Why is the parametrization of the helix is $s \mapsto (\cos 2 \pi s, \sin 2 \pi s, s )$?

Question

Why is the parametrization of the helix is $s \mapsto (\cos 2 \pi s, \sin 2 \pi s, s )$?

Could anyone explain to me how we can parametrize the helix to get this parametrization?

Answer 1

Here is an intuitive explanation : Imagine you have a parametrization for a circle $(sin(2\pi t), cos(2\pi t))$, but when you move along the circle, you are also going upwards with some constant velocity with regard to the angle velocity, and you go up for exactly $1$ when you complete one round of the circle.

Answer 2

First, one should know that in 2D, as $t$ runs from $0$ to $2\pi$, the point $(\cos t, \sin t)$ traces the unit circle counterclockwise, starting from and ending at the point $(1,0)$. Thus, to form a helix, simply generalize this to 3D by moving up along the $z$-axis as you trace the circle around.