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This question is a little theoretical. Say I have 3 points that are bound within a certain 3D volume of some shape (i.e. box, cylinder, cone, etc). By using PCA, I find the most two dominant directions and reduce the data dimension to 2D.

All of this is done by the following:

  1. Compute 3x3 covariance matrix of the 3D data.
  2. Take eigenvectors and eigenvalues of the covariance matrix.
  3. Sort by eigs by descending order of eigenvalues.
  4. Take first two largest eigenvectors and multiply them with the data. The data is now 2 dimensions.

My question is: What properties are maintained through the projection? For example, if the original 3D data was bound by a radius of size r (say in the X and Y directions only), does this apply also to the 2D data? What happens to the range of the data? The distances? What can I infer about the new 2D data from knowing the characteristics of the 3D data?

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The new 2D data is the projection of the 3D data onto the tangential plane of the underlying 3D shape. This should allow you to answer most of your questions by a simple drawing.

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