Spatial Pythagorean-Hodograph B–Spline curves and 3D point data interpolation

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), Cn-continuous PH B–Spline curves can be computed by optimizing several scale-invariant fairness measures with interpolation constraints.