Representation And Approximation Of Curves And Surfaces Based On The Bézier Method In CAGD

Computer Aided Geometric Design (CAGD) becomes a new cross-cover subject following the development of aviation, shipbuilding and industrial design and manufacture. Representation and approximation of curves and surfaces are important task of CAGD. Bézier curves and surfaces are modeling tools widely used in CAD/CAM systems. The Bézier representation and approximation of arbitrary shaped curves and surfaces by Bernstein polynomials is of great practical importance. In this dissertation, we have made deep researches on these topics and obtained abundant and innovative results as follows.On representation and shape adjustment for curves and surfacesRepresentation and shape adjustment for curves and surfaces is an important research topic which requires profuse geometric modeling handles, which can help designers to modify shape intuitively and conveniently in computer aided geometric design.In order to control the shape of curves and surfaces more flexibly in geometric modeling, three new design techniques with multiple shape parameters are presented.(1) A class of blending functions of degree n+1 with multiple shape parameters is proposed, including the common Bernstein basis function of degree n. We define a class of adjustable generalized Bézier curves and surfaces based on this blending function, and study their properties. Moreover the shape of the generalized Bézier curves can be adjusted entirely or locally by changing the values of the shape parameters when the control polygon is maintained.(2) A class of polynomial basis functions of degree n with multiple shape parameters is defined. The same degree Bernstein basis function is its special case and they share the same properties. The basis functions are used to construct polynomial curves and surfaces with shape parameters. The curves and surfaces share the same properties as the same degree Bézier curves and surfaces. Moreover, we point out that it is very convenient to adjust locally or entirely the shape of curves and surfaces by modifying the values of shape parameters. Furthermore, by the conversion formulae between the basis functions with multiple shape parameters and Bernstein basis, the polynomial curves with shape parameters can be expressed in the same degree Bernstein form. It is more suitable for use in shape design systems.(3) Curves with shape parameters can easily be extended to tensor product surfaces by tensor product method. However, this method is not suitable for constructing non-tensor product surfaces with shape parameters. In this dissertation, we also present a set of Bernstein polynomial basis functions with three parameters of degree three, which are the extensions of the cubic Bernstein bases over the triangular domain. Surfaces constructed by the polynomial basis functions are extension of cubic Bézier surfaces over the triangular domain. The surfaces with parameters presented have similar properties to those of cubic triangular Bézier surfaces. Moreover, they are shape adjustable under the fixed control points because of the adjustable shape parameters. Modification or deformation of the surface is more flexible.On free-form deformation of curves and surfaces based on the polynomial factorDeformation of curves and surfaces can be treated as a mapping from R~3 to itself. Deformationmethods have been widely used in the fields of both geometric modeling and computer animation. In the field of geometric modeling, the advent of non-uniform rational B-splines brought us a nearly perfect approach for mathematical description of free-form shape. However the interactive technique (changing weight factor or knot vector, moving control points) accompanying it for shape modification is limited. Therefore, to generate complex shape, people have to draw support from other techniques for shape modification or space deformation. Some of them have become the core of certain commercial CAD/CAM softwares. Previous deformation methods still have room for improvement in such aspects as exact control of deformation region and guarantee of continuity between deformed and undeformed region in local deformation or shape modification. So finding new, effective and intuitive deformation approaches is still one increasingly significant research field in computer graphics. We develop the techniques of deformation based extension function for parametric surfaces. Unlike traditional methods, its main thoughts are acting on curves and surfaces' equations with the operator matrices constructed by so-called extension function to alter the shape of the curves and surfaces. It is fit for any other curves and surfaces than those expressed by implicit forms. Experiments show that the method is very simple, intuitive and easy to control and without any auxiliary tool. The results have practical value for promoting and reinforcing functions in geometric design systems.On conditions of convexity for Bernstein-Bézier surfaces over trianglesIn CAD and CAGD, the convexity of surfaces is an important and interesting mathematical topic with application to modeling objects. In order to simplify B-nets weak convex conditions, a group of convexity preserving conditions of Bézier triangle surfaces are further improved. On this basis, we put conditions into an infinite convexity preserving area. In this area, using piecewise linear interpolation method, we obtain some Bézier triangle surface convexity preserving linear sufficient conditions. Furthermore, a group of linear sufficient conditions of Bézier triangle convexity is constructed. These conditions are stronger than B-nets convex weak conditions, but weaker than known linear convexity preserving conditions. Geometric interpretations are provided.On approximation of circular arcs by quartic Bézier curves Optimal approximation of parametric curves and surfaces is one of the most important problems in CAGD. Modern CAD / CAM systems do not dispose the circles represented by the implicit equations or the parameter equations with trigonometric functions. Parameter polynomials are usually used to approximate the circular arcs. In order to effectively approximate the circular arcs, we propose three approximation methods for circular arcs by quartic Bézier curves. Using an alternative error function, we give the closed form of the Hausdorff distance between the circular arc and the quartic Bézier curve. We also show that the approximation order is eight and confirm that the approximation proposed in this dissertation has a smaller error than previous quartic Bézier approximations. By subdivision of circular arcs with equi-length, our method yields the curvature continuous spline approximation of the circular arc.On construction of closed spline curves with given tangent polygonThe mathematical description of chain-wheel drives with two fixed non-circular wheels, a transmission chain of constant length and a given non-constant velocity-ratio leads to a system of nonlinear functional equations and it seems hopeless to solve this problem exactly. It is, however, possible to construct a set of tangents of these wheels by kinematic methods, i.e., a planar convex polygon, and then constructed closed spline curves with given tangent polygon, that is, the approximation of the demand curve. Closed adjustable quadratic, cubic and quartic Bézier curves with all edges tangent to a given control polygon are constructed by using of the C~1-continuity and C~2-continuity geometric relationship of combination quadratic, cubic and quartic Bézier curves. The control points of the Bézier curve segments are computed simply by the vertices of the given tangent polygon directly. Spline curves constructed are of C~1-continuity and C~2-continuity, and are shape preserving to tangent polygon. The shape of curves can be controlled by adjusting the values of shape parameters. A generation algorithm of closed adjustable quartic Ball curve with all edges tangent to a given control polygon is proposed using this method. Experiments show that the method given in this dissertation is simple, intuitive, effective and easy to control. It will suit better for modeling demands in CAGD.