Not HW, but seems to fit here.
I compute $$n.S = \frac{(-1+\cos(c s))}{c^2} \sin(c s) \neq 0$$
I use the following in Mathematica:
r[s_, \[Alpha]_] := Sin[Cos[\[Alpha]] s]/Cos[\[Alpha]]
z[s_, \[Alpha]_] := (1 - Cos[Cos[\[Alpha]] s])/Cos[\[Alpha]]
x[s_, \[CurlyPhi]_, \[Alpha]_] := r[s...