Abstract— The science of the gear mechanisms, which is an independent direction of “Applied Mechanics” covers two main scientific directions: theory of gearing and geometric theory of gear transmissions. The main tasks, which are solved by this direction of the science of the gear mechanisms are: development of adequate approaches to the synthesis and related to it generation of the active tooth surfaces of different types of gear mechanisms; choice of rational form of active tooth surfaces; creation of calculating methods for design and elaboration of innovative drives and etc. The current study is dedicated to the synthesis of one less studied class of spatial gear transmissions – cylindrical rack drives. This type of drives serves to transform the rotation of a cylindrical worm into translational displacement of a gear rack. An analytical dependencies, defining the basic geometric characteristics of this type three-link mechanisms, for the cases when the rotating link is a cylindrical worm, which active tooth surfaces are asymmetric cylindrical convolute, Archimedean and involute helicoids, are presented in the study. This motions transformer should be treated as a special case of spatial three-link rack drive, which rotating link is a convolute, Archimedean and involute worm with a constant axial pitch.