We present a formulation of nonpenetration constraint between pairs of polytopes which accounts for all possible combinations of active contact between geometric features. This is the first formulation that exactly models the body geometries near points of potential contact, preventing interpenetration while not overconstraining body motions. Unlike many popular methods, ours does not wait for penetrations to occur as a way to identify which contact constraints to enforce. Nor do we overconstrain by representing the free space between pairs of bodies as convex, when it is in fact nonconvex. Instead, each contact constraint incorporates all feasible potential contacts in a way that represents the true geometry of the bodies. This ensures penetration-free, physically correct configurations at the end of each time step while allowing bodies to accurately traverse the free space surrounding other bodies. The new formulation improves accuracy, dramatically reduces the need for ad hoc corrections of constraint violations, and avoids many of the inevitable instabilities consequent of other contact models. Although the dynamics problem at each time step is larger, the inherent stability of our method means that much larger time steps can be taken without loss of physical fidelity. As will be seen, the results obtained with our method demonstrate the effective elimination of interpenetration, and as a result, correction-induced instabilities, in multibody simulations.