HI..

Dividing into n number of points is another alternative option.. I mean user can redistribute vertices either by specifying spaces of by specifying number of points.. Anyhow, i need that also later. Thanks in advance ...

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Convert the vertex near the number 1 to an r theta offset from the neighbouring node,
how to find theta..??

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But if you know the spacing then the division isn't necessary; you still need theta so the r-p and p-r conversions are still both needed, but if vertex 1 isn't at an r that is evenly divisible by the spacing then you have the previously mentioned problem, what if, say, vertex 1 is at (11,45) and the spacing of 4 is requested? Would you then place the next vertex at r=1 for the next segment, i.e.:
In this case the vertex closest to P can be slightly adjusted so that old P will coincide with new vertex..

Thanks n regards