I have a data point at (0, 0) where measurements of a tank's shell are taken from. I have used this data point to plot the circle in a graph.
However, this data point is not the true centre of the tank. I need to calculate the true centre from this data point and x,y coords in relation to this data point. Once the true centre is found, I would like to adjust the x,y coords so that the centre is shifted to (0, 0).
Problem
The unknown circle has an equation $$ (x - x_0)^2 + (y - y_0)^2 = r^2 $$ with unknown center $(x_0, y_0)$ and unknown radius $r$.
Radius
Assuming you can sample the rim of a circle in the form $r:r(\alpha)$ for a range of angles $\alpha$. It would be nice to have the rim given as \begin{align} x &= r \cos \alpha \\ y &= r \sin \alpha \end{align} In fact the $i$-th measurement at angle $\alpha_i$ gives a distance $r_i$ and should give \begin{align} x_i &= r_i \cos \alpha_i \\ y_i &= r_i \sin \alpha_i \end{align}
The curvature is then something like $$ \kappa = \frac{|x'y''-y'x''|}{(x'^2+y'^2)^{3/2}} $$ where the values for $x'$, $x''$, $y'$, $y''$ need to be determined from the $x_i$, $y_i$, $\alpha_i$. That will boil down to estimating the derivatives with finite differences of $x_i$ and $y_i$ values. As those values are derived from measurements they will suffer from errors.
From the estimated curvature $\kappa$ the radius $r$ of the circle should be the inverse $$ r = \frac{1}{\kappa} $$
Center
That seems tricky as well, as you will be able only to see part of the rim, so just averaging all known rim positions will not give the center coordinates.
You might go for the best fit of $$ f_i(x_0, y_0) = (x_i - x_0)^2 + (y_i - y_0)^2 - r^2 $$ and try to find a pair $(x_0, y_0)$ that minimizes $$ \delta(x_0, y_0) = \sum_{i=1}^N \left| f_i(x_0, y_0) \right| $$ I am not sure what a good method would be to minimize that function.
Probably your set of $(x_i, y_i)$ and the estimated $r$ might give a nice bounding box, so the problem might be suitable for some bisection algorithm.
Alternative
Another approach might be to use the data to best fit all three unknowns at once $$ f_i(x_0, y_0, r) = (x_i - x_0)^2 + (y_i - y_0)^2 - r^2 $$ and try to find $(x_0, y_0, r)$ that minimize $$ \delta(x_0, y_0, r) = \sum_{i=1}^N \left| f_i(x_0, y_0, r) \right| $$