LINE TRACED ON A SURFACE

Lines (or curves) traced on a surface can be classified into three categories:

- the geometric lines, that only depend on the geometry of the surface:

 Line curvature asymptotic geodesic pseudogeodesic geodesic circle Definition (see the notations) extreme normal curvature zero normal curvature - normal curvature = curvature of the curve - minimal length normal curvature proportional to the curvature constant geodesic curvature Differential condition  or  - the topographic lines, based on a given vertical direction, supported by .

 Line level slope helix crest, or thalweg Definition Constant altitude Maximal slope Constant slope Differential condition   - the gravitational lines, depending on a given gravitational field .

 Line flow catenary brachistochrone Definition line followed by a massive point. - shape assumed by a homogeneous wire placed on the surface - homogeneous line the center of gravity of which has minimal altitude. - minimal travel time Differential condition   where 