ALGEBRAICALLY RECTIFIABLE PARAMETRIC CURVES

Sufficient and necessary conditions for the arc length of a polynomial parametric curve to be an algebraic function of the parameter are for mulated. It is shown that if the arc length is algebraic, it is no mor e 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 prope r subset. The algebraically rectifiable cubics, other than Pythagorean -hodograph curves, constitute a single-parameter family of cuspidal cu rves. The implications of the general algebraic rectifiability criteri on are also completely enumerated in the case of quartics, in terms of their cusps and intrinsic shape freedoms. Finally, the characterizati on and construction of algebraically rectifiable quintics is briefly s ketched. These forms offer a rich repertoire of curvilinear profiles, whose lengths are readily determined without numerical quadrature, for practical design problems.