We present an algorithm that determines the point on a convex parametric surface patch that is closest to a given (possibly moving) point. Any initial point belonging to the surface patch converges to the (possibly moving) closest point without ever leaving the patch. Thus the algorithm renders the patch invariant and is globally uniformly asymptotically stable. The algorithm is based on a control problem formulation and solution via a switching controller and common control Lyapunov function. Analytic limits of performance are available, delineating values for control gains needed to out-run motion (and shape) and preserve convergence under discretization. Together with a top-level Voronoi diagram-based switching algorithm, the closest point algorithm treats parametric models formed by tiling together convex surface patches. Simulation results are used to demonstrate invariance of the surface patch, global convergence, limits of performance, relationships between low-level and top-level switching, and a comparison to competing Newton-iteration based methods.