Cycloidpoints were recently introduced [1]. The present paper demonstrates the existence of Burmester point trios at which the rotation of the moving plane between precision positions is a constant multiple of the rotation of the circlepoint centerpoint vector. A new geared linkage is introduced to mechanize this commensurate relationship. Equations for finite-motion generation and for both finite and infinitesimal path generation are developed. Cognates of this mechanism are shown to exist.

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