Conformal arc-length as $\frac 1 2$-dimensional length of the set of osculating circles

The set of osculating circles of a given curve in \boldsymbol S^3 forms a lightlike curve in the set of oriented circles in \boldsymbol S^3 . We show that its “ \frac 1 2 -dimensional measure” with respect to the pseudo-Riemannian structure of the set of circles is proportional to the conformal arc-length of the original curve, which is a conformally invariant local quantity discovered in the first half of the last century.