Curvature Using Circles

Curvature Using Circles

Given the equation $(x - h)^2 + (y - k)^2 = r^2$ representing the family
of all circles of radius r at the point $(h,k)$ if we try to form the
differential equation representing this family we find an equation of the
form
$$\kappa = \frac{1}{r} = \frac{y''}{\sqrt{(1 + y'^2)^3}}$$
which is surprisingly the equation for the curvature of a plane curve
(ignoring absolute values which arise in the derivation).
But this expression was derived interpreting $y$ as part of a circle, how
in the world can one justify plugging in other functions & saying this
represents the curvature of that curve at the point? I know you're trying
to say that, locally, the curve moves as though it were moving along the
arc of a circle of radius $r$, but there seems to be a jump in my mind as
to how you get that interpretation out of what I've written.