Research On B-spline Curve And Surface Modeling

Computer Aided Geometric Design (CAGD) is a subject of mordern applied technologies, which has gradually developed with the development of navigation, shipbuilding, mechinical design and manufacture, and it is also an important branch of applied mathematics. As the important content of the graphics computer and Computer Aided Geometric Design, curve and surface modeling mainly study the representation, design, display and analysis of curve and surface in the computer graphics system.In recent years, B-spline curves and surfaces have found their wider and wider applications in engineering design due to their good local support property. Quality of curve and surface is the key problem to designers. Its evaluation criteria is fairing of curves and surfaces.Therefore, the fairing of curve and surfaces is an important reseach subject in Computer Aided Geometric Design. The fairing of curves and surfaces is involved in geometric characteristic of shapes, practicality and easy processing, and it is an aesthetic concept difficult to define.In this paper, the following problems were investigated:Firstly, the conditions between two adjacent quartic B-spline curves of smooth connection are researched. The continuous conditions of G~1 and G~2 between two quartic B-spline curves are obtained according to the relationships of derivative vectors of corresponding orders at djacent endpoint between two curves. The method can be used to solve the problem of smooth connection of two B-spline curves with respect to geometric continuity.Secondly, we deal with the fairing of the "new" curve produced after smooth connection of two adjacent quartic B-spline curves. An optimization method combining selection-point with minimum energy methods is adopted after analysising advantages and disadvantages of the existing fairing methods. The main idea is that the knots to be removed are automatically detemrined such that the resulted curve possesses the least energy on condition that deviation is within given tolerance.This algorithm not only is automatical and objective,but also achieves data compressing at the time of fairing curves.Finally, the smooth G~1 joining conditions in public boundary line of two pairs of biquartic B-spline surfaces was investigated and the eigenequation of the public the boundary line was obtained, based on definition of geometric continuity of parametric surface. The conditions are directly expressed by the control points of the two B-spline surfaces.