Smooth Connection Of Bezier Surface Patches And Approximation Of Circles With Polygons

Smooth connection of Bezier surface patches, regularity and convexity of curves and generation of curves are important content of Computer Aided Geometric Design and Computer Graphics. Generally speaking, three methods of connecting surface patches are often used in practice, which are connection, connection with continuity of tangent planes (called G’continuity) and connection with curvature continuity (called G2continuity). Curves can be generated with line segments or dots. Seven problems are studied in this thesis. Problems1,2,3,4are studied in Chapter3and problems5,6,7are studied in Chapters4,5,6respectively.The four problems studied in Chapter3are as follows:1. The problem of smooth connection of triangular Bezier surface patches around a common comer with G’continuity is further studied. Zhang Renjiang has studied this problem, in his paper, the degree of constructed surface patches is3, but all the surface patches have the same constant terms, terms of the first degree and the second degree, only terms of the third degree can be changed to adjust shapes of surfaces. Based on the geometric feature and consistence conditions of smooth connection, a new method of the connection of surface patches with G’continuity is presented. In this method, only constant terms and terms of the first degree of the surface patches are the same, the terms of the second degree and the third degree can be different. Therefore, it is significant in theory and practice. Firstly, in Zhang’s method when a surface patch is fixed, the others have only four coefficients which can be chosen according to the requirement. However, in my thesis when a surface patch is given, seven coefficients of the others can be chosen. The proportion of the number of shape parameters, which are used for adjusting the shape of surfaces, is four to seven. Secondly, if the real surface is formed out of several surface patches and two of them are different quadratic ones, the constructed surface in Zhang’s method can’t coincide with the real surface on the parts of quadratic surface patches, while mine can do it. Hence, the method in this thesis is more efficient and flexible for local shape adjustment.2. A new conception called Gaussian curvature continuity is presented. Connection with Gaussian curvature continuity is based on the G1continuity and two adjacent surface patches have the same Gaussian curvature along their common boundary. This is a new method of connection. The condition of this method is stronger than that of G1continuity but weaker than that of curvature continuity. According to the definition of Gaussian curvature and condition of G1continuity, the condition and the method for Gaussian curvature continuity of triangular Bezier surface patches around a common comer are acquired. Contour curves of Gaussian curvature of a surface are very important in fairness examination of surfaces. Fairness of surfaces will be affected badly if contour curves are not continuous. The fairness of surfaces with Gaussian curvature continuity is better than that of G1continuity. The constructed surface patches have low degree and the degree of surface patches is four.3. The sufficient conditions and method for curvature continuity of triangular Bezier surface patches around a common comer are obtained. The degree of the surface patches is5, which is lower than what has been ever seen in other papers. For this reason, the calculation of the algorithm is low and construction of surface is easier by using our method.4. A method of interpolation in two directions is put forward. In general process of connecting surface patches around a common corner, the steps are usually as follows:construct the first surface patch on a domain, and then determine the others in turn according to the condition and the former surface patch clockwise (or anticlockwise). But the last surface patch may not be connected smoothly with the first one, and some deviation might be caused. The idea of interpolation in two directions is that when n surface patches have been constructed clockwise, other n surface patches can be constructed in the opposite direction. The result is that there are two surface patches on each domain. The two surface patches on the same domain form a new surface patch. The last new surface patch and the first new one can be connected smoothly along their common boundary. By using this method, surface patches are constructed and connected with G1continuity, G2and Gaussian curvature continuity respectively. Since factors for adjusting shape of surfaces are increased, the method in this thesis is more flexible for local shape adjustment.According to the relationship between rectangular coordinates and barycentric coordinates, the methods of smooth connection of triangular Bezier surface patches are obtained by transforming polynomial patches to triangular Bezier surface patches.5. Chapter4presents the condition and the method of curvature connection of Bezier surface patches at four-patch corner based on Kahrnann condition. Up to now, no one has presented the method of connecting n surface patches with curvature continuity around a common corner. Several authors have studied curvature connection of bicubic Bezier surface patches.In this thesis, the degree of Bezier surface patches is free and can be chosen according to the requirement. The problem of curvature connection of Bezier surface patches at four-patch corner is solved.6. The methods of determining regularity and convexity of Bezier and NURBS curves are given in Chapter5. The problems of determining regularity and convexity of curves are converted into those of detecting the existence of roots of equations. Calculating the values of resultants or solving equations are general methods of determining regularity and convexity of curves. In this thesis, calculating values of resultants and solving equations are not needed, while regularity and convexity of curves and number of irregular points can be got at the same time by the signs of function values at two end points of the interval. This method is simple and practical.7. An algorithm for approximation of circles with polygons is studied in Chapter6. In traditional algorithm, circles are approximated with inscribed polygons. Liu Yongkui presented another algorithm called intersect polygon algorithm, his algorithm is the best in terms of distance. Based on the extremum principle and best area approximation, a new algorithm for generating circles is obtained, which is the best in terms of area Then the three algorithms are compared with one another. Through the comparison it can be learnt that by using polygons with the same number of edges to approximate circles, the minimal area between the circle and the polygon in traditional algorithm and intersect polygon algorithm are2.7times and1.2times as large as that in our algorithm respectively, while under the same requirement of area, the number of edges of polygons in traditional algorithm and intersect polygons algorithm are1.63times and1.1times as large as that in our algorithm respectively. Therefore, in terms of computation and convergence velocity, our algorithm is the best of these three algorithms.