Question 10.6.8: Sketching a Hyperbolic Paraboloid Sketch the graph of the qu...

We first consider the traces in planes parallel to each of the coordinate
planes:

\text { parallel to } x y \text {-plane }(z=k): 2 y^2-x^2=k \text { (hyperbola, for } k \neq 0 \text { ) },

\text { parallel to } x z \text {-plane }(y=k): z=-x^2+2 k^2 \text { (parabola opening down) }

and  \text { parallel to } y z \text {-plane }(x=k): z=2 y^2-k^2 \text { (parabola opening up). }

We begin by drawing the traces in the x z \text { – and } y z \text {-planes, as seen in Figure 10.61a. } Since the trace in the x y \text {-plane is the degenerate hyperbola } 2 y^2=x^2 \text { (two lines: } x=\pm \sqrt{2} y \text { ), } we instead draw the trace in several of the planes z=k \text {. Notice that for } k>0 \text {, } these are hyperbolas opening toward the positive and negative y \text {-direction and for } k<0 \text {, } these are hyperbolas opening toward the positive and negative x \text {-direction. } We indicate one of these for k>0 \text { and one for } k<0 \text { in Figure } 10.61 b \text {, } where we show a sketch of the surface. We refer to this surface as a hyperbolic paraboloid. More than anything else,
the surface resembles a saddle. In fact, we refer to the origin as a saddle point for this graph. (We’ll discuss the significance of saddle points in Chapter 12.)

A wireframe graph of z=2 y^2-x^2 \text { is shown in Figure } 10.61 c \text { (with }-5 \leq x \leq 5 and -5 \leq y \leq 5 \text { and where we limited the } z \text {-range to }-8 \leq z \leq 12 \text { ). } Note that only the parabolic cross sections are drawn, but the graph shows all the features of Figure 10.61b.
Plotting this surface parametrically is fairly tedious (requiring four different sets of equations) and doesn’t improve the graph noticeably.