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I am trying to figure out how to fit a plane passing through the origin in [tex]\mathbf{R}^3[/tex], given a cloud of N points.

The points are vectors of the form [tex](x_1^{(k)}, x_2^{(k)}, x_3^{(k)})[/tex] , wherekstands for the k-th point. I want to minimize the sum of squared distances point-plane.

What I came out with was the following:

- solve a 3x3 homogeneous linear system AX=0 in which the (i,j) element of A is:

[tex]\sum_{k=1}^N x_i^{(k)} x_j^{(k)}[/tex]

Now I have two questions:

1) I have read somewhere that this turns out to be an eigenvalues problem. Basically I need to find the eigen-values/vectors in order to solve the system. Why?

2) I found another method which instead builds a rectangular Nx4 matrix in which, in the first column there are all thex's of the points, in the second column all they's, in the third all thez's, and in the fourth all 1's. Then they compute the SVD and extract (I think) the last column of the rightmost output matrix. Why does that work??? What's the difference?

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# Plane Fitting with Linear Least Squares

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