# How to easily calculate lengths (or relative lengths) of paths between symmetry points in BZ

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I am trying to easily calculate the length between special kpoints within the BZ of the 32 point groups in a crystal system. I am calculating the lengths in order to scale k point sampling along these paths for electronic structure calculations. I could use the reciprocal lattice vectors to calculate each and every special symmetry point for every Bravais Lattice to get the lengths but I am wondering if there is an easy way to at least find the ratio between the lengths that I am just missing. I for example want to find the lengths of the paths for Rhombohedral Bravais Lattice crystal so I use the paths

but for shorter parts of the paths I want to sample less k points and for paths that are longer I sample more (so that they're for the most part evenly spread out)

I am under the impression that since the fractional coordinates of some these points are dependent to the cell's reciprocal lattice vectors, that some of the path lengths change when the lattice vectors change. So I am wanting to have an easy way to get these lengths without hardcoding all of the equations with if/else statements. I fear I might just have to do that. Hopefully you can tell me otherwise!

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