The Shape Preserving Interpolation Of Quadratic Trigonometric Bézier Curves

This thesis makes main use of quadratic trigonometric polynomial to study theproblem of shape preserving interpolation for planer curves in CAGD area,and itprovides some algorithms of the shape preserving interpolation.The main work andresults are outlined as follows:First,it gives the explicit expressions of quadratic trigonometric Béziercurves,and studies the properties of the curves,which are similar to that of cubicBézier curves: properties of endpoints,geometrical invariability,symmetry,convexhull property,the variation diminishing properties and convexity-preserving.Thecorresponding examples show that quadratic trigonometric Bézier curves,which arecloser to the control polygon compared with the cubic Bézier curves,have a bettershape preserving effect.Secondly,it discusses the joining between adjacent quadratic trigonometricBézier curves,and generates two algorithms to construct the shape preservinginterpolation curves which reach C3continuity:one that takes advantage of thejoining between adjacent curves meets the global adjustment of the curves’ shape;theother using the piecewise curves allows the local modification of curves’ shape.Thecorresponding graph examples are given respectively.Thirdly,it constructs rational quadratic trigonometric Bézier curves based on thequadratic trigonometric Bézier curves,and studies the properties of the curves,whichare similar to that of rational cubic Bézier curves:properties ofendpoints,geometrical invariability,convex hull property,the variation diminishingproperties and convexity-preserving.Then,it discusses the geometric continuous andparameters continuous joining between adjacent curves and provides the shapepreserving interpolation algorithms which achieve C~2continuity and G3continuityrespectively.The approaches don’t require solving the system of equations,thecalculations are simple,and the shape of the curves can be locally modified.Thecorresponding examples illustrate that the curves which can achieve G3continuityhave the best shape preserving effect,and have great application value.Finally,it makes a summary in the end,and presents some problems to be studiedfurther.