-bar motion

A new approach is adopted to study the geometry of the coupler curves associated to geared 5-bar motion. The key idea is to think of a configuration of the mechanism as a point in a higher -dimensional configuration space; the family of all configurations is then represented by an algebraic curve in that space. Coupler curves appear naturally as projections of this curve, so their properties can be deduced by projection, independent of any explicit knowledge of their equations. Introduction The present paper has its genesis in the general problem of elucidating the algebraic geometry of coupler curves for planar mechanisms. Much depends on knowing the number and nature of the singular points, so one seeks a better understanding of how these arise. A first small step in this direction was taken in Given the above context, the discussion in [4] of geared 5-bar motion assumes particular interest, in view of the authors' comment that here again the coupler curve has just five singular points, lying on a conic. In this paper we take the same viewpoint of geared 5-bar motion as was taken in [6] for the planar 4-bar. The net result is that one is able to say rather more about the geometry of coupler curves than appeared in (/) In [6] it was shown that the Grashof equations correspond precisely to the natural geometric condition that the linkage curve has a singularity off the hyperplane at infinity. The latter condition makes perfect sense for any planar mechanism so provides a sensible general definition of the term Grashof equation. In particular, we can adopt this viewpoint for th