This article deals with the spatial counterpart of the recently introduced class of planar Pythagorean-Hodograph (PH) B–Spline curves. Spatial Pythagorean-Hodograph B–Spline curves are odd-degree, non-uniform, parametric spatial B–Spline curves whose arc length is a B–Spline function of the curve parameter and can thus be computed explicitly without numerical quadrature. After giving a general definition for this new class of curves, we exploit quaternion algebra to provide an elegant description of their coordinate components and useful formulae for the construction of their control polygon. We hence consider the interpolation of spatial point data by clamped and closed PH B–Spline curves of arbitrary odd degree and discuss how degree-$(2n+1)$, $C^n$-continuous PH B–Spline curves can be computed by optimizing several scale-invariant fairness measures with interpolation constraints.

本文介绍了最近推出的一类平面勾股勾线图（PH）B–样条曲线的空间对应关系。空间勾股-Hodograph B-样条曲线是奇数度，非均匀，参数化的空间B-样条曲线，其弧长是曲线参数的B-样条函数，因此可以明确地进行计算，而无需数值正交。在为此类新曲线给出了一般定义之后，我们利用四元数代数提供了它们的坐标分量的优美描述以及用于构造其控制多边形的有用公式。因此，我们考虑通过任意奇数度的钳制和闭合PH B-样条曲线对空间点数据进行插值，并讨论$（2ñ+\mathrm{1个}）$， $C^{ñ}$-连续的PH B样条曲线可以通过优化具有插值约束的几个尺度不变的公平性度量来计算。