Geodetic Surfaces: Understanding Their Geometry and Topology

Ohio State University Research Foundation -Do Not Use, Columbus OH

Investigators

Abstract

DMS-0310642 Alan J. Saalfeld and Tamal K. Dey This collaborative project funded under DMS-0310642 and DMS-0310354 is a CARGO full team award made under solicitation http://www.nsf.gov/pubs/2002/nsf02155/nsf02155.htm. Geodesists and computational geometry specialists use different mathematical tools for building surfaces from sampled points. One part of this project (DMS-0310642) encompasses cross-fertilization of tool development along with merging and extension of the underlying mathematical theories. The mathematical development will address issues in statistical adjustment theory, meshing and other computational geometry applications, and modified polyhedral Morse theory. Another part of the project (DMS-0310354) deals with exposing and equipping undergraduates to work on such problems by offering courses and summer REU workshops. The mission is to involve research at all levels, based on the interplay of fundamental ideas, such as curvature, polyhedral geometry, graphs on surfaces, and combinatorial topology, eventually leading to advanced topics such as noise handling in geodetic data, Morse theory, mesh generation of surfaces, and smoothing. Using statistical adjustment methods, traditional geodesists fit smooth surfaces to noisy surface data. Using computational geometry, our computer scientists have begun developing and will continue to develop efficient algorithms for finding meshed surfaces that are close to and provably homeomorphic to densely sampled smooth surfaces. The interplay between differential and discrete geometry has sparked a renewed interest among mathematicians and theoretical computer scientists to find algorithms for computing topological invariants and for examining properties of surface decompositions and of graphs on surfaces. The research offers challenges to graduate students, undergraduate students, and research faculty alike.