Contour lines are used when maps are designed to show the physical nature of the land. They do this by linking all points which are the same height above sea level. The width between the contour lines indicates the steepness of gradients or slopes in the area. The closer the lines are together, the steeper is the slope.
On physical maps giving the height of mountains, rivers, lakes and principal towns all areas between certain heights are generally shown in the same color. This is known as layer coloring.
Other methods for indicating heights include relief maps moulded in plastic to the physical feature raised as on a model. Spot heights may be shown, but these merely give the heights above sea level of certain points of the map. It does not follow that the ground rises evenly from one point to another. Very old maps have mountains drawn on them. Later ones have lines called hachures radiating from a central point, with longer lines to show gentler slopes.
Another system is to show the form of land by hill shading. But none of these methods is so effective as the use of contour lines. More generally, a contour line for a function of two variables is a curve connecting points where the function has the same particular value.
The gradient of the function is always perpendicular to the contour lines. When the lines are close together the magnitude of the gradient is large: the variation is steep. A level set is a generalization of a contour line for functions of any number of variables.
