The curvature combs are confusing to me. Is this equivalent to saying that Apple’s rounded corners have a continuous second derivative while naïve rounded corners only have a continuous first derivative but a discontinuous second derivative? Or am I misunderstanding this?
That is roughly correct. For a curve with smooth parametrization f(t) = (x(t), y(t)), with f'(t) ≠ 0 for all t, the curvature at the point f(t) is defined to be κ := |(f'(t)/|f'(t)|)'| -- the magnitude of the first derivative of the unit tangent vector. This is a nonnegative quantity, and is well defined, that is, independent of the specific parametrization of the curve (as legosexmagic said, you need to do something…
Oops, my mistake: that is not the definition of curvature! The outer derivative must be taken with respect to arclength, rather than with respect to t.