Canonical frame of a curve on a conformal plane

It is shown how one can investigate a differential geometry of smooth curve on conformal plane by the Elie Cartan method of exterior forms and moving frame. We find the canonical form of the derivation equations of a curve (the latter not being a circle) in case of semi-isotropic frame. We give a new proof of the theorem that the constant (specifically, zero) conformal curvature curves are the rhumb line. We integrate a system of structure equations of the isotropy subgroup of a point.