Determining eigenvectors about pca_transform

[mkobos]

First, that's an interesting application of PCA that you have in mind. It seems that you want to apply it in some computer graphics problem. I'm more of a data analysis guy, so this is something new to me.

If I understand correctly, you have a cloud of points in a 3-dimensional space and you want to find a 2-dimensional plane that "fits" them best. In such case, you can compute the eigenvectors and select the first two ones (the vectors are ordered according to importance, i.e. from the ones along which the variance is the largest to the ones along which it is the smallest). These two vectors will span the plane that you're looking for. In order to compute the normal vector of the plane, you need to compute the cross product of these two vector.

Replace the last three lines in your snippet with the lines below to see how this works in practice:

          Vector3d v1 = new Vector3d(vectors[0][0], vectors[1][0], vectors[2][0]);
          Vector3d v2 = new Vector3d(vectors[0][1], vectors[1][1], vectors[2][1]);
          System.out.println("v1="+v1+", v2="+v2);
          Vector3d vCross = new Vector3d();
          vCross.cross(v1, v2);
          System.out.println("cross="+vCross);

The significant part of output produced by this is

v1=(1.0, 0.0, 0.0), v2=(0.0, 0.0, 1.0)
cross=(0.0, -1.0, 0.0)

BTW, the eigenvectors correspond to columns and not rows of the matrix returned by pca.getEigenvectorsMatrix().

I'm assuming the components that aren't useful are being dropped.

Yes, the components along which the variance of the training points is very small are removed.