We were told to consider a spiral car park ramp that looked like a helix. We were also told that the radius of the spiral is $r$ and the distance between two subsequent levels is $\delta$.
Here is an example of a helix. I am aware that the equation for a helix is normally of the form:
$\gamma(t)= (r\cos (t), r\sin (t), ht)$
Where $r$ is the radius and $2\pi h$ is the pitch length.
From the jpg link, $\lambda= \delta /2$. How can I include $\delta$ in the paramaterised equation for the unit speed helix?
Say you have the radius $r$ and the pitch $\delta$. A possible equation for it would be $$t \mapsto ( r \cos 2 \pi t, r \sin 2 \pi t, \delta t)$$
This curve has constant speed, but not $1$. In one unit of time $1$, you went around the circle one time, and raised yourself with height $\delta$. Total length is $\sqrt{(2 \pi r)^2 + \delta^2}$ ( if you unroll the cylinder on which you are moving you are movind on a line, with slope $\frac{\delta}{2\pi r}$ ). To have constant speed $1$, you need to rescale the parameter. You get $$s \mapsto (r \cos \frac{ 2 \pi s}{\sqrt{(2 \pi r)^2 + \delta^2}}, r \sin \frac{ 2 \pi s}{\sqrt{(2 \pi r)^2 + \delta^2}}, \frac{\delta s}{{\sqrt{(2 \pi r)^2 + \delta^2}} })$$