The Extension And Shape Analysis Of A Class Of Trigonometric Polynomial Curves

The modeling and shape analysis of trigonometric polynomial curves are hottopics in the field of Computational Geometry in recent years, and it has greattheoretical significance and application value. The shape of curve always needs somelocal adjustment according to the actual situation in shape modeling. However, curveswith one shape parameter have some limitations in the local adjustment when thecontrol points are fixed. In addition, we always need to judge if there are singularityand inflection points on a curve or not, and try to avoid extra singularity and inflectionpoints in actual application. Thus the study of singularity and inflection points oncurves can help us control the shape of curves in curve modeling. However, there isless study on the shape of trigonometric polynomial curves, and most of them usealgebraic method, the results are not intuitive. To solve these problems, trigonometricpolynomial curves with two shape parameters and its extensions are put forward inthis paper firstly; the properties, continuously joining and application of them aredeeply studied. The shape characteristics of quadratic trigonometric polynomialBézier curves with a shape parameter and quasi-quartic trigonometric polynomialBézier curves are studied by using the method based on the theory of enveloping andtopological mapping secondly. The main contents of this paper are as follows:In the first chapter, the development history, research status and applications oftrigonometric polynomial curves are discussed. The advantages and disadvantages ofvarious methods of shape analysis are summarized in detail; the significance of shapeanalysis is elaborated. In addition, the main contents of the paper are introduced.In the second chapter, the definitions and properties of a class of trigonometricpolynomial curves are given, which lay a foundation for theoretical research insubsequent chapters.In the third chapter, the definitions and properties of a class of quasi-quartictrigonometric polynomial Bézier curves with two parameters and its extended onesare given. The continuously joining with two segments of quasi-quartic trigonometricpolynomial Bézier curves or two extended ones in general situation and theirapplications are studied. The necessary and sufficient conditions forG1(C1), G2(C2) continuously joining with two segments of quasi-quartic trigonometric polynomialBézier curves or two extended ones are obtained; the applications of them in curvemodeling are studied.In the fourth chapter, the shape characteristics of a class of quadratictrigonometric polynomial Bézier curves are investigated. According to therelationship between control polygons and its edge vectors, a result on the singularityand inflection points of spacial quadratic trigonometric polynomial Bézier curves isobtained by calculation and deduction, and then the necessary and sufficientconditions for cusps, loops, inflection points and locally or globally convex region ofthe planar quadratic trigonometric polynomial Bézier curves are obtained respectivelyby using the method based on the theory of enveloping and topological mapping. Thetheory is confirmed by some numerical examples. In addition, the influences of shapeparameter to the shape distribution chart on uv plane are discussed.In the fifth chapter, the geometric characteristics of a class of quasi-quartictrigonometric polynomial Bézier curves in plane under a class of special controlpolygons are studied. The necessary and sufficient conditions for cusps, loops andinflection points are obtained under these control polygons, and its distribution chartin uv plane is given. In addition, the influences of shape parameter to the shapedistribution chart and to the shape adjustment of curve are studied. Its correctness isconfirmed with examples.In the sixth chapter, the summary of the paper is given and the future researchwork is put forward.