[nequo]

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?

[yshklarov]

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 like this to get a quantity independent of the parametrization, but it is essentially a "normalized" second derivative).

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.

[yshklarov]

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.

[legosexmagic]

thats pretty much it yes. the reason they dont just say 2nd derivative is that the derivative isnt a geometry property. you can have two curves with the same shape but different derivatives.

curvature only depends on the shape of the object.

[phkahler]

It's called C2 continuity.

[kelsolaar]

The Continuity of Splines: https://youtu.be/jvPPXbo87ds

[harywilke]

I came here to post exactly this excellent video on splines by Freya Holmér. This discontinuity of reflections effect is directly mentioned at 29 minutes in chapter 4 on geometric continuity.

[milosmns]

Thank you.

[jsjohnst]

Actually, you only need G2 continuity for the reflection to be continuous (yes, C2 is also G2 for regular curves).

[yshklarov]

Do you know whether it's usual nowadays in industrial design to make curves that are actually smooth (I mean, derivatives of all orders are continuous) and not just C2? Or is it technically challenging for some reason, or not useful?

[amelius]

Isn't this sort of thing common in the automotive industry?

[nine_k]

It's very common on railways.

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 :)

[pmahoney]

See also "Why roller coaster loops aren’t circular anymore"

https://news.ycombinator.com/item?id=32005421