Net of Parametric Curves on a Surface

a pair of one-parameter families of curves on a surface. For example, on a hyperboloid of one sheet, the two families of rulings constitute a net of parametric curves. In differential geometry, such nets are studied primarily in the small—that is, on a sufficiently small piece of a surface, within which neither the surface nor the curves forming a parametric net have singular points. Here, the curves are assumed to be sufficiently smooth and to be arranged in such a way that precisely two curves of the net—one from each family—pass in two different directions through each point of the given region.

Every system of curvilinear coordinates (u, v) on a surface determines a net of coordinate curves consisting of the two families u = const and v = const. The form of the formulas of the theory of surfaces depends on the choice of the net of coordinate curves. Thus, if this net is orthogonal, then in the expression of the first quadratic form

ds² = E du² + 2F du dv + Gdv²

the coefficient F = 0; as a result, many formulas are simplified. Besides nets of coordinate curves that can be superimposed on a surface in infinitely many ways but are not necessarily connected with the surface by any geometric relations, there exist on every surface nets of coordinate curves that are determined by the surface itself.