The G~3Spline Basis Functions

The continuity of the curve and surface is an important subject in ComputerAided Geometric Design and Computer Graphics. Because of the parametric con-tinuity’s excessively strict defnition cannot solve the problem of visually smoothof the curve and surface very well, researchers proposed a measure to the conti-nuity which has nothing to do with the parameter selection and can provide moredegrees of freedom for curve and surface modeling. The measure is geometriccontinuity, also named Gncontinuity.The Gncontinuity can describe the parameter curves’ and surfaces’ smoothobjectively and accurately. Curves and surfaces based on Gncontinuity are morefexibility and practicability. It facilitates the free-form curve and surface modelingdesign, modifcation, and processing. In addition, the spline functions both keepsome advantages of the polynomial, and overcome the local properties of poly-nomial. With the development of Computer technology, the spline function hasalso made rapid development and a wide range of applications in Computer AidedGeometric Design, Computer Aided Design and other felds. Most literatures arefocusing on G1、 G2continuity. The conclusions and results have been satisfying.For the stitching of the surface, the requested visually smooth are mainly the G2continuity, because for the general design of model satisfying the G2continuity isenough for the most applications. However, with the development of the societyand the needs of actual production, the low order of geometric continuity cannotmeet the increasing demand of the visual efect of product design and performancerequirements. This thesis mainly studies the basic problems such as the expressionand the constructing method of higher order geometric continuous curve urgently in the engineering practice, the main content includes:1. This thesis proposes a kind of general explicit expression of quartic β-spline basis function for the frst time, which take a general forms of the specialshape parameters.Diferent from its original method, the new method is basedon semi-G2spline basis function, at the same time the B′ezier basis function isintroduced, with the aid of the advantage of B′ezier functions.2. This thesis puts forward geometric constructing method of G3continuityspline function. The adoption of the B′ezier basis functions, makes the constructionprocess of G3continuous spline function to be the process of constructing theB′ezier curve. By using spline control vertex constructs the G3control vertex,this method will simplify the constructing method, which make the constructionprocess straightforward. Therefore the application of G3continuous spline in thefelds of Computer Aided Geometric Design, Computer Aided Design is furtherimproved.