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7.6 A Warp Saddle

Let's look at the warp saddle
made by graphing the function
\[ \begin{align} z &\depends (x, y) \\ z &= x y. \end{align} \]
We'll take a few vertical slices by fixing \( y \)-values. Each of these slices can be seen by looking down the \( y \)-axis when standing in front of the surface.
\( \{ {y = -1} \} \)-slice.
Setting \( y \) equal to negative one,
\[ \begin{align} z &\depends x \\ z &= - x \end{align} \]
we find a line with slope negative one.
\( \{ {y = 0} \} \)-slice.
Setting \( y \) equal to zero has an interesting effect.
\[ \begin{align} z &\depends x \\ z &= 0 \end{align} \]
This slice is allowed to depend on \( x \), but is just the constant, zero. The graph consists of all points on the \( x \)-axis.
\( \{ {y = 1} \} \)-slice.
By fixing \( y \) equal to one,
\[ \begin{align} z &\depends x \\ z &= x \end{align} \]
we find a line with slope one.
Notice how the vertical slices twist as we progress \( y \)-values. The \( \{ {y = 0} \} \)-slice is level: it has zero slope. Increasing \( y \) increases the slope, and decreasing \( y \) decreases the slope.
Let's finish our exploration of the warp saddle by taking a look at its contour map.
Most contours are hyperbolas, with the notable exception of the \( \{ {z = 0} \} \)-contour.
\( \{ {z = 0} \} \)-contour.
Setting \( z \) equal to zero gives us
\[ \begin{align} &\{ {x y = 0} \} \\ &\{ {x = 0 \orSpaced y = 0} \}. \end{align} \]
In the \( (x, y) \)-coordinate plane, these are the equations for the \( y \)-axis and the \( x \)-axis, respectively.
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