Contact between complex bodies and simple counterparts like straight lines or circles occur in many two-dimensional mechanical models. The corresponding contact detection problems are complicated and thus far, no explicit formulas have been available. In this paper, we address the contact detection problem between two planar bodies: one being either a straight line or a circle and the other an almost arbitrary geometry—the only requirement is a unique contact point for all possible contact situations. To solve this general problem, a novel procedure is applied which provides necessary conditions for the description of the geometry based on the special case of a rolling contact. This results in a parameterization of the geometry which gives the potential contact point depending on the relative orientation between the two bodies. Although the derivation is based on a rolling contact, the result is valid in general and can also be used for efficient contact detection when the bodies are separated. The derived equations are simple and easy to implement, which is demonstrated for two examples: a foot-ground contact model and a cam-follower mechanism.