Some rigid-body constraint varieties generated by linkages

© Springer Science+Business Media Dordrecht 2012. The set of rigid-body displacements allowed by three simple open-chain linkages are studied. These linkages consist of a cylindrical and spherical joint: The CS dyad, a revolute, a prismatic and a spherical joint: The RPS linkage, two revolutes and a spherical joint: The RRS linkage. Using the Study quadric to represent the group of all rigid-body displacements the constraint varieties for these examples are found. In the case of the CS and RPS linkages these are found to be quartic hypersurfaces while the constraint variety for the RRS linkage is a hypersurface of degree 8. Finally it is shown that all three constraint varieties are linear projections of a Segre variety in P15