Niveaukurven und Höhenlinienkarten

Consider a topographic map that shows the elevation of a function f.

On this map, each curved line represents a fixed elevation above the xy plane. These lines are level curves, also known as contour lines.

Mathematically, each level curve corresponds to the set of points where f of x, y equals k, where k represents a constant height.

Suppose a point moves along one of these contour lines. As long as the point stays on that line, its elevation remains constant.

This means the function value remains constant at every point along the path.

These level curves result from slicing the three-dimensional surface at different heights.

The slices are then projected onto the flat xy plane, forming a two-dimensional contour map, which shows how the surface height varies across horizontal positions.

This concept can be used to draw a 2D map of a 3D mountain, providing a detailed visualization of surface variations on a flat plane.

Where the lines are close together, the slope of the terrain is steep; where the lines are farther apart, the slope is gentler.