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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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Hermite interpolation by Pythagorean hodograph quintics
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by R. T. Farouki and C. A. Neff PDF
Math. Comp. 64 (1995), 1589-1609 Request permission


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 $\sigma (t)$. 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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Additional Information
  • © Copyright 1995 American Mathematical Society
  • Journal: Math. Comp. 64 (1995), 1589-1609
  • MSC: Primary 65D17; Secondary 53A04, 65Y25, 68U07
  • DOI:
  • MathSciNet review: 1308452