A circular cubic curve called a center-point curve is central to kinematic synthesis of a planar 4R linkage that moves a rigid body through four specified planar positions. In this paper, we show the set of circle-point curves is a non-linear subset of the set of circular cubics. In general, seven arbitrary points define a circular cubic curve; in contrast, we find that a center-point curve is defined by six arbitrary points. Furthermore, as many as three different center-point curves may pass through these six points. Having defined the curve without identifying any positions, we then show how to determine sets offour positions that generate the given center-point curve.
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