How do I evenly distribute a grid of dots within an amorphous shape?

I have this amorphous blob shape outlined by evenly spaced dots. I am looking for a way to fill the shape with dots so that they are evenly spaced, and possibly on a grid. The filled shape on the right is along the lines of what I'm shooting for, but with the dots spaced more perfectly. I'm okay with the dots changing scale if they need to accommodate each other in the areas where the blob gets thinner — so long as the dots on the outside remain perfectly on the path. Any ideas?
