Rotated Paraboloid Surface Fitting
Do you have enough data points to fit all 10 parameters - you will need at least 10?
I also suspect that 10 parameters are to many to describe a general paraboloid, meaning that some of the parameters are dependent. My fealing is that a translated and rotated paraboloid needs 7 parameters (although I'm not really sure)
I have a set of experimentally determined (x, y, z) points which correspond to a parabola. Unfortunately, the data is not aligned along any particular axis, and hence corresponds to a rotated parabola. 
I have the following general surface:
Ax^2 + By^2 + Cz^2 + Dxy + Gyz + Hzx + Ix + Jy + Kz + L = 0
I need to produce a model that can represent the parabola accurately using (I'm assuming) least squares fitting. I cannot seem to figure out how this works. I have though of rotating the parabola until its central axis lines up with z-axis but I do not know what this axis is. Matlab's cftool only seems to fit equations of the form z = f(x, y) and I am not aware of anything in python that can solve this.
I also tried solving for the parameters numerically. When I tried making this into a matrix equation and solving by least squares, the matrix turned out to be invertible and hence my parameters were just all zero. I also am stuck on this and any help would be appreciated. I don't really mind the method as I am familiar with matlab, python and linear algebra if need be. 
Thanks