The Rational Cubic DP And Generalized Ball Representation Of Arc

This thesis discusses two types of curves——generalized Ball curves and rational DP curves which are of great importance in CAGD researches, and makes a study in the representation of arc by rational cubic Wall-Ball curves, rational cubic Said-Ball curves and rational cubic DP curves. These curves have shown greatly efficient effects on evaluating, degree elevation and degree reduction, and they are shape preserving except the rational cubic Wall-Ball curves. The purpose of this thesis is to exert their range of representation, and use the conclusion to identify if a curve is arc and to design a curve to represent arc. Based on a systematic discussion on the content, characteristics and the up-to-now accomplishments of these curves in CAGD, we present our researches in four ways as follows:1. The time complexity analysis for the evaluating algorithms of these curvesAt first, we provide the time complexity analysis and compare for the evaluating algorithms of rational n degrees DP curves, n degrees Wall-Ball curves, n degrees Said-Ball curves and rational n degrees Bézier curves. This indicates that it's necessary to study the conditions for these curves representation of arc and exert their range of representation.2. The rational cubic Wall-Ball curves representation of arcAs a kind of generalized Ball curves, Wang-Ball curves have shown greatly efficient effects on evaluating parametric curves, degree elevation. In order to exert its effect on geometric design, we give the sufficient and necessary conditions of the rational cubic Wall-Ball curves representation of arc.This thesis use two different method to obtain two groups of the sufficient and necessary conditions, one method is to use the properties of arc and use the theories of analytic geometry and linear algebra, and the other method is to transform the rational cubic Wall-Ball curves to the rational cubic Bézier curves and use the conditions for the rational cubic Bézier curves representation of arc. At last we prove these two groups of the sufficient and necessary conditions are equivalent.3. The rational cubic DP curves representation of arcRational DP curves not only have an evaluation algorithm of linear complexity, but also are composed by normalized totally positive (NTP) bases., we give the sufficient and necessary conditions of the rational cubic DP curves representation of arc.We use the method as Wall-Ball curves', also obtain two kinds of conditions, and we prove they are equivalent.4. geometric construction and some examplesWe obtain the geometric construction for the representation of arc by rational cubic Wall-Ball curves and rational cubic DP curves, and based on the geometric construction, we give some examples for designing arc of different central angles. We also use the sufficient and necessary conditions for the rational cubic Wall-Ball curves representation of arc and the rational cubic DP curves representation of arc to identify if a rational cubic DP curve or a rational cubic Wall-Ball curve is arc, and give some examples.