Hermite interpolation by Pythagorean hodograph quintics

Authors:
R. T. Farouki and C. A. Neff

Journal:
Math. Comp. **64** (1995), 1589-1609

MSC:
Primary 65D17; Secondary 53A04, 65Y25, 68U07

MathSciNet review:
1308452

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Abstract: The *Pythagorean hodograph* (PH) *curves* are polynomial parametric curves whose hodograph (derivative) components satisfy the Pythagorean condition for some polynomial . 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.

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DOI:
https://doi.org/10.1090/S0025-5718-1995-1308452-6

Article copyright:
© Copyright 1995
American Mathematical Society