I can’t post images but imagine a circle centered on $0$ with radius $5$ and a point at $(3,3)$ and a point at $(4,4)$.
I know that this is a circle equation but there something I can’t figure out. If I want the distance between the point $(4,4)$ and the circle, I would use Pythagorean’s theorem and so the distance is $\sqrt{4^2+4^2} - r$ right? But there is another equation that gives a number equal to $0$ on the circle, increases positively as we go far away from the circle and increases negatively as we step inside the circle. This equation is $x^2+y^2=r^2$. With the point $(4,4)$, we get a distance of $4^2+4^2-5^2=7$, but this is not the real distance which is $\sqrt{4^2+4^2} - 5= 0.65$ in this case. So the question is: what does the result of the circle function applied to a point not on the circle mean?
If we apply the formula with the point $(3,3)$ we get a negative number but what the heck does this number mean? The only thing I know is that if it’s $0$, it’s on the circle, if it’s negative it’s inside the circle and positive outside. But it doesn’t seem related the to the real distance so what does it represent?
The point of the circle which is closest to $(4,4)$ is$$5\left(\frac4{\sqrt{4^2+4^2}},\frac4{\sqrt{4^2+4^2}}\right)=\left(\frac5{\sqrt2},\frac5{\sqrt2}\right)$$and therefore the distance from $(4,4)$ to the circle is$$\left\lVert(4,4)-\left(\frac5{\sqrt2},\frac5{\sqrt2}\right)\right\rVert=4\sqrt2-5.$$You can apply the same idea to $(3,3)$; you will get $5-3\sqrt2$.