How to find out if a set of point3D are in the same plane?

Question

I have a set of point3D (X, Y, Z). I need to check if they are coplanar with some sort of tolerances. My way of doing it is that: I convert all points from the Global Coordinate System to Local one, where local x,y is in the same plane of the plane defined by 3 points in the set, and z is normal to that plane. And then, all I need to do is to check if all points in the set have approximately similar local z values.

However, the tricky part is how to pick the 3 points to define the reference plane. If picked randomly, this would result in sometimes the set of points are coplanar, sometimes not. Do you have any suggestion?

Answer 1

Probably the most common way to do this would be with Principal Component Analysis: https://en.wikipedia.org/wiki/Principal_component_analysis

The short description is:

1. Calculate the 3x3 covariance matrix of the input points
2. Extract the smallest Eigenvector. That is the normal vector for the plane that is the Least-Squared-Error fit to all your points.