Numerical variational methods in differential geometry and applications to computer graphics

In Chapter 1, we define discrete objects like $\delta$-tangents, $\delta$-normals and $\delta$-curvatures which are analogues of smooth objects in differential geometry. These are incorporated into numerical methods for computer simulation of geometric objects. Some methods to generate geodesic curves on surfaces are discussed.In Chapter 2, we develop several methods to generate approximate models of minimal surfaces, using the tools developed in Chapter 1. Models generated using average methods show quick convergence but some deviations from the actual solution. Normal variation methods require heavier computations, but show less deviation from the actual solution.In Chapter 3, Rubel's Quasi-solution methods are discussed and as applications, methods are developed to find out explicit solutions of partial differential equations defining minimal surfaces and surfaces with restrictions on the man curvature.In Chapter 4, Transversal Scalar Curvature is defined and some relations between scalar curvatures in tangential, transversal and ambient spaces are discussed.The listing of a computer program which demonstrates the methods of Chapter 2 is include in the Appendix.U of I OnlyETDs are only available to UIUC Users without author permissio