Oh cool! I was working on something similar (but not the same) a couple years back when trying to figure out the bounding box of an arc along the circumference of a circular watch face: https://www.desmos.com/calculator/4e5c41c2ab

I don't remember exactly what I was doing but I think you might also be able to skip the for loop by directly calculating the point on the axis by using mod or floor/ceil.