The set of osculating circles of a given curve in S3 forms a lightlike curve in the set of oriented circles in S3. We show that its “½-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.
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Rémi Langevin, Jun O'Hara, Conformal arc-length as ½-dimensional length of the set of osculating circles. Comment. Math. Helv. 85 (2010), no. 2, pp. 273–312DOI 10.4171/CMH/196