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Zbl 0847.68125
Farouki, R.T.; Neff, C.A.
Hermite interpolation by Pythagorean hodograph quintics. (English)
[J] Math. Comput. 64, No.212, 1589-1609 (1995). ISSN 0025-5718; ISSN 1088-6842/e

Summary: The Pythagorean hodograph (PH) curves are polynomial parametric curves $\{x(t), y(t)\}$ whose hodograph (derivative) components satisfy the Pythagorean condition $x^{\prime 2}(t)+ y^{\prime 2}(t)\equiv \sigma^2(t)$ 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.

MSC 2000:: *68U07 Computer aided design
53A04 Curves in Euclidean space
68U05 Computational geometry, etc.
41A05 Interpolation

Keywords: Pythagorean hodograph

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Zentralblatt MATH Berlin [Germany]