Construction of rational curves with rational arc lengths by direct integration

摘要

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'(xi) = (xi) - v(2)(xi), 2u(xi) v (xi)) / w(2)(xi), where w(xi) is a monic polynomial defined by prescribed simple roots, we identify conditions on the polynomials u(xi) and v(xi) which ensure that integration of r'(xi) produces a rational curve with a rational arc length function s(xi). 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. (C) 2019 Elsevier B.V. All rights reserved.