Abstract
Given five pairs of attachment points of a planar platform, there exists a sixth point pair so that the resulting planar architecturally singular platform has the same solution for the direct kinematics. This is a consequence of the Prix Vaillant problem posed in 1904 by the French Academy of Science. The theorem discusses the displacements of certain or all points of a rigid body that move on spherical paths. Borel and Bricard awarded the prizes for two papers in this regard, but they did not solve the problem completely. In this paper, the theorem is extended to the elliptic paths in order to determine the displacements of certain or all points of a rigid body that move on super-ellipsoid surfaces. The poof is based on the trajectories of moving points which are intersections of two implicit super-ellipsoid surfaces.
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