I have this math problem:
Let $S$ be the surface that consists of all points $(x,y,z)$ that satisfy the equation $x^2+y^2=z^2$.
1) What are the intersections of $S$ with horizontal planes $z=h$?
2) What is the intersection of $S$ with the $yz$-plane $x=0$?
3) Describe $S$.
4) Describe the space curve $C$ defined by $r(t)=tcost(t)i+tsin(t)j+tk$ and verify that $C$ lies on S.
I have no idea where to start. Thanks for the help.
Note that $\phi_{+}: \mathbb{R}^2\to S$, defined by $$\phi_{+}(x,y)=(x,y,\sqrt{x^2+y^2}),$$ is a local map of $S$, as well as $\phi_{-}: \mathbb{R}^2\to S$, defined by $$\phi_{-}(x,y)=(x,y,-\sqrt{x^2+y^2}).$$
1) The intersection of $S$ and the horizontal plane $z=h$ is a circle of radius $h$.
2) You can to think like the item 1), but you go to found two lines.
3) $S$ is described by your local maps.
4) You can to see that $C$ is a spiral curve.
Errata: We have a problem when $(x,y,z)=(0,0,0)$, because we don't' have the tangent plane in this point. So, $S-\{(0,0,0)\}$ is a smooth surface, non-connected.