You never see a flat rollercoaster track go into a perfectly circular loop-the-loop because while that track is continuous in the first derivative, it is not continuous in the second derivative, and would make for a very jarring ride.
Entering the circular loop from a horizontal track would imply an instant onset of the maximum g-force (as would a direct transition a circular path with smaller radius of curvature). An immediate transition from one radius of curvature to another would give a continuous, smooth track, but with discontinuous second derivatives. Clearly, a function with continuous higher derivatives would be preferable.
Roller Coaster Loop Shapes
So is there a special curve with which to make the optimal photo studio wall?
This is why car surface designers, who are in many ways reflection pattern designers, worry about continuity and smoothness in the higher derivatives.
- A Periodic Table of Form: The secret language of surface and meaning in product design by Gray Holland
