The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic spline biarcs is proposed. To facilitate real-time computations of $C^2$ PH quintic splines, an efficient local data stream interpolation algorithm is introduced. Each spline segment interpolates second and first order Hermite data at the initial and final end-point, respectively. In the spline extension of the scheme a $C^2$ smooth connection between successive spline segments is obtained by taking the locally required second-order derivative information from the previous segment. Consequently, the data stream spline interpolant is globally $C^2$ continuous and can be constructed for arbitrary $C^1$ Hermite data configurations. A simple and effective selection of the free parameters that arise in the interpolation problem is proposed. The developed theoretical analysis proves the fourth approximation order of the local scheme while a selection of numerical examples confirms the same accuracy of its spline extension. In addition, the performances of the algorithm are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.

翻译：提议用Pythagorean-hodlog(PH) 二次曲线来构建平滑的空间路径。 为了方便实时计算$C $2$PH 的二次曲线, 引入了一个高效的本地数据流内插算算法。 每个样条段分别在初始和最终终点将Hermite 数据相接第二和第一顺序。 在该计划的样条扩展中, 通过从上一个段获取本地所需的第二阶衍生信息, 获得连续各样段之间的平滑连接 $C $2美元。 因此, 数据流样条内插器是全球性的 $C $2$, 并且可以构建任意的 $C $1$ Hermite 数据配置 。 提议简单而有效地选择在内插图问题中产生的自由参数。 开发的理论分析证明了本地图的第四近似顺序, 而选择的数字示例证实了其样条扩展的准确性。 此外, 算法的性表现也通过考虑将其应用于点流间插图和自动生成的一级衍生物信息进行验证。