It is shown how to determine those points, in a system under planar motion, which have trajectories which over a given range best approximate circles and straight lines. These points are best approximations in the sense of having a minimum error-norm. In this work a general norm is used which results in an approximation theory which includes the least-square and mini-max approximations as special cases. Several special motions are considered in detail, and some applications to linkage design are given.

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