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Hermite interpolation by rotation-invariant spatial Pythagorean-hodograph curves. (English) Zbl 1001.41003
A parametric curve $𝐫\left(t\right)=\left({x}_{1}\left(t\right),{x}_{2}\left(t\right),{x}_{3}\left(t\right)\right)$ is a Pythagorean-hodograph (PH) space curve if each component ${x}_{i}$ is a polynomial and the derivative ${𝐫}^{\text{'}}$ satisfies the Pythagorean condition $\sum {\left({x}_{i}^{\text{'}}\left(t\right)\right)}^{2}={\sigma }^{2}\left(t\right)$. The purpose of this paper is to solve the Hermite interpolation problem using quaternion representation for spatial PH curves with hodographs of the form ${x}^{\text{'}}\left(t\right)={u}^{2}\left(t\right)+{v}^{2}\left(t\right)-{p}^{2}\left(t\right)-{q}^{2}\left(t\right)$, ${y}^{\text{'}}\left(t\right)=2\left[u\left(t\right)q\left(t\right)+v\left(t\right)p\left(t\right)\right]$, ${z}^{\text{'}}\left(t\right)=2\left[v\left(t\right)q\left(t\right)-u\left(t\right)p\left(t\right)\right]$, $\sigma \left(t\right)={u}^{2}\left(t\right)+{v}^{2}\left(t\right)+{p}^{2}\left(t\right)+{q}^{2}\left(t\right)$. The complex-variable model is here considered to facilitate the solution procedure and analysis of the resulting interpolants. It is formulated the first-order Hermite interpolation for PH quintic space curves in scalar form, and the corresponding quaternion version to solve the problem of characterizing the set of spatial rotations by certain combinations of four polynomials which admit a characterization in terms of quaternions. This solution is applied to the construction of spatial PH quintic Hermite interpolants. The quaternion solution of the first order Hermite interpolation problem considered in this paper, admits a natural extension to the construction of spatial ${C}^{2}$ PH quintic splines which interpolate a sequence of points in space with prescribed end conditions. Examples of rotation-invariant PH quintic space curves are given.

##### MSC:
 41A05 Interpolation (approximations and expansions) 53A04 Curves in Euclidean space