Interpolation of 3D data streams with C^2 PH quintic splines

The construction of smooth spatial paths with Pythagorean-hodograph (PH) quintic splines is proposed. To facilitate real-time computations, an efficient local data stream interpolation algorithm is introduced to successively construct each spline segment as a quintic PH biarc interpolating second- and first-order Hermite data at the initial and final end-point, respectively. A C2 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 C2 continuous and can be constructed for arbitrary C1 Hermite data configurations. A simple and effective selection of the free parameters that arise in the local interpolation problem is proposed. The developed theoretical analysis proves its fourth approximation order while a selection of numerical examples confirms the same accuracy for the spline extension of the scheme. In addition, the performances of the method are also validated by considering its application to point stream interpolation with automatically generated first-order derivative information.