Let s be the unit parameter, c(s)-P = f(s)c'(s), f is some unknow function of s. So

c'(s) = f'(s)c'(s) + f(s)c''(s)

That is (1-f'(s))c'(s) = f(s)c''(s)

since c'(s) is unit, their inner product <c'(s), c''(s)>=0, so we must have

1-f'(s)=0, that is f(s)=f(0)+s. f(s) is not 0 when s is not -f(0).

So c''(s)=0 when s is not -f(0)