This paper for the first time investigates a family of line-symmetric Bricard linkages by means of two generated toroids and reveals their intersection that leads to a set of special Bricard linkages with various branches of reconfiguration. The discovery is made in the concentric toroid–toroid intersection. By manipulating the construction parameters of the toroids, all possible bifurcation points are explored. This 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 toroid–toroid intersection. Therefore, a way to reconfigure the Bricard linkage to a pair of different types of Bennett linkage 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 the family of special line-symmetric Bricard linkage studied in this paper. The method is applied to the plane-symmetric case in the following paper published together with this paper.

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