There is a set of points (in cartesian coordinates), that all lie on the plane. The problem is in rolling up the plane into cylinder in selected direction.
I tried transformation of the coordinates into cylindrical coordinate system. First I defined the value of radius (basing on some not related considerations). Then, I got following coordinates in cylindrical system:
$$\begin{Bmatrix} \rho_i= \sqrt{x_i^2+y_i^2} \\ \varphi_i= atan(y_i/x_i) \\ z_i=z_i \\ \end{Bmatrix}$$
and substituted $\rho_i$ by the value of calculated radius and performed transformation back to cartesian coordinates: $$\begin{Bmatrix} x_i = \rho_icos\varphi_i \\ y_i = \rho_isin\varphi_i \\ z_i=z_i \\ \end{Bmatrix}$$
But as expected, it gave only half-cylinder. Unfortunately, I am very weak in math, so, I suppose, that there is a simple solution for this problem, but I can not formulate it correctly.
Thanks in advance. MuKeP.
Suppose the unit vector $\mathbf{u}$ is the direction in which one wishes to roll up the plane into a cylinder. If we let the radius of the cylinder $r=\frac{1}{2\pi}$ then the transformation is essentially a rotation of the plane by an angle $\phi$ satisfying $\cos\phi=\mathbf{u}\cdot(1,0)$. Let $\mathbf{v}$ denote the unit vector in the direction of the rotated $y$-axis orthogonal to $\mathbf{u}$. The rotated axes $x^\prime$ and $y^\prime$ are represented by the dotted lines in the diagram.
Construct a cylinder about the $x^\prime$ axis with radius $r=\frac{1}{2\pi}$. Then a point $(x,y)$ will be transformed to a point $(a,b)$ in the rotated system where, using the standard rotation formulas,
\begin{eqnarray} a&=&\phantom{-}x\cos\phi-y\sin\phi\\ b&=&-x\sin\phi+y\sin\phi \end{eqnarray} In the cylindrical coordinates, $r=a,\,\theta=b$. So the transformation is
\begin{eqnarray} r&=&\phantom{-}x\cos\phi-y\sin\phi\\ \theta&=&-x\sin\phi+y\sin\phi \end{eqnarray}
It is a many-to-one transformation since the entire plane is wrapped around the cylinder. If a cylinder with a radius different from $\frac{1}{2\pi}$ is desired, then $\theta$ must be scaled accordingly.