Hermite interpolation by Pythagorean hodograph quintics
Authors:
R. T. Farouki and C. A. Neff
Journal:
Math. Comp. 64 (1995), 15891609
MSC:
Primary 65D17; Secondary 53A04, 65Y25, 68U07
MathSciNet review:
1308452
Fulltext PDF Free Access
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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 lowestorder PH curves that are sufficiently flexible for general interpolation/approximation problems are the quintics. While the PH quintics are capable of matching arbitrary firstorder Hermite data, the solution procedure is not straightforward and furthermore does not yield a unique resultthere are always four distinct interpolants (of which only one, in general, has acceptable "shape" characteristics). We show that formulating PH quintics as complexvalued 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 freeform curves, and allows a oneforone substitution of PH quintics in lieu of the widelyused "ordinary" cubics.
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 R. S. Millman and G. D. Parker, Elements of differential geometry, PrenticeHall, Englewood Cliffs, NJ, 1977. MR 0442832 (56:1208)
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 B. Pham, Offset curves and surfaces: a brief survey, Comput. Aided Design 24 (1992), 223229.
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 I. J. Schoenberg, On variationdiminishing approximation methods, On Numerical Approximation (R. E. Langer, ed.), Univ. of Wisconsin Press, Madison, WI, 1959, pp. 249274. MR 0102167 (21:961)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00255718199513084526
PII:
S 00255718(1995)13084526
Article copyright:
© Copyright 1995
American Mathematical Society
