Developable mechanisms on regular cylindrical surfaces can be described using cyclic quadrilaterals. Mechanisms can exist in either an open or crossed configuration, and these configurations correspond to convex and crossed cyclic quadrilaterals. Using equations developed for both convex and crossed cyclic quadrilaterals, the geometry of the reference surface to which a four-bar mechanism can be mapped is found. Grashof mechanisms can be mapped to two surfaces in open or crossed configurations. The only way to map a non-Grashof mechanism to a cylindrical surface is in its open configuration. Extramobile and intramobile behavior can be achieved depending on selected pairs within a cyclic quadrilateral and its position within the circumcircle. Selecting different sets of links as the ground link changes the potential behavior of the mechanism. Different cases are tabulated to represent all possibilities. A non-Grashof developable mechanism can only exhibit extramobile or intramobile behavior if all of the joints lie on one half of the circumcircle.