Half-symmetrical 6-bar curves produced by focal linkages or their derivatives

Abstract

Six-bar linkages, generating symmetrical 6-bar curves may be extracted from a symmetrical, overconstrained linkage, such as Kempe's focal linkage or its generalized form, as demonstrated in the Figs. 1 and 2. Such an extracted linkage, therefore, is a sub-chain or, otherwise, its coupler cognate that produces the same coupler motion. Further, all 6-bars contain a straight-line attached to the crucial link of them, that obtains symmetrical positions with respect to an axis of symmetry. (In the Appendix it is shown that kinematic inversion does not destroy this property.) The symmetry-conditions, finally, are all expressed in terms of the dimensions of the linkage, producing the symmetrical positions of the line.