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

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摘要

A polynomial Pythagorean-hodograph (PH) curve r(t)=(x1(t),…,xn(t)) in Rn is characterized by the property that its derivative components satisfy the Pythagorean condition x1′2(t)+⋯+xn′2(t)=σ2(t) for some polynomial σ(t), ensuring that the arc length s(t)=∫σ(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R2 and R3, 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 Rn 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.

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

论文评审过程:Received 13 August 2011, Revised 2 April 2012, Available online 11 April 2012.

论文官网地址:https://doi.org/10.1016/j.cam.2012.04.002