This paper presents closed-form solutions to the problem of defining the attachment of a spherical joint (a ball and socket) in a spatial linkage. The defined attachment should minimize the opening angle of the socket, maximizing the ball retention capability. Previous solutions to this problem have used iterative searching techniques, which are computationally intense. In this paper the problem is reformulated as a surface fitting problem. Two formulations are presented, both of which require the solution of the associated eigenvalue problems. The eigenvectors correspond to the central axis of symmetry for the socket opening. A numerical example is presented to compare the results obtained by these formulations with those available in the literature.

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