This demo shows the three slice surfaces of the function f(x,y,z) = x2 - y3 + z5 at the hotspot P. In the "4D Color Graph" window, we slice the color graph by a horizontal xy-hyperplane and two vertical xz- and yz-hyperplanes and the slice surfaces are depicted in the corresponding 3D Graph.
You can rotate the slice surfaces around to see how the color of point (x,y) in the plane corresponds to the function value f(x,y,z0). Move the hotspot around to look at the slice curves of different point in the domain.
Even though the hyperplanes appear as planes, they are really 3-dimensional objects: One of the three domain coordinates is fixed, and at each point, the fourth coordinate is free. Thus, any point in the xy-hyperplane can take on all possible colors.
Finally, the "All Slice Curves" window shows how the hyperplanes intersects the function graph: at each point f(P) in the intersection set with the function graph, we get the color that the function graph has at P.