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 $5$.

翻译：在本文中,我们讨论了建造 $G$2$ Pythagorean-hodraphic(PH) 样条曲线的问题,这种曲线是内插点、正切方向和弯曲,并指定了弧长度。 内插方案完全是局部性的。 每个样条线段的定义是7美元的PH双曲线, 其结果是根据4个自由参数, 以封闭形式解决$G$2美元内插方程式。 通过将其中2个设置为零, 事实证明, 两个边切线之间的任何数据和所选比率都可满足长度限制。 长度内插方程式会减少为一个长方程, 通常有4种解决办法。 要选择最好的方程, 就要观察弯曲能量的价值。 提供了几个数字例子, 以说明所获得的理论结果, 并用数字证实近似值为5美元。