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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

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

The 2020 MCQ for Mathematics of Computation is 1.78.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


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