Rational space curves are not "unit speed"

A method is developed to solve the problem of spatial curves (n = 3) by invoking a sufficient-and-necessary characterization for Pythagorean quartuples of polynomials. The method shows that the curve degenerates to a straight line parallel to the x-axis if the hodograph components ý and ź of the polynomials vanish identically. It is necessary to first identify a sufficient-and-necessary form for Pythagorean (n+1)-tuples of polynomials to extend the argument to &ℝn, with n > 3. The method shows that the existence of curves in &Rdbl;3, which is parameterized by rational functions of the arc length, is transformed into a problem of identifying four polynomials u(t), v(t), p(t), and q(t) so that the three indefinite integrals will yield rational functions.