Algebraically rectifiable parametric curves

Abstract

Sufficient and necessary conditions for the arc length of a polynomial parametric curve to be an algebraic function of the parameter are formulated. It is shown that if the arc length is algebraic, it is no more complicated than the square root of a polynomial. Polynomial curves that have this property encompass the Pythagorean-hodograph curves-for which the arc length is just a polynomial in the parameter-as a proper subset. The algebraically rectifiable cubics, other than Pythagorean-hodograph curves, constitute a single-parameter family of cuspidal curves. The implications of the general algebraic rectifiability criterion are also completely enumerated in the case of quartics, in terms of their cusps and intrinsic shape freedoms. Finally, the characterization and construction of algebraically rectifiable quintics is briefly sketched. These forms offer a rich repertoire of curvilinear profiles, whose lengths are readily determined without numerical quadrature, for practical design problems. © 1993.