Variational problems arise as a natural consequence of modeling the physical world. Stretching an elastic membrane or bending a piece of metal can be mathematically described as a minimization problem. Perhaps the simplest example of this idea is the "spline." Originally, a spline was a drafting tool consisting of a long, thin piece of wood that could be clamped to drafting table at certain fixed points. Since the deformed wood naturally assumed a minimum energy state, tracing its length produced a smooth curve. In the 1950's, mathematicians redefined a spline as the mathematical solution to the following problem: Find a smooth function F(t) such that the "bending energy" of F(t) is minimized subject to the constraint that F(t) interpolates user-specified values at the integers (the spline's knots). The space of functions that minimizes this problem is the natural cubic splines. In the 1970's, Lane and Riesenfeld discovered that cubic splines can also be expressed as the limit of a sequence of functions, each related by a simple averaging rule. This averaging process, known as subdivision, is an increasingly popular modeling technique in geometric design. In recent work, the PI has shown that the solution process to a wide range of variational problems can be expressed concisely as a subdivision scheme. This project investigates application of this approach to three important topics: - Subdivision rules for thin-plate splines - Fast numerical solution of variational problems - Creation of splines with local knot insertion rules
