  We solve the so far open problem of constructing all spatial rational curves
with rational arc length functions. More precisely, we present three different
methods for this construction. The first method adapts a recent approach of
(Kalkan et al. 2022) to rational PH curves and requires solving a modestly
sized system of linear equations. The second constructs the curve by imposing
zero-residue conditions, thus extending ideas of previous papers by (Farouki
and Sakkalis 2019) and the authors themselves (Schr\"ocker and \v{S}\'ir 2023).
The third method generalizes the dual approach of (Pottmann 1995) from planar
to spatial curves. The three methods share the same quaternion based
representation in which not only the PH curve but also its arc length function
are compactly expressed. We also present a new proof based on the quaternion
polynomial factorization theory of the well known characterization of the
Pythagorean quadruples.
