trace

The cross-section left when a surface is sliced by a plane that freezes one input variable at a constant.

A trace is the intersection of the surface $z=f(x,y)$ with a plane parallel to one of the coordinate planes. Setting $y=b$ and reading off $z=f(x,b)$ produces a single-variable curve in the plane $y=b$ — the trace in that direction; setting $x=a$ produces the analogous trace with $y$ left free. A trace is not the same object as a level curve: a trace fixes an INPUT coordinate and lets $z$ vary, while a level curve fixes the OUTPUT $z$ and lets both inputs vary.

Examples

For $z=4-x^2-y^2$ , the trace at $y=0$ is the parabola $z=4-x^2$ , vertex $(0,0,4)$ .