Parameterizing a wierd surface.

Question

I need to parameterize the following surface so I can display it in Mathematica. $$ x = -\cos(yz) $$ Help on setting the bounds so that it fits within the following bounds is also appreciated. $$ \begin{array}{rcccl} 0 &\leq& x &\leq& 0.5 \\ -1 &\leq& y &\leq& 1 \\ -1 &\leq& z &\leq& 1 \end{array} $$ Thanks!

Answer 1

Your plot is empty: for $-1 \le y \le 1$ and $-1 \le z \le 1$, we have $-\pi/2 < -1 \le yz \le 1 < \pi/2$, consequently $\cos yz > 0$ and $-\cos yz < 0$. Your stated range for $x$ does not result in any such points satisfying the given equation.

Ignoring your stated range for $x$, we can use the Mathematica command

ParametricPlot3D[{-Cos[u v], u, v}, {u, -1, 1}, {v, -1, 1}, PlotRange -> All]

to get a plot with the natural parametrization $y = u$, $z = v$.