How to transfer a 2D sheet into Cylinder?

Question

I know the coordinates of all points on a 2D sheet. Now I want to map these points according to function/rule that change the positions of the points & turns it into a Cylinder.

Answer 1

Cylindrical coordinates are:

$$\begin{bmatrix} x_c \\ y_c \\ z_c \\ \end{bmatrix}=\begin{bmatrix} r\,\cos(\phi) \\ r\,\sin(\phi) \\ z \\ \end{bmatrix}$$

with:

$$2\,\pi\,r=H\quad ,\Rightarrow\quad r=\frac{H}{2\,\pi}$$

$$r\,\phi=y\quad ,\Rightarrow\quad \phi=\frac{y}{r}=2\,\pi\frac{y}{H}$$

$\Rightarrow$

$$\begin{bmatrix} x_c \\ y_c \\ z_c \\ \end{bmatrix}=\begin{bmatrix} \frac{H}{2\,\pi}\,\cos(2\,\pi\frac{y}{H}) \\ \frac{H}{2\,\pi}\,\sin(2\,\pi\frac{y}{H}) \\ z \\ \end{bmatrix}\quad,-H/2\le y\le H/2,\quad 0\le z \le L$$

Answer 2

You can map a cartesian coordinate system to a cylindrical one as shown here: http://mathworld.wolfram.com/CylindricalCoordinates.html

The question itself is fairly complicated. If you imagine your cylinder having the lowest point on the curve at the x-axis with the x-axis value itself being length then you're mapping the y-axis as a distance of circumference around your cylinder, with some set maximum circumference dependent upon the size of your plane. So if your plane is $y=[-10,10]$ a perfect cylinder would have radius $r=\frac{20}{2\pi}$. Your $y_c$ and $z_c$ values on the cylinder would then be mapped by: $y_x=r \sin(\frac{y*\pi}{10})$ and $z_c=r-r \cos(\frac{y*\pi}{10})$.

Hope this helped!