A synthesis procedure is developed to envelope a given plane curve by a tangent line carried by the coupler plane of a planar four-bar linkage. The synthesis procedure is based on a modification of the planar rigid-body displacement matrix developed by Suh [1]. The approach is based on considering a given curve as the envelope of a moving tangent line and synthesizing mechanisms to envelope the given curve. This is a novel approach in mechanism design and adds a new dimension to the path-generation problem of mechanism synthesis. The foregoing procedure is also extended to synthesize eight-link mechanisms to simultaneously coordinate the motion of two tangent lines. In addition to finite position synthesis, both infinitesimal and mixed position synthesis are considered.

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