If a catenary rolls on a straight line its focus traces out a parabola and vice versa.
Is it true? Are there more such examples and how are they co-related?
In case of a circle rolling on a fixed straight line we have a cycloid trace for a point on circle periphery and, when a rigid straight line rolls on a fixed circle one obtains an involute for locus for an initially fixed peripheral contact point.
Can a reciprocity be established? In other words ... if (x,y) are cartesian coordinates and (s,R) natural coordinates of a rigid curve ( arc length and radius of curvature) by means of differential calculus/geometry or otherwise could some sort of a differential reciprocal relation exist for the pair? like e.g.,
$$ f(x,y) \rightarrow g(s,R) ; \; g(x,y) \rightarrow f(s,R)? $$
Thanks in advance for all thoughts on the topic.