高精度三次参数样条曲线的构造

构造参数样条曲线的关键是选取节点，该文讨论了GC^2三次参数样条曲线需满足的连续性方程，提出了构造GC^2三次参数样条曲线的新方法，在讨论了平面有序五点确定一组三次多项式函数曲线，平面有序六点唯一确定一条三次多项式函数曲线的基础上，提出了计算相邻两区间上的节点的算法，构造的插值曲线具有三次多项式函数精,该文还以实例对新方法与其它方法构造的插值曲线的精度进行了比较。

Constructing of Cubic Parametric Spline Curve with High Precision

The construction of parametric spline curves and surfaces plays a very important role in the fields of CAGD, CG, scientific computing and so on. The crux of constructing parametric spline curves is the parameterization of the data points. This paper discusses the continuity equations which the cubic parametric spline curves have to satisfy, and presents a new method to construct GC 2 cubic parametric spline curves. The span of each two adjacent intervals is normalized as one, and then the corresponding three knots can be taken as . Making them be GC 2 continuity at the join point s i sets up the continuity equation of the two curve segments on . The parametric spilne curve is constructed by solving the continuity equations. To set up the continuity equations, an approach for computing knots is presented. The basic idea for computing the knots on the two adjacent intervals is as follows. The five ordered data points in a plane determine a set of cubic polynomial function curves and the six ones determine a cubic polynomial function curve uniquely. The knot s i at the two adjacent intervals is computed by a way that the consecutive six data points are supposed to be taken from a cubic polynomial curve, then a linear relation among s i and other knots is set up, the knot s i is obtained by straightforward constructing the cubic polynomial curve. The constructed parametric spline curve has the precision of cubic polynomial function, i.e., if the given data points are taken from a cubic polynomial function f(t) , then the constructed cubic parametric spline curve reproduces f(t) exactly. The comparisons of the precision of the new method with other ones showed that when used to construct cubic parametric spline curves, the new method in general gives better approximation than other methods.