Answer
Please see the figure attached.
Notes:
- Red dashed curves are the vertical traces parallel to the $xz$-plane
- Black dashed lines are the horizontal traces
Work Step by Step
We have $f\left( {x,y} \right) = \sin \left( {x - y} \right)$. By setting $y=a$ we fix the $y$-coordinate and obtain the vertical trace curve $f\left( {x,a} \right) = z = \sin \left( {x - a} \right)$ that lies in the plane parallel to the $xz$-plane. Similarly, we can set $x=a$ to fix the $x$-coordinate and obtain the vertical trace curve $f\left( {a,y} \right) = z = \sin \left( {a - y} \right)$ that lies in the plane parallel to the $yz$-plane.
The horizontal traces can be obtained by setting $c = \sin \left( {x - y} \right)$.
${\sin ^{ - 1}}c = x - y$,
$y = x - {\sin ^{ - 1}}c - 2\pi n$, ${\ \ }$ for $n = 0,1,2,3,...$
Thus, the horizontal traces are the lines $y = x - {\sin ^{ - 1}}c$ that lie in the plane $z=c$.
