A local $C^2$ Hermite interpolation scheme with PH quintic splines for 3D data streams

TL;DR

This paper introduces an efficient local $C^2$ Hermite interpolation scheme using Pythagorean-hodograph (PH) quintic splines for real-time 3D data stream applications, ensuring smoothness and high accuracy.

Contribution

It presents a novel local interpolation algorithm for $C^2$ PH quintic splines that guarantees smoothness, high approximation order, and applicability to arbitrary Hermite data configurations.

Findings

The scheme achieves fourth-order approximation accuracy.

Numerical examples confirm the scheme's high precision.

The algorithm performs well in point stream interpolation tasks.

Abstract

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.