Canonical frame of a curve on a conformal plane

We show how differential geometry of smooth curves on the conformal plane can be studied by Élie Cartan’s method of exterior forms and moving frames. We find the canonical form of the derivation equations of a curve (which is not a circle) in the case of a semi-isotropic frame. We give a new proof of the theorem that states that curves of constant (in particular, zero) conformal curvature are loxodromes. We integrate the system of structure equations of the isotropy subgroup of a point.