两种带形状参数的曲线

Two Kinds of Curves with Shape Parameters

本文构造了两种带参数的三角样条基,基于这两组基定义了两种三角样条曲线。与二次B样条曲线类似,这两种曲线的每一段都由相继的三个控制顶点生成。这两种曲线具有许多与二次B样条曲线类似的性质,但它们的连续性都比二次B样条曲线更好。对于等距节点,在一般情况下,这两种曲线都整体C3连续,在特殊条件下,它们都可达C5连续。两种曲线中的形状参数均有明确的几何意义,参数越大,曲线越靠近控制多边形。另外,当形状参数满足一定条件时,这两种曲线都具有比二次B样条曲线更好的对控制多边形的逼近性。运用张量积方法,将这两种曲线推广后所得到的曲面也具有较好的连续性。

Two kinds of trigonometric spline bases are constructed in this paper.Based on these bases,two kinds of trigonometric spline curves are defined.As each piece of these trigonometric spline curves are generated by three consecutive control points,these curves retain many properties of the quadratic B-spline curve,but they have a higher order of continuity than the quadratic B-spline curve.For equidistant knots,these curves are C3continuous,and they are C5 continuous under special conditions.The shape parameters of the curves have an explicit geometric meaning.The curves approach the control polygon as the parameter increases.Besides,these curves are closer to the control polygon than the quadratic B-spline curve when the shape parameters are under special conditions.By using the tensor product method,the two kinds of curves can be extended to surfaces.The surfaces have a higher order of continuity than the bi-quadratic B-spline surfaces.