Does there exist a transformation between a loop and a close loop ?

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SUMMARY

The discussion centers on the transformation between a line structure and a closed loop structure, specifically in the context of studying vibration modes. Yen asserts that a transformation exists, but it is not linear, referencing the Riemann sphere as an example. The challenge lies in the mapping of poles, where one pole is finite and the other is at infinity, complicating the transformation process.

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Does there exist a transformation between a line and a closed loop ?

Dear All:
For a line structure(say a very long atomic chain) and a closed loop structure( connect the head and tail of this atomic chain). Does there exists a transformation between these two structures?
For example if we want to study the vibration mode of these two cases. If we already know all the modes of one case then are we able simultaneously know the information of the other modes without any further calculation?

Best Yen
 
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