Subdivision Schemes and their Applications in Geometric Modeling

Abstract

Subdivision is a basic tool to describe smooth curves and surfaces in computer aided geometric design. Since no single subdivision scheme can be adequate for every situation so, there is always a space to present new schemes. The main purpose of this dissertation is to introduce different kinds of subdivision schemes for curve and surface designing based on arity and complexity. Several explicit formulae for generation of mask of subdivision schemes are presented. Many well known existing schemes are special cases of our proposed schemes. Convergence and smoothness of stationary and non-stationary subdivision schemes are evaluated by using Laurent polynomial method and asymptotic equivalence technique respectively. Some of remarkable properties of proposed subdivision schemes like Hölder regularity, support of basic limit function, error bounds, total absolute curvature, artifact, shrinkage effect, limit stencil, convexity preservation, affine invariance and reproduction are discussed. The applications of the schemes developed have also been depicted through different examples.