The extended plane \(\mathbb{C}\cup\{\infty\}\) is considered as a 2-dimensional angle shape space associated to the Euclidean plane [\textit{J. A. Lester}, 52, No. 1--2, 30--54 (1996; Zbl 0860.51009) and Aequationes Math. 52, No. 3, 215--245 (1996; Zbl 0860.51010)]. The equivalence problem between two spherical curves with respect to the group of conformal transformations on the sphere preserving a fixed point is investigated. The group corresponds to the group of similarities of the Euclidean plane. The Shirokov invariant determines a plane curve up to direct similarity. Selfsimilar curves are studies.