Previously the authors have considered various types of approximations which can be used in the design of planar linkages. The idea behind these works has been to determine points in a moving system which best approximate a circle or a line, and thereby help to provide the dimensions of a specific mechanism. In this context the idea of "best" has been relative and dependent on the variable norm of the approximating spaces. We have considered least-squares and general L{sub}D approximations. A companion work recently published by the American Society of Mechanical Engineers deals directly with the Chebychev approximation as a case in its own right. The question we put was: given a discrete set of points, determine the circle or line which best approximates them in the sense of minimum maximum-error. In the present work we extend these ideas to three dimensions. In this paper we formulate Chebychev approximations of spheres, planes and of line congruences. These results are useful in the Kinematic synthesis of spatial linkages with joints formed from spheres, revolutes, circles and cylinders. It is interesting to note that the application of Chebychev approximations on finite sets in three-dimensions seems to not have been heretofore attempted. The relative simplicity of the formulations and the resulting solutions indicate much potential for these new techniques. The method has the advantage of being relatively independent of the number of points being considered, and so the same technique is useful for any number of design positions. It also has the advantage of allowing for an analytical formulation which provides an understanding of the structure of the solution spaces as well as a basis for computational techniques.
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