Whether the curve closes or not, we know from the curvature function
for which parameter values 180 degree rotation about the principal normal
of the curve is a symmetry. The construction of closed curves proceeds
in two steps: first we choose parameters so that two neighboring symmetry
normals intersect. Then all symmetry normals pass through this point. Then
the parameters are further restricted so that the angle between neighboring
symmetry normals is a rational number times pi, for 3-fold symmetry we
adjust to pi/3, for 4-fold symmetry we adjust to pi/4.