Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3

JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS

卷:236

Article

Sakkalis, Takis²  Farouki, Rida T.¹ 

[1] Univ Calif Davis, Dept Mech & Aerosp Engn, Davis, CA 95616 USA

[2] Agr Univ Athens, Math Lab, Athens 11855, Greece

关键词: Pythagorean-hodograph curves;    Parameterization of n-tuples;    Complex numbers;    Quaternions;    Octonions;    Hopf map;   

DOI  :  10.1016/j.cam.2012.04.002

来源: Elsevier

【 摘 要 】

A polynomial Pythagorean-hodograph (PH) curve r(t) = (x(1)(t). . . . . x(n)(t)) in R-n is characterized by the property that its derivative components satisfy the Pythagorean condition x(1)('2)(t) + . . . + x(n)('2)(t) = sigma(2)(t) for some polynomial sigma(t), ensuring that the arc length s(t) = integral sigma(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R-2 and R-3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in R-n for n > 3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n + 1)-tuples when n > 3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n = 5 and n = 9, and investigate some of their properties. (C) 2012 Elsevier B.V. All rights reserved.

【 授权许可】

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