calculus - Intuitive characterization of the graph of a twice differentiable function

In high school textbooks, the following characterizations are often found:

A function is continuous if its graph can be drawn without lifting the pencil.

and

A function is differentiable if its graph has no sharp bends.

Is there a similarly intuitive characterization for twice differentiable functions based on the shape of the function graph alone? Of course, we can say things like "a function is twice differentiable if the graph of its rate of change has no sharp bends" but that does not really help when trying to recognize the graph of a twice differentiable function.

A notable example for where this could be relevant is cubic B-splines, which are widely used in design and architecture and often appear as "perfectly smooth" shapes, although they are in general only twice differentiable at the joints between the individual spline polynomials.

Answer

At a point where a function is not $C_2$ (i.e. not twice differentiable), there will be a discontinuity in the curvature of its graph. In other words, the curvature will jump abruptly from one value to another. Designers can see these jumps, if they are large enough, but many people can't. You can try some experiments with circular arcs that join tangentially, to judge your own sensitivity and powers of observation.

However, the jumps become very visible if the curve is used to create a surface, and then light is reflected from that surface. Designers like to look at the reflections of straight lines in their surfaces. These are typically called "reflects" or "reflection lines" in the design business. If a surface has a non-$C_2$ join, then its reflection lines will have sharp corners where they cross this join, which will look pretty ugly. So, surfaces of car bodies (for example) are always $C_2$. Actually, they are often $C_3$, or better, which ensures that the reflection lines are $C_2$.

So, cubic b-splines are not used very much in the design of car bodies. They are OK for small surfaces or non-glossy surfaces, where the reflections are not so visible, but not for large shiny ones. Car designers typically use higher degree splines, not cubics.

There's another wrinkle. The curves used in design applications are usually parametric ones. For these, the relationship between continuity of derivatives and continuity of curvature is quite complex. Of course, designers only care about curvature; they don't care about derivatives.

For further info, see this question/answer and this paper.