Lines are the most basic curves in Euclidean geometry and the plane is covered by a 2-dimensional family of lines. Circles are the most basic curves in non-Euclidean geometry and the 2-dimensional unit-sphere is covered by a 3-dimensional family of circles. How is an embedded surface the union of "simple curves"? We consider a curve to be "simple" if it is a rational curve in a covering family, such that the curve is of lowest possible degree. Thus lines in the plane and circles in the unit-sphere are simple curves. For a given surface we can construct a labeled graph whose vertices are families of simple curves. Two vertices in this "simple family graph" are connected by an edge, if general simple curves in the corresponding two families intersect in at least two points. We label the edge with this intersection product. We label a vertex with the dimension of its corresponding family. We classified the graphs that are the simple family graph of some real surface. Our goal is to determine what a simple family graph says about the surface itself. Three disjoint vertices in the graph define a hexagonal web of curves. A discretized hexagonal web divides the underlying surface into triangles such that many hexagons appear. If a simple family graph of a smooth surface contains a vertex with label at least 3, then the projective closure of this surface is biregular isomorphic to the projective closure of the unit-sphere. This explains why the projective closure of the unit-sphere is a natural model for non-Euclidean geometry. If a simple family graph contains two disjoint vertices, then the surface admits a parametrization of minimal bidegree (d,d) where d is the degree of a simple curve on the surface.