与给定多边形相切的三角多项式曲线

Trigonometric polynomial curves with given tangent polygon

给定控制多边形和控制多边形边上的切点,给出了与控制多边形相切的三角均匀多项式曲线,所得曲线是C3连续,形状可调的,且构造的三角均匀多项式曲线对原来曲线是保形的.除了通过切点参数,还可以通过三角均匀多项式曲线参数来调整曲线形状,使所得曲线更加逼近多边形,并可进一步、类似地可构造与给定多边形相切的C2m-1(m=1,2,3)连续的m次三角多项式曲线.利用给出的三角均匀多项式曲线来逼近多边形,主要有2个特点:一是曲线能达到连续,并且在切点固定时曲线的形状可以进行调整;二是只需增加一个新节点就可以通过切点,减少了额外点.此外,还通过图例说明研究方法的可行性.

Given control polygon and tangents to the control polygon, we propose an approach to construe trigonometric uniform polynomial curve with all edges tangent to the given control polygon. The curve segments are joined together with C^3 continuity and the shape can he adjusted. Also the constructed curve is a shape-preserving curve for the original curve. We can also adjust the shape of curves by parameters of trigonometric uniform polynomial curve to approach the polygon closer. The general trigonometric polynomial curves with m degree and C^2m-1 （m = 1,2,3） continuity are further presented. The trigonometric uniform polynomial curve constructed in this paper has two advantages. One is that the curve can be continuous and the shape can be adjusted with the fixed tangents. The other is that only adding one new knot, the curve can pass through the tangent, which reduces excess points. Finally, we illustrate that this method is effective by using examples.