Ok, I stumbled accros a repeated problem while unwrapping to a trimsheet texture. I often want to unwrap a “bent” mesh and lay it out straight to have the texture bend with the mesh, but blender refuses to stretch the texture like I want it to. Here’s a screenshot pointing out the issue. Any Ideas?
Hello,
I think you need some vertical subdivisions (Try Ctrl+R). If your poligons are too narrow the texture has wrong interpolation along them.
Hope it helps.
Here’s what happens with extra edge loops. I changed the texture to clarify the effect. As I said, it happens on multiple objects. I was kind of hoping for a hidden button somewhere that magically fixes this issue, but I havent been able to find it yet 
I have a mind to import the mesh in another 3d tool to see if it is a blender only thing.
The basic primitive is a triangle, even if what you see are 4-gons. The deformed mesh keeps the UV map in a triangle basis, thats why you see the zigzag lines.
In this image is more evident (the tri is clearly visible in the tweaked polygon):
This is a common issue with triangle based 3d apps.
Subdivision minimizes the effect.
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Even, no need for Edit the mesh, just add a Subdivision Surface modifier:
Check the Keep Corners in the Advanced settings of the modifier.
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Wow, I see. Thanks for the excellent answer!
I was experementing with a worklow with low polycounts. At least I know now that this isue is to be expected, then. I’ll keep working on it, thanks for the help.
LoPoly is ok, you only have to deal with this issues when deforming the mesh after UV unwrapping.
Good Luck.
See also wikipedia Texture Mapping → Affine texture mapping… it’s faster but incorrect… if you really wanna know 
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That´s the real technical explanation of whats happening. I have a tendency to simplify.
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I wasn’t implying any un-knowledge… i came across it lately and wanted anyway to reasure me about the correct tech term… so it was pure egoism 