Given a certain Triangle in 3D (Kinda irrelevant to the problem since it can be solved in a plane with just transforming coordinates), I'd like to generate a set amount of points (which is set obviously by just dividing the area of the tri by a certain value) that are equidistant between each other, while filling up as much as possible of the given area of the triangle.

There should be some separation from the edges, otherwise when I do this on the polygon right next to it I'll get some overlapping points. I'd rather generate the points on the edges myself with another method.

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Generating on the edges of the triangle is simple enough, covering the surface of it is my biggest problem tho. I'm using this to generate a bunch of sampling points to take in reference for some colored lighting on objects (light reflected off surfaces), and having sampling points that cover all my meshes in a consistent manner would be of big help.

I've googled a fair bit but I haven't found much of use yet that isn't for super accurate stuff. Since this is just for calculating some lighting, accuracy is not really needed and it doesn't matter if there's much holes on the shape. It should just be fairly consistent in covering the surface in most of these polies.

You seem to choose complex way to solve some simple problem. What exactly are you trying to accomplish? Is it cube map? Specular light? Bump map?

Okay, how about you obtain these points from hardware rasterized triangle and perform the whole thing via shader?

Couldn't you do this with barycentric coordinates?(or something like it?)

ie:
Having a triangle ABC, you might get good results when going with A + i/n*AB + j/m*AC with AB and AC being the vectors from A to B or C respectively (This is just my guess, don't know what it would look like)