Rotated Paraboloid Surface Fitting

Question

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

Answer 1

Dont use any toolboxes, GUIs or special functions for this problem. Your problem is very common and the equation you provided may be solved in a very straight-forward manner. The solution to the linear least squares problem can be outlined as:

1. The basis of the vector space is x^2, y^2, z^2, xy, yz, zx, x, y, z, 1. Therefore your vector has 10 dimensions.
2. Your problem may be expressed as Ap=b, where p = [ABCDEFGHIJKL]^T is the vector containing your parameters. The right hand side b should be all zeros, but will contain some residual due to model errors, uncertainty in the data or for numerical reasons. This residual has to be minimized.
3. The matrix A has a dimension of N by 10, where N denotes the number of known points on surface of the parabola.

A = [x(1)^2 y(1)^2 ... y(1) z(1) 1

...

x(N)^2 y(N)^2 ... y(N) z(N) 1]

4. Solve the overdetermined system of linear equations by computing p = A\\b.

Answer 2

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)