One of the basic theories of kinematic synthesis, namely, Burmester’s classical centerpoint-circlepoint theory, is shown to be one of several special cases of a broader, more general new theory, involving points of the moving plane whose several corresponding positions lie on cycloidal curves. These curves may be generated by “cycloidal cranks.” Such “cycloidpoints,” centers of their generating circles (“circlepoints”) and base circles (“centerpoints”) are proposed to be called “Burmester point trios” (BPT’s). In case of 6 prescribed arbitrary positions, such BPT’s appear to lie, respectively, on three higher plane curves proposed to be called “cycloidpoint,” “circlepoint” and “centerpoint curves,” or, collectively, “generalized Burmester curves.” In the case of hypocycloidal cranks with “Cardanic” proportions, the hypocycloids become ellipses. For 7 prescribed positions, the number of BPT’s is finite. Application to linkage synthesis for motion generation with prescribed order and timing is presented and cognate-motion generator linkages, based on multiple generation of cycloidal curves, are shown to exist. Analytical derivations are outlined for the equations of the “generalized Burmester curves,” and possible further specializations and generalizations are indicated.