Well, in terms of an actual FME-based implementation, I have no idea (other than implementing the whole thing in Python and sticking it in a Python Caller); however, in general, the solutions, mathematically, are mostly related to 3D Least Squares Surface Fitting. You have a cube of known geometry that you want to position in space so as to minimize the distance of points from the surface of the cube, i.e. minimizing the residuals, just like in 2D linear Least Squares (an oversimplification, I realize, but not too far from the truth). Your problem is of a type that is bread-and-butter in industry applications, like gaming graphics and industrial parts and equipment laser scanning, so there is a HUGE amount of literature on the subject. Here is a list of a few examples as a starting point (from a Google search on "3D analog of least squares fitting surfaces"):

• TypeConstrained Total Least Squares Fitting of Curved Surfaces to 3D Point-Clouds -- Seems to me to be exactly what you're trying to do, including initial guess for fitting surface, which in your case is a known cube. Paper is dense but very relevant. Just brush up on Lagrange multipliers, the Newton method, and eigenvectors!
• Least Squares 3D Surface Matching
• Least squares matching of 3D surfaces
• Direct Least-Squares Fitting of Algebraic Surfaces - a slightly different approach
• A Survey of Methods for Moving Least Squares Surfaces -- Moving Least Squares methods - another way to go
• Least Squares Fitting of Analytic Primitives on a GPU -- taking advantage of computer GPUs - practical
• Surface Reconstruction from Points
• An As-Short-As-Possible Introduction to the Least Squares, Weighted Least Squares and Moving Least Squares Methods for Scattered Data Approximation and Interpolation

HI Tcrossman,

Thank you for the feedback.

There several approaches to tis plane-fitting challenge based on point clouds.

They all seem to need some solid mathematical understanding of the problem.

Cheers,

j