Construction of rational curves with rational arc lengths by direct integration

Abstract 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 2 ( ξ ) − v 2 ( ξ ) , 2 u ( ξ ) v ( ξ ) ) / w 2 ( 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.