Linear Forms for Projectivized Dyads Computed via the Grobner Basis and Elimination Theory

Dyad structures are fundamental to every mechanism for kinematic synthesis and design. In this paper, a complete exposition of the constraint varieties containing the kinematic image of all dyad types ($2 \mathrm{P}, \mathrm{RP}, \mathrm{PR}$ and 2 P) of arbitrary geometry and its corresponding inverses in the sevendimensional projective space computed via the Grobner basis and projective elimination theory is provided. This technique gives a more systematic and general method for determining an implicit representation of the constraint varieties of the dyads from its parametrization determined through the projectivized dual quaternions representing the rigid transformations. The results show that the constraint varieties are linear forms of dimension 1 or 3, and the kinematic images are contained in the intersection of a linear n-space, where n is 1 or 3, and the Study quadric minus the exceptional generator.