Planar and spherical mechanisms can be mapped to regular cylindrical and circular conical developable surfaces using cyclic quadrilaterals. The link lengths and link angles can be used in equations to calculate the circumcircle of a cyclic quadrilateral to find the radius or cone angle of a developable surface. Two other numerical and graphical methods to map mechanisms to these surfaces are discussed and expanded. Useful properties of developable mechanisms are also identified using cyclic quadrilaterals. While Grashof mechanisms can be mapped to developable surfaces in either their open or crossed configurations, the only way to map a non-Grashof mechanism to a cylindrical or conical surface is in its open configuration. Extramobile and intramobile behaviour can be predicted based the cyclic quadrilateral's position within the circumcircle and the mechanism's grounded link. The possible configurations are tabulated and analysed.