The purpose of this paper is two-fold. First, we describe the Beta-spline curve and surface technique and derive the equations governing the splitting of Beta-spline curves and surfaces. Second, we present a very general adaptive subdivision algorithm that can be used with a variety of surface techniques. It incorporates splitting criteria based on flatness and prevents cracks from occurring between approximating polyhedra. The testing required to determine when surface splitting can stop can be very expensive and methods for reducing these costs are examined. The tolerance controlling the splitting process may itself be adaptive, so that as an object moves farther away the tolerance is automatically increased. These ideas have all been used in the implementation of a surface design and rendering system.