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 of four positions that generate the given center-point curve.

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