level curve

The set of points where $f(x,y)$ equals one fixed output value $k$ , drawn flat in the $xy$ -plane.

A level curve of $f(x,y)$ is the set of every point where $f$ takes one fixed output value $k$ : the curve $f(x,y)=k$ , drawn flat in the $xy$ -plane. It is exactly the set the surface $z=f(x,y)$ crosses when sliced by the horizontal plane $z=k$ , then projected straight down — the curve carries no height information of its own, only the label $k$ written beside it does. Level curves for different values of $k$ never cross, since one point can have only one output.

Examples

The level curve of $f(x,y)=y-x^2$ at $k=3$ is the parabola $y=x^2+3$ .