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I have two points on a one-dimensional Euclidean submanifold, say the x-axis.

I want to assume that this subspace is kind of "cyclic". This is often accomplished with the compactification [tex]R\cup \{ \infty \}[/tex]

The question is: How can I compute distances (up to some constant factor) between two points taking into account this sort of "cyclicness" ?

My idea was to use the complex plane, translate the x-axis vertically so that it passes through the point [tex](0,i)[/tex] and apply a Möbius transformation [tex]1/z[/tex]. Now all the points [tex]z=x+i[/tex] where [tex]x\in R[/tex] are mapped onto a circle, and I could use the shortest arc between the two corresponding points.

- Is this actually correct?

- Is the "shortest arc" length the correct metric to use?

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# Distances, compactification & Möbius transformations

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