Lets say I want to create a material similar to the one below.
I’m using UV maps so that I can simply “rig” the pupil by moving the mapping values.
However, My texture looks like it is clearly an unwrapped UV, which it is, but I’d like to make it look more like the first image, more akin to the normal anatomy of an eye where the iris curves inward. How can I do this with a UV Image?
By displacement and/or normal mapping. Make a “high poly” (not all that high) and bake to a normal map. You can still shift or scale the UV for a tangent space normal map. (Just don’t rotate it.) Could also do something similar with a bump map. Real displacement would require a lot of verts to look good, which is doable with adaptive subdivision.
Note that real eyes don’t just have an iris + pupil, they also have a lens. You could do that with the same kind of normal mapping technique described, where you basically mix with a glossy shader with a different normal/bump map (but you can’t use displacement to double-up the mesh, you’d need two separate meshes for that.)
I apologize for the late reply! I tried using an uv map to give the eyes some dimension, but it turned out like this.
I guess I’m trying to go for something like in this tutorial below where the iris is curved inwards.
The only problems is that the character, Sonic, has eyes that are not shaped like a sphere.
The angle you picked is basically the worst one possible to show off the possibility of normal mapping. Normal mapping works well when seen head on, and it works well when at a distance. If that camera angle/scale isn’t representative of a camera angle you intend to use, find one that is.
Well, that’s a bit difficult. This is an animation I’m doing so there are many angles that are being used, front and side ones included.
Use the info from these tutorials, and you should be able to get the results you want…
Okay, thanks! 
