In CAGD, the Said-Ball representation for a polynomial curve has two advantagesover the B′ezier representation, since the degrees of Said-Ball basis are distributed in a step type.One advantage is that the recursive algorithm of Said-Ball curve for evaluating a polynomialcurve runs twice as fast as the de Casteljau algorithm of B′ezier curve. Another is that theoperations of degree elevation and reduction for a polynomial curve in Said-Ball form are simplerand faster than in B′ezier form. However, Said-Ball curve can not exactly represent conics whichare usually used in aircraft and machine element design. To further extend the utilizationof Said-Ball curve, this paper deduces the representation theory of rational cubic and quarticSaid-Ball conics, according to the necessary and su？cient conditions for conic representation inrational low degree B′ezier form and the transformation formula from Bernstein basis to Said-Ballbasis. The results include the judging method for whether a rational quartic Said-Ball curve is aconic section and design method for presenting a given conic section in rational quartic Said-Ballform. Many experimental curves are given for confirming that our approaches are correct ande？ective.
Many works have investigated the problem of reparameterizing rational B^zier curves or surfaces via MSbius transformation to adjust their parametric distribution as well as weights, such that the maximal ratio of weights becomes smallerthat some algebraic and computational properties of the curves or surfaces can be improved in a way. However, it is an indication of veracity and optimization of the reparameterization to do prior to judge whether the maximal ratio of weights reaches minimum, and verify the new weights after MSbius transfor- mation. What＇s more the users of computer aided design softwares may require some guidelines for designing rational B6zier curves or surfaces with the smallest ratio of weights. In this paper we present the necessary and sufficient conditions that the maximal ratio of weights of the curves or surfaces reaches minimum and also describe it by using weights succinctly and straightway. The weights being satisfied these conditions are called being in the stable state. Applying such conditions, any giving rational B6zier curve or surface can automatically be adjusted to come into the stable state by CAD system, that is, the curve or surface possesses its optimal para- metric distribution. Finally, we give some numerical examples for demonstrating our results in important applications of judging the stable state of weights of the curves or surfaces and designing rational B6zier surfaces with compact derivative bounds.
为使得求值简单且具保形性的DP-NTP曲线增加一个形状调节的功能,将文献(Delgado J,Pe a J M.Ashape preserving representation with an evaluation algorithm of linear complexity.Computer Aided GeometricDesign,2003,20(1):1-10)中给出的一类全新的标准全正(NTP)基进行推广,提出了带多个形状参数的DP-NTP基,并在此基础上定义了带形状参数的DP-NTP曲线.在分析DP-NTP曲线基本几何性质的同时,给出了这类带形状参数的DP-NTP基到Bernstein基的转换公式,并对各个形状参数的几何意义进行讨论,给出了形状调节的一些实例.实例结果表明,形状参数可以较好地起到外形调节的作用.
In this paper, we improve the generalized Bernstein basis functions introduced by Han, et al. The new basis functions not only inherit the most properties of the classical Bernstein basis functions, but also reserve the shape parameters that are similar to the shape parameters of the generalized Bernstein basis functions. The degree elevation algorithm and the conversion formulae between the new basis functions and the classical Bernstein basis functions are obtained. Also the new Q-Bezier curve and surface constructed by the new basis functions are given and their properties are discussed.
