In this demo, we show the slice curves of the function f(x,y,z) = x2 - y3 + z5. The x-slice curve is the curve obtained by fixing y = y0 and z = z0, and it corresponds to the collection of points (x,y0,z0,f(x,y0,z) in 4-space. The graph of this slice depends only on one parameter, x, so it is a curve. In the X-Slice Curve window, we map this curve from 4-space into the collection of points (x,f(x,y0,z0) in the plane y = z = 0.
Similarly, we obtain the y-slice curve by fixing x = x0 and z = z0, and then projecting down into the plane x = z = 0; and we obtain the z-slice curve by fixing x = x0 and y = y0, projecting the curve into the plane x = y =0.