- #1
mnb96
- 715
- 5
Hi,
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?
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?