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Construction of rational curves with rational arc lengths by direct integration

  • Author(s): Farouki, RT
  • Sakkalis, T
  • et al.

A methodology for the construction of rational curves with rational arc length functions, by direct integration of hodographs, is developed. For a hodograph of the form r (ξ)=(u (ξ)−v (ξ),2u(ξ)v(ξ))/w (ξ), where w(ξ) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(ξ) and v(ξ) which ensure that integration of r (ξ) produces a rational curve with a rational arc length function s(ξ). The method is illustrated by computed examples, and a generalization to spatial rational curves is also briefly discussed. The results are also compared to existing theory, based upon the dual form of rational Pythagorean-hodograph curves, and it is shown that direct integration produces simple low-degree curves which otherwise require a symbolic factorization to identify and cancel common factors among the curve homogeneous coordinates. ′ 2 2 2 ′

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