This first demonstration starts with the graph the function f(x,y) = x² - y² along with its x- and y-slice curves. In the control panel there are two tape decks that allow you to choose x₀ and y₀. Then, in the window labeled Domain, the lines (x₀,y) and (x, y₀) are drawn.

Under the function f(x,y), these two lines are mapped to the red and green slice curves that each appear in their own windows and on the surface itself. They are called slice curves because they can be thought of as the intersection of the function graph with vertical slicing planes. To view the vertical slicing planes, check the two Checkboxes marked (x,y₀,z) and (x₀,y,z).

The x- and y-slices are produced by slicing the function graph through the point f(x₀,y₀) along the x and y-directions, but in general, we can slice along any direction in the xy-plane.