Derivatives of the curvature of a plane curve

Question

I would like to know what is the name of the derivative of the curvature of a plane curve. It should be called "sharpness", but I cannot find a reference. There should also be a connection with the jerk. I have the same question for the second derivative of the curvature, which has a connection with the snap. These topics are involved in the $G^4$ interpolation with splines. Thanks.

Answer 1

Different terms are used in different fields.

I have often heard car stylists refer to the "acceleration" of a curve, by which they mean the rate at which the curvature (or radius of curvature) is changing. When they say a curve has "a lot of acceleration", they mean that there is a large change in curvature over the length of the curve, which in turn means that the rate of change of curvature must be large.

They never say whether they are thinking of the rate of change of curvature with respect to arclength or some other parameter. Of course, arclength is the only parameter that makes much physical sense.

In physics, dynamics, and design of machinery, roads, and railway tracks, rate of change of acceleration is called "jerk". Since acceleration is closely related to curvature (especially when a curve is being traversed at constant speed), jerk is related to the derivative of curvature. In the design of railway tracks, people use special transition curves to avoid discontinuities of curvature.

For some related material, please see my answer to this other question.