On the relevance of the differential expressions $f^2+f^{\mathrm{\prime }2}$, $f+f\mathrm{"}$ and $ff\mathrm{"}-f^{\mathrm{\prime }2}$ for the geometrical and mechanical properties of curves

We present a unified approach to known and new properties of curves by showing the ubiquity of the expressions in the title in the analytic treatment of their mechanical and geometric properties