Circle wrapped on a cylinder

Question

I'd like to know the parametric equations of a circle(r) wrapped on a cylinder(R).

$x(t)= r\times cos(t)$
$y(t)=?$
$z(t)=?$

Which are the parametric equations?
Thanks

Answer 1

It seems that you are wrapping a unit disk of radius $r>0$ isometrically around a cylinder $Z$ of radius $R>0$. Let the axis of the cylinder be the $z$-axis.

Bevor the wrapping the disk lies in a $(s,z)$-plane and is determined by the condition $$s^2+z^2\leq r^2\ .$$ Its boundary circle $\gamma$ has a parametric representation of the form $$\gamma:\quad t\mapsto(s,z):=(r\cos t,r\sin t)\qquad(-\pi\leq t\leq\pi)\ .$$ The wrapping $f$ maps points $(s,z)$ onto points $(x,y,z)$ on $Z$ having the same $z$-coordinate and having a polar angle $\phi=\arg(x,y)$ such that $R\phi=s$. It follows that $f$ is given by $$f:\quad (s,z)\mapsto\left(R\cos{s\over R},\ R\sin{s\over R},\ z\right)\ .$$ It follows that a parametric representation of the wrapped circle is given by $$f\circ \gamma:\quad t\mapsto\left(R\cos{r\cos t\over R},\ R\sin{r\cos t\over R},\ r\sin t\right)\qquad(-\pi\leq t\leq\pi)\ .$$