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?
The curvature comb consists (it seems to me) of a normal line segment at each point, whose length is directly proportional to the curvature there.
curvature only depends on the shape of the object.
You don't want a train to suddenly experience a sideways force as it enters a curve. So the 2nd derivative of the position, acceleration, should not instantly go from 0 to a constant, instead, it should grow smoothly, like pictured in TFA. So you want a non-zero 3rd derivative, and possibly even a non-zero 4th derivative.
Cubic splines of course give you a nice 3rd derivative. In fact, you can't make a perfect circle using cubic splines, because of that. Draw your device's shape in Illustrator or Inkscape, and you likely make it Apple-like :)