This is going to be my first question on this forum, I'll try to be as clear as possible!
The problem starts narrating that I am interested on obtaining the 3D coordinates of 50 points on a cylinder. For this, I have taken a picture of the cylinder as if standing on the Y axis of the following diagram:
I take one picture, then rotate the cylinder $\theta_1$ degrees and take another, then rotate again $\theta_2$ degrees and repeat. I manage to take 50 pictures.
My problem starts when I try to solve it using linear algebra. So far I have two matrices $\mathbf{X}_{50\times50}$ and $\mathbf{Z}_{50\times50}$ and I also have a $\mathbf{\Theta}_{50\times1}$ matrix where I have stored the angles (which are given to me). They ask me to obtain the original XYZ coordinates from all the given information.
As I understand it, the projection of an original XYZ coordinate $(x_i, y_i, z_i)$ onto the XZ plane can be given by the following: $$\begin{pmatrix} x^{'}_{1} \\ z^{'}_{1} \\ \end{pmatrix} = \begin{pmatrix} x_{1} \\ y_{1} \\ z_{1} \\ \end{pmatrix} \begin{pmatrix} \cos(\theta) & -\sin(\theta) & 0 \\ \sin(\theta) & \cos(\theta) & 0 \\ 0 & 0 & 1 \\ \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ \end{pmatrix}$$
I understand the I have to rotate and then project (otherwise i'd be losing information). I know this system can be solved using least squares ($Ax=b$) where $A$ would be a combination of the Rotation and Projection matrix, something like :
$$A = \begin{pmatrix} \mathbf{PR_{\theta_1}} & \mathbf{0}_{2x3} & \cdots & \cdots &\mathbf{0}_{2x3} \\ \mathbf{0}_{2x3} & \mathbf{PR_{\theta_2}} & \mathbf{0}_{2x3} & \cdots & \vdots\\ \vdots & \mathbf{0}_{2x3} & \ddots & & \vdots \\ \vdots & \vdots & & & \vdots \\ \mathbf{0}_{2x3} & \cdots & \cdots & \cdots & \mathbf{PR_{\theta_{50}}}\\ \end{pmatrix}$$
But once I solve that system, I obtain a solution matrix that has 50 different XYZ points for one angle, then another 50 different XYZ points for another angle, and that's where I don't really know what to do.
I'd really appreciate any advice, even if it implies starting over! Thanks in advance everyone.
EDIT: I apologize if I've broken any specific format rules or otherwise, I'm new to this forum and come from a very poor maths background.
EDIT2: I'll try to explain the process a bit better:
I take a picture of the XZ plane as if I was standing on the Y axis, so each one of the 3D points in the XYZ space gets projected onto the XZ plane with coordinates $x^{'}_{1}$ and $z^{'}_{1}$. Afterwards, there is a rotation of the cylinder around the Z-axis, with angle $\theta_1$. I take another picture, and obtain a different set of (x', z') coordinates. I do this for the 50 theta angles.