Hermite interpolation by Pythagorean hodograph quintics
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 by R. T. Farouki and C. A. Neff PDF
 Math. Comp. 64 (1995), 15891609 Request permission
Abstract:
The Pythagorean hodograph (PH) curves are polynomial parametric curves $\{ x(t),y(t)\}$ whose hodograph (derivative) components satisfy the Pythagorean condition $xâ€™{}^2(t) + yâ€™{}^2(t) \equiv {\sigma ^2}(t)$ for some polynomial $\sigma (t)$. Thus, unlike polynomial curves in general, PH curves have arc lengths and offset curves that admit exact rational representations. The lowestorder PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary firstorder Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique resultâ€”there are always four distinct interpolants (of which only one, in general, has acceptable "shape" characteristics). We show that formulating PH quintics as complexvalued 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 freeform curves, and allows a oneforone substitution of PH quintics in lieu of the widelyused "ordinary" cubics.References

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Additional Information
 © Copyright 1995 American Mathematical Society
 Journal: Math. Comp. 64 (1995), 15891609
 MSC: Primary 65D17; Secondary 53A04, 65Y25, 68U07
 DOI: https://doi.org/10.1090/S00255718199513084526
 MathSciNet review: 1308452