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Hermite interpolation by Pythagorean hodograph quintics. (English) Zbl 0847.68125
Summary: The Pythagorean hodograph (PH) curves are polynomial parametric curves $\left\{x\left(t\right),y\left(t\right)\right\}$ whose hodograph (derivative) components satisfy the Pythagorean condition ${x}^{\text{'}2}\left(t\right)+{y}^{\text{'}2}\left(t\right)\equiv {\sigma }^{2}\left(t\right)$ for some polynomial $\sigma \left(t\right)$. Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowest-order PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary first-order Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique result – there are always four distinct interpolants (of which only one, in general, has acceptable “shape” characteristics). We show that formulating PH quintics as complex-valued functions of a real parameter leads to a compact Hermite interpolation algorithm and facilitates an identification of the “good” interpolant (in terms of minimizing the absolute rotation number). This algorithm establishes the PH quintics as a viable medium for the design or approximation of free-form curves, and allows a one-for-one substitution of PH quintics in lieu of the widely-used “ordinary” cubics.
MSC:
 68U07 Computer aided design 53A04 Curves in Euclidean space 68U05 Computer graphics; computational geometry 41A05 Interpolation (approximations and expansions)
Keywords:
Pythagorean hodograph