APPROXIMATION METHODS FOR POLYGONAL CURVES

Abstract

The work presented in this thesis is to develop novel methods for the approximation of non convex polygons with a large number of vertices and to develop approximation of parametric curves. A flexible log linear algorithm to approximate a given set of input points that form a simple closed polygon is first presented. The algorithm is based on the polar angle of the ver-tices with respect to their centroid. Since the proper representation of such shapes is not easy because of a large number of vertices, it would be easier to store such shapes as a subset of their points that closely resembles the original shape. The algorithm approximates a polygon by giving weights to its vertices with respect to a base coordinate system which is the cen-troid of the polygon. The level of approximation is an user driven input which specifies the number of vertices to be present in the approximated polygon. Results of the implementation are provided. The approximation of free form curves from the perspective of manufacturing is also presented. The concept of a special class of polynomials called Pythagorean Hodographs is used to approximate parametric curves of degree three and higher. The arc length of the curves is used as a measure of approximation by using a suitable quadrature rule. Results of the approximation are provided for a number of cases.