Interpolation of 3D data streams with C² PH quintic splines

Abstract

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.

Appendix

In this appendix, we recall the basic rules of the noncommutative quaternion algebra $\mathrm{I}\mskip -4.0 mu \text{H}\mskip -4.0 mu \text{I}$, used in the paper. Each quaternion $\mathcal {Q} \in \mathrm{I}\mskip -4.0 mu \text{H}\mskip -4.0 mu \text{I}$ can be defined as (q₀,q₁,q₂,q₃)^T, with q_i $\in\mathrm{I}\mskip -3.0 mu \text{R}$, and with q₀ and  $\mathbf {q}\ :=$ (q₁,q₂,q₃)^T respectively referred to as scalar and vector part of the quaternion $\mathcal {Q}$. With this notation, a short scalar/vector representation can also be adopted $\mathcal {Q} = q_{0} + \textbf q ,$ where, if $q_{0} = 0 , \mathcal {Q}$ is named a pure vector quaternion and can be shortly denoted just as $\mathbf {q}$. Conversely, when $\mathbf {q}$ vanishes, $\mathcal {Q}$ is a pure scalar quaternion and can just be denoted as any real number. The sum in $\mathrm{I}\mskip -4.0 mu \text{H}\mskip -4.0 mu \text{I}$ is the standard sum in $\mathbb{R}^4$ but the quaternion product has a specific noncommutative definition that can be compactly defined as

$$\mathcal{A} \mathcal{B} = (a_{0} + \textbf{a})(b_{0}+\textbf{b}) = (a_{0}b_{0} - \textbf{a} \cdot \textbf{b}) + (a_{0} \textbf{b} + b_{0} \textbf{a} + \textbf{a} \times \textbf{b}) ,$$

where standard notation to denote scalar and cross vector products is used. Denoting with $\mathcal {Q}^{*} := q_{0} - \mathbf q$ the conjugate of $\mathcal {Q},$ the product $\mathcal {Q}\mathcal {Q}^{*} = \mathcal {Q}^{*}\mathcal {Q} ={q_{0}^{2}} + \textbf {q}^{T}\textbf {q}$ is just a nonnegative pure scalar quaternion, while, for any pure vector v, the product $\mathcal {Q} \textbf {v} \mathcal {Q}^{*}$ remains a pure vector quaternion. The module $\vert \mathcal {Q} \vert$ of a quaternion is defined as $\vert \mathcal {Q} \vert := \sqrt {\mathcal {Q}\mathcal {Q}^{*}}$ and $\mathcal {Q}$ is a unit quaternion if $\vert \mathcal {Q} \vert = 1$. Unit quaternions allow a compact representation of spatial rotations. For any pure vector quaternion v and unit quaternion $\mathcal {Q} = \cos \limits ({\theta }/{2}) + \textbf {w} \sin \limits ({\theta }/{2})$, the product $\mathcal {Q} \textbf {v} \mathcal {Q}^{*}$ always defines the vector obtained by rotating v through the angle 𝜃 about the axis defined by w.