I need to add something that isn't totally clear, the geometry I'm working on has thickness, a front and a back, and I'm trying to map one side only. So faces have been selected and UV'd only on one side.

The squared off outer edge of my UV map I presume are because that edge joins the rest of the geometry. If I separate the side from the rest of the object then that outer edge becomes smooth.

My next step in this is to remake the geometry from scratch on a perfect X,Y or Z, and see if I can get it to work.

Obviously it's solvable or else what have people been doing to make eye balls all these years.

Dave, yeah I did delete the sides when I made the disc originally.