# Compute rotation matrix from fitting a plane with its own coordinate system

I'm trying to find rotation matrix for a plane from given 4 points $$X_1 ... X_4$$ that form a square. These points are fixed on a rigid square that can take any position in world coordinate system. I have only 3D coordinates (in world CS) $$X_1 ... X_4$$ of these points.

The algorithm that I am aware of is following:

1. From each point subtract centroid
2. Calculate SVD
3. Find normal as 3rd column of matrix $$U$$

It allows to compute a plane that fits that 4 points and a normal for that plane.

In my case, I need to compute rotation matrix (from plane coordinate system to world coordinate system).

The plane/local (more exactly, that rigid square) coordinate system is defined as:

• X axis goes in the same direction as vector from $$X_1$$ to $$X_2$$
• Y axis goes in the same direction as vector from $$X_3$$ to $$X_2$$
• Z axis goes "up", as in right-handed coordinate system
• origin is located at the centroid

Now, having $$X_1 ... X_4$$ and $$U$$, how can I compute $$R$$ that will transform any point from plane (local) coordinate system to world coordinate system?

• You’ve basically already done it. – amd May 8 at 6:57
• @amd, unfortunately, I cannot see it ;) – Simon May 30 at 9:50