The mathematicians for centuries have researched the surfaces theory. In this paper, we consider strophoidal surface in three dimensional Euclidean space E^3. We present notations of a geometry. addition, stating helicoidal surface, define and calculate its Gauss map, Gaussian curvature, mean curvature. Finally, give some relations curvature that kind surfaces.

Let $$\Gamma _{g}$$ be the fundamental group of a closed connected orientable surface genus $$g\ge 2$$ . We introduce combinatorial structure core surfaces, that represent subgroups These structures are (usually) 2-dimensional complexes, made up vertices, labeled oriented edges, and 4g-gons. They compact whenever corresponding subgroup is finitely generated. The theory surfaces we initiate here...