We prove that if f is an orientation-preserving homeomorphism of a closed orientable surface M whose singular set Σ(f) is totally disconnected, then f is topologically conjugate to a conformal transformation.

In this paper we introduce the notion of singular evolutoid set which is all points evolutoids a fixed smooth planar curve with at most cusp singularities. By Gauss-Bonnet Theorem for Coherent Tangent Bundles over Surfaces Boundary (Theorem 2.20 in [4]) applied to extended front hedgehog obtain an integral equality periodic curves.