IBM Journal of Research and Development
The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Hermite interpolation by Pythagorean hodograph quintics
Mathematics of Computation
Minkowski pythagorean hodographs
Computer Aided Geometric Design
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
Journal of Symbolic Computation
Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
Journal of Symbolic Computation
Advances in Computational Mathematics
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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)=@s^2(t) for some polynomial @s(t), ensuring that the arc length s(t)=@!@s(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 n3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n3. 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.