I think one of the best way is to intersect $f(x,y,z)=0$ with some planes. I am doing that by using Maple:
While $f=0$ intersects with $z=1,2,3,\cdots,15$, the following shapes are created:
[> with(plots); with(student); f := (1/16)*x^2-(1/9)*y^2-z^2 = 1; for i to 15 do a[i] := subs(z = i, f) end do; implicitplot([seq(a[i], i = 1 .. 15)], x = -45 .. 45, y = -45 .. 45);

While $f=0$ intersects with $x=1,2,3,\cdots,15$, the following shapes are created:
[> with(plots); with(student); f := (1/16)*x^2-(1/9)*y^2-z^2 = 1; for i to 15 do a[i] := subs(x = i, f) end do; implicitplot([seq(a[i], i = 1 .. 15)], z = -45 .. 45, y = -45 .. 45);

And finally, we have the following curves while intersecting $y=1,2,3,\cdots,15$:

Considering all cases in a $xyz$ system of coordinates we get:
