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Hermite Interpolation by Pythagorean Hodograph Quintics

R. T. Farouki and C. A. Neff
Mathematics of Computation
Vol. 64, No. 212 (Oct., 1995), pp. 1589-1609
DOI: 10.2307/2153373
Stable URL: http://www.jstor.org/stable/2153373
Page Count: 21
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Hermite Interpolation by Pythagorean Hodograph Quintics
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Abstract

The Pythagorean hodograph (PH) curves are polynomial parametric curves {x(t), y(t)} whose hodograph (derivative) components satisfy the Pythagorean condition $x'{^2}(t) + y'{^2}(t) \equiv \sigma^2(t)$ for some polynomial σ(t). 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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