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

A polynomial Pythagorean-hodograph (PH) curve r(t)=(x1(t),…,xn(t)) in RnRn 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)σ(t), ensuring that the arc length s(t)=∫σ(t)dt is simply a polynomial in the curve parameter tt. PH curves have thus far been extensively studied in R2R2 and R3R3, 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 RnRn for n>3n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)(n+1)-tuples when n>3n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5n=5 and n=9n=9, and investigate some of their properties.