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

A polynomial Pythagorean–hodograph (PH) curve r(t) = (x1(t), . . . , xn(t)) in R n is characterized by the property that its derivative components satisfy the Pythagorean condition x′2 1 (t)+ · · ·+x ′2 n (t) = σ 2(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 R 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.