The Fairing Extension Of TC-bézier Curves And Surfaces

Curve and surface extensions have been an extensive research problem in the fieldof Computer Aided Geometric Design (CAGD). It is of great significance in theoreticaland practical applications. Many scholars did a lot of researches on this issue. Thesestudies are mainly focused on parametric curves and surfaces, specially for (rational)Bézier、(rational) Bspline and NURBS curves and surfaces. Meanwhile, T-Bézier andT-Bspline curves and surfaces of the trigonometric polynomial curves and surfaces arealso researched. However, relevant studies of the TC-Bézier curve and surface, whichare mainly specific to its nature and joint, rarely refer to the extension of TC-Béziercurves and surfaces. So, this thesis studies the extension of the quadratic and cubicTC-Bézier curves and surfaces.Firstly, based on the condition ofG1continuity and the physical deformationenergy function, an extension algorithm for a quadratic TC-Bézier curve extending to atarget point is presented. Secondly, via generalizing this algorithm to the quadraticTC-Bézier surface’s extension, aG1continuity extension surface to the originalsurface is obtained. Finally, based on the condition ofG2continuity and taking theminimal physical deformation energy of the curve as the objective function, anextension algorithm for a cubic TC-Bézier curve has been studied. At the same time, theauthor extends the algorithm to the cubic TC-Bézier surface’s extension. What’s more,some experimental examples for all cases mentioned above are given.This thesis is mainly divided into five chapters, the main contents of each chapterare as follow:In the first chapter, it mainly introduces the significance and development course ofextensions of curves and surfaces. Moreover, the advantages and disadvantages ofdomestic and international research in curve and surface extensions are analyzed andsummarized in detail. Besides, it gives a brief introduction to the research contents ofthis thesis.In the second chapter, the definitions and properties of the quadratic and cubicTC-Bézier curve and surface as well as the definitions of physical deformation energiesof the curve and surface are introduced, which lay a foundation of theoretical researchand practical application for subsequent chapters. In the third chapter, takingG1continuity as the constraint conditions and takingthe minimal physical deformation energy of the curve as the objective function, itpresents a fairing extension algorithm for a quadratic TC-Bézier curve to a target point.In addition, it generalizes this algorithm to the situation where a quadratic TC-Béziersurface extends to a target curve. Meanwhile, applications of this extension algorithm inthe modeling of complex curve and surface engineering are given.In the fourth chapter, a fairing extension algorithm for the cubic TC-Bézier curveor surface to a target point or curve is presented according to the fairing extensionalgorithm of the quadratic TC-Bézier curve and surface. Moreover, the applications ofthe algorithm in the modeling of curve and surface engineering are given.In the last chapter, the summary and inadequacies of this thesis are given and theproblems for further studies are put forward.