论文标题
$ g^2 $ planar hermite interpolants具有规定的弧形长度
Construction of $G^2$ planar Hermite interpolants with prescribed arc lengths
论文作者
论文摘要
在本文中，我们解决了构建$ g^2 $平面毕达哥拉斯（PH）的问题，即样条曲线，插入点，切线方向和曲线，并规定了弧形长度。插值方案是完全局部的。每个样条线段定义为$ 7 $的pH BIARC曲线，这导致具有$ G^2 $插值方程的封闭式解决方案，具体取决于四个免费参数。通过将其中两个固定为零，可以证明可以满足两个边界切线之间的任何数据和任何选择比率的长度约束。长度插值方程通常用四个溶液简化为一个代数方程。为了选择最佳选择，可以观察到弯曲能的值。提供了几个数值示例来说明所获得的理论结果，并在数值上确认近似顺序为$ 5 $。
In this paper we address the problem of constructing $G^2$ planar Pythagorean--hodograph (PH) spline curves, that interpolate points, tangent directions and curvatures, and have prescribed arc-length. The interpolation scheme is completely local. Each spline segment is defined as a PH biarc curve of degree $7$, which results in having a closed form solution of the $G^2$ interpolation equations depending on four free parameters. By fixing two of them to zero, it is proven that the length constraint can be satisfied for any data and any chosen ratio between the two boundary tangents. Length interpolation equation reduces to one algebraic equation with four solutions in general. To select the best one, the value of the bending energy is observed. Several numerical examples are provided to illustrate the obtained theoretical results and to numerically confirm that the approximation order is $5$.
