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Cylinder UVs

Here are 4 cylinders with a wave texture and their UV unwraps. From left to right:

  1. Cylinder with default unwrap
  2. The same cylinder tapered with proportional editing. Tip vertices are all at the same location. No change to UVs, texture tapers to tip properly.
  3. The same, but with doubles removed. The UVs have changed, but the texture still works.
  4. The same, but re-unwrapped with Cylinder projection. The texture is distorted.

My question is: How can I make UV unwrapping like 3 for 4, or other similar shapes that didn't start with clean UVs? Can I somehow split the tip UVs of 4 to be like 3? Can it be done without Cylinder Unwrap?

I am modeling mesh hair where each strand is roughly a tapered cylinder, but wasn't necessarily formed in such a way that I have clean UVs. But I need my hair texture to taper to the tips properly.

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In mesh edit mode, mark edges as seams (in the edge menu -> mark seam). Wherever you have a seam the unwrap operator will unstitch the seam in the UV map and unfold accordingly.

How you pick edges to mark as seams is dependent on what you're unwrapping. In this case you'll want to select the vertical edges towards the tip then mark seam.

enter image description here

When you unwrap these should end up like your version 3.

enter image description here

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