Comment by fouronnes3

With a fuzzy graph, what the essay shows, we plug each point into a recipe, the equation, and see how badly it fails. Big failure → dark region (like a bitter taste). Small failure → light region (almost right). It’s just showing the raw mistake you get after plugging in x and y.

With a signed distance function, instead of looking at the recipe’s mistake, we measure how far the point is from the perfect curve—like pulling out a ruler and measuring the nearest distance to the “correct” line.

It always has units of length and behaves nicely (positive outside, negative inside, zero on the curve).

So the fuzzy graph is about “how wrong is the equation here?”

A signed distance function is “how far away from the exact solution am I?”

They’re related ideas (both start from equations written as something = 0), but they’re not the same thing.

Reasonably, I ask myself: "but isn't dark region far distance and light region close distance?"

Sitting with that:

In the fuzzy graph, we’re coloring by equation error—how badly the equation is satisfied at each point.

In a signed distance field, we color by actual geometric distance to the curve.

Those two numbers aren’t the same unless the equation is written in a very special way. If you multiply an equation by some huge factor (say, multiply everything by 1,000, or divide by something small), the shape of the solution curve doesn’t change—distance hasn’t changed—but the equation’s error suddenly becomes 1,000 times larger (or smaller). That would completely change the shading in the fuzzy graph while leaving the real distances untouched.

Signed distance is measured with a ruler (pure geometry). The fuzzy graph is measuring algebraic error (how close the formula comes to zero). Both can get lighter near the curve and darker away from it, but they’re doing it for different reasons, so they’re not interchangeable.