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I'm looking for a way to convert a set of lines in R^3 into points in R^n so that distance between any pair of points points is a good approximation of the distance between corresponding pair of lines. Can anyone see a practical way of doing this?

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This won't be easy, since it is possible for line A to be close to both lines B and C, while lines B and C are far from each other --- that can't happen with points. –  Gerry Myerson Oct 6 '12 at 2:40
Maybe you could try transform this set in a manifold. For example, the set of all lines in $\mathbb{R}^{2}$ is a mobius band. –  Tomás Oct 6 '12 at 2:52

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