Discrepancy and isotopy for manifold approximations

Date of Completion

January 2009


Applied Mathematics|Computer Science




This thesis examines manifold approximation, specifically in one and two dimensions, by constructing an efficient set of sample points. These points are chosen so that, they are uniformly distributed with respect to the curvature of the manifold. We utilize the tool of discrepancy to measure how uniformly distributed the sample sets are. We are able to prove geometric bounds on the distance and tangential deviation of the approximant. This framework in turn allows us to demonstrate conditions upon which we can guarantee the approximant is ambient isotopic to the manifold. ^