Let’s assume we want to approximate a point cloud with a second degree polynomial : .
The point cloud is given by points with coordinates . The aim is to estimate , and where will fit the point cloud as mush as possible. We want to minimize for each point the difference between and , ie. we want to minimize . The matrix form of the system is given by:
Let’s define , and :
The system is now given by:
The optimal solution is given by:
Where is the pseudoinverse of given by .