This paper for the first time investigates a disintegrated Bricard mechanism in two toroids and reveals their intersection that leads to a set of Bricard variations with various branches of reconfiguration. The discovery is made in the concentric toroid-toroid intersection. By manipulating the construction parameters of the toroids any possible bifurcation point is explored. This casts light on the reconfiguration of these linkages and leads to the common bi-tangent planes that present singularities in the intersection set.

The study reveals the presence of Villarceau and secondary circles in the toroids intersection. Therefore, a way to reconfigure the Bricard linkage to two different types of Bennett mechanism is uncovered. Further a linkage with two Bricard and two Bennett motion branches is explored. In addition, the paper reveals the Altmann linkage as a member of this Bricard variation family.

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