fitting a linear surface with numpy least squares

Question

So I want to solve the equation z= a + b*y +c*x,. getting a,b,c. ie: making a (plane) surface fit to a load of scatter points in 3D space.

Answer 1

The constants a, b, and c are the unknowns you need to solve for.

If you substitute your N (x, y, z) points into the equation, you'll have N equations for 3 unknowns. You can write that as a matrix:

[x1 y1 1]{ a }   { z1 }
[x2 y2 1]{ b }   { z2 }
[x3 y3 1]{ c } = { z3 }
    ...
[xn yn 1]        { zn }

Or

Ac = z

where A is an Nx3 matrix, c is a 3x1 vector and z is a 3xN vector.

If you premultiply both sides by the transpose of A, you'll have an equation with a 3x3 matrix that you can solve for the coefficients you want.

Use LU decomposition and forward-back substitution.