Ummmm... You're just plotting a function of 2 variables (R^2 → R) as a heat map.

"Note that the Shadow Circle is invisible in the conventional graph. In fact, the conventional graph looks identical to a conventional graph of the x=0 equation (as if the denominator was not there)."

Ummm... Yeah, because the equation x / (x^2 + y^2 - 1) = 0 simplifies to x = 0. Your "fuzzy graph" is actually just a plot of the function z(x, y) = |x / (x^2 + y^2 - 1)|, where z is encoded as a color.