A new extremal distance tracking algorithm is presented for convex parametric curves and surfaces undergoing rigid body motion. The geometric extremization problem is differentiated with respect to time to produce a dynamical system that incorporates dependence on both surface shape and rigid body motion. Extremization then takes place by integrating these dynamical equations, but with a feedback controller in place to stabilize the solution. A controller design using feedback linearization is developed that simultaneously accounts for surface shape and motion while asymptotically achieving (and maintaining) the extremal pair. Collision detection then takes place in a framework fully analogous to that used for multibody simulation. Local stability results are extended to provide global stability for body shapes composed of pieced-together convex parametric surface patches using a switching algorithm.