Reciprocal relations in Roulette /glissette rollings 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.

It is an old problem that goes back to James Gregory 1668 in "Geometriae pars universalis". He invented a transformation between polar  and orthonormal coordinates 
$$ y=\rho, \,\, x= \int  \rho\, d \theta$$
There is identity of arc length between the polar curve and $(x,y) $ curve by a rolling motion and the pole runs along x-axis. A theorem of Steiner-Habich is important in the theory (pp 3-4 of the paper I Gregory's transformation). 
You can view examples here http://christophe.masurel.free.fr/#s9
All papers are open-access. 
There are also many informations in "Nouvelles annales de mathematiques" (1842-1927) -but in french language- http://www.numdam.org/numdam-bin/feuilleter?j=nam
or on Gallica.fr and also in Mathesis 
C. Masurel
