Factorization of Rational Curves in the Study Quadric and Revolute Linkages

The research on this paper started with an attempt to understand the geometry of Bennett linkages [1–6] from the point of view of dual quaternions. The group of Euclidean displacements can be embedded as an open subset of the Study quadric in the projectivization of the dual quaternions regarded as a real vector space of dimension eight. Rotation subgroups and and their composition then get an algebraic meaning (see Hao [7], Selig [8, Chapter 9 and 10]). This was exploited in [4] to devise an algorithm for the synthesis of a Bennett linkage to three pre-assigned poses. The key observation there was that the coupler curve is the intersection of a unique 2-plane with the Study quadric. Here, we translate the synthesis problem entirely into the language of dual quaternions and we show that the problem is equivalent to the factorization of a left quadratic polynomial into two linear ones.