I've messed about with that sort of thing before: https://www.youtube.com/watch?v=pe4UAmb2wJw.

I never got that far - I was primarily considering it as a point location definition for an image format, and I wanted a symmetrical curve.

You can actually do that, if you relax the constraint that no grid point is shared (it's not really that big a problem, you just need to base the transform on the location between pixels, and not the centres of the pixels).