The Extension Of A Class Of Trigonometric Polynomial Curves And Surfaces

Parametric curve and surface extensions have been one of hot topics of study in thefield of CAGD. It has important significance in theory research and practical application.In the application of parametric curve and surface design, it’s often needed to extend theknown curve or surface to a given target point or curve, and also need to construct anextended curve or surface between two non-adjacent curves or surfaces. Through theextension method of curves or surfaces, several pieces of curves or surfaces can becombined together, and then more complex shape of curves or surfaces can beexpressed. Currently, parametric curve and surface extension problems are mainlyconcentrated on the research of Bézier and B-spline or rational Bézier and rationalB-spline curves and surfaces. Because Bézier and B-spline curves can not preciselyrepresent some quadratic curves and surfaces, and there are still some problems in thechoice of weight factors in the construction and joint of rational Bézier and rationalB-spline methods, so many researchers studied trigonometric polynomial curves andsurfaces, such as T-Bézier and T-B-spline curves and surfaces. These curves andsurfaces can not only precisely represent some quadratic curves and surfaces, but alsohas simple construction, as well as stronger convex hull character, and can be jointedeasily. However, few people study the extension problem of T-Bézier and T-B-splinecurves or surfaces. In order to make T-Bézier and T-B-spline methods became a betterand more effective design tool, this paper studies the extension problems of cubicT-Bézier and T-B-spline curve and surface. Firstly, a fairing extension algorithm for acubic T-Bézier curve or surface to a target point or curve is presented based on thecondition ofG~2continuity and physical deformation energy. Secondly, a new blendingextension algorithm for two cubic T-Bézier curves or T-Bézier surfaces is given basedon the curve extension algorithm and a class of rational trigonometric blending function.Thirdly, a fairing extension algorithm for two cubic T-B-spline curves and surfaces isput forword based on the condition ofG~2continuity and approximate formation ofstrain energy. Moreover, some expramantal examples are given in each kind of theabove cases. The main contents of this paper are as follows:In the first chapter, the development history of free-form curves and surfaces isintroduced. Research status and applications of parametric curve and surface extension are also discussed. In addition, the main contents of the paper are introduced.In the second chapter, the definitions and properties of cubic T-Bézier andT-B-spline curve and surface，and physical deformation energies of the curve andsurface are introduced, which lay a foundation for theoretical research and practicalapplication in subsequent chapters.In the third chapter, a fairing extension algorithm for a cubic T-Bézier curve to atarget point is presented based on the condition ofG~2continuity and physicaldeformation energy. This method is also applied to extend a cubic T-Bézier surface to atarget curve. Meanwhile, some expramantal examples in the curve and surface designare given.In the fourth chapter, a new extension algorithm ofC2continuity between twocubic T-Bézier curves is presented based on the curve extension algorithm and a class ofrational trigonometric blending function. This method is also applied to constructextension surface between two pieces of cubic T-Bézier surfaces. Moreover, someexpramantal examples in the curve and surface design are given.In the fifth chapter, a fairing extension algorithm for two cubic T-B-spline curvesand surfaces is put forword based on the condition ofG~2continuity and approximateformation of strain energy. This method is also applied to construct extension cubicT-B-spline surface between two pieces of cubic T-B-spline surfaces. Meanwhile, someexpramantal examples in the curve and surface design are given.In the last chapter, the summary of this paper is given and the problems for furtherstudy are put forward.