  In classical curve theory, the geometry of a curve in three dimensions is
essentially characterized by their invariants, curvature and torsion. When they
are given, the problem of finding a corresponding curve is known as 'solving
natural equations'. Explicit solutions are known only for a handful of curve
classes, including notably the plane curves and general helices.
  This paper shows constructively how to solve the natural equations explicitly
for an infinite series of curve classes. For every Frenet curve, a family of
successor curves can be constructed which have the tangent of the original
curve as principal normal. Helices are exactly the successor curves of plane
curves and applying the successor transformation to helices leads to slant
helices, a class of curves that has received considerable attention in recent
years as a natural extension of the concept of general helices.
  The present paper gives for the first time a generic characterization of the
slant helix in three-dimensional Euclidian space in terms of its curvature and
torsion, and derives an explicit arc-length parametrization of its tangent
vector. These results expand on and put into perspective earlier work on
Salkowski curves and curves of constant precession, both of which are
subclasses of the slant helix.
  Keywords: Curve theory; General helix; Slant helix; Curves of constant
precession; Frenet equations; Frenet curves; Frenet frame; Bishop frame;
Natural equations; Successor curves; Salkowski curves; Closed curves.
