Please, help me! How I can parametrize this surface:
I have a parabolic hyperboloid. $ H := (z = xy) $ intersected by a cylinder whose base is a unit circle centered at the point $(0, 0, 1)$ I tried to use hyperboloic functions, paraboloic coordinates and e.t.c. The main difficulty for me is that I don't know how a cross-section of this surfaces looks like. Ok, maybe I can draw it, but it will not give me anything.
Also, if it is easier to find the projection on the coordinate planes than to find the parametrization, then it will even be enough for me to know the projection.
Thank you in advance!
Making the parameterization
$$ x = r\cos t\\ y = r\sin t\\ z = r^2\sin t\cos t $$
for $0 \le t \lt 2\pi$ we get in blue for $r=1$
We have also for the area element
$$ dx dy \equiv r dr dt $$
and the surface normal vector for $S(x,y,z) = z-x y = 0$
$$ \nabla S(x,y,z) = (-y,-x,1)\equiv (-r \sin t,-r \cos t, 1) $$