Level Lines and Surfaces

(also contour lines and surfaces), sets of points at which the function u(P) of point P of a plane or three-dimensional space assumes constant values. The equation u(P) = const in a two-dimensional domain defines a level line; in a three-dimensional domain it defines a level surface. Level lines and surfaces are used extensively to represent functions in meteorology, such as isotherms and isobars, and in geodesy and topography. Level lines and surfaces degenerate into points at the extrema of the function u(p). The gradient of the function u(P) is perpendicular to the level line or surface at the corresponding point.