Is it provable that you can't have a spline construction with all three of these properties? Given the sheer number of spline techniques available, I assume there is no clear 'winner,' but I'd be interested in seeing if there is a formalization of that.
I'm still unclear why C2 is desirable over C1. I don't think a C1 spline would look bad, is there a good reason people want this?
@zbp I seem to remember that it is. In fact, it would seem to basically follow from the examples shown in these slides (and some basic linear algebra). If you ask for interpolation + 2nd order continuity, you end up with a natural spline, which requires a global solve. If you ask for 2nd order continuity + locality, you're essentially forced into B-splines, which don't interpolate. And if you ask for locality + interpolation you get Bézier/Hermite splines, which only ensure 1st-order continuity.
@tcl1 Great question. For curves, it's actually quite easy to motivate from the perspective of dynamics/animation: the 2nd derivative of a curve corresponds to the acceleration. For instance, imagine this curve describes the path of a camera. If you don't have C2 continuity, you will experience "jerky" motion. Since F = ma, this means that if a doesn't change continuously, there can be a sudden change in force. Something like someone slamming on the breaks! Or sudden turbulence on an airplane. Doesn't make for very nice animation---and certainly undesirable if you're using splines to drive motion for a vehicle/rollercoaster/robot/etc.
From a geometric/visual point of view it's maybe a bit easier to motivate for surfaces rather than curves. If you have only 1st order continuity for surfaces, then you notice it in the reflection lines. For instance, if you have a car body with a shiny coat of paint, there will be a "jump" in the reflection along curves of only C1 continuity.
Another fascinating (though less "real world") example is that if you have only C1 continuity, surfaces can behave in weird ways---for instance you can embed the flat torus into three dimensional space.