understanding projections. I would like some help trying to understand the projection on a plane in $R^3$.
As an example let's say I have a cone given by the set $\{(x,y,z) \in \mathbb{R}^3 : x^2 + y^2 = z^2, z>0\}$
Why is it that if I write the $z$ explicitly and take the set of points given by $(x,y,\sqrt{x^2+y^2})$  (substituting in the $z$ I found) I have the projection of the cone on the $xy$ plane? This set of points are still in 3 dimensional space and belong to the cone.
What is the source of my confusion? How should I be looking at it?
 A: It sounds rather like you are talking about the projection $(x,y,z)\mapsto (x,y,0)$, which projects all of 3 space onto the plane. When restricted to the cone, it still maps onto the plane (excuse me for throwing in the origin.)
The other map $(x,y)\mapsto (x,y,\sqrt{x^2+y^2})$ maps the plane onto the cone. You can, of course, consider the domain to be $\Bbb R\times \Bbb R \times \{0\}$.
A: First of all, you are missing half of the solution when you solve for $z$, above. Remember, when we take a square root, there is always a positive and negative answer. Also, the ordered triple that you have, $(x,y,\sqrt{x^2+y^2}) \in \mathbb{R}^3$ will not be just in the $x$, $y$ plane; if we rewrote what that vector is in $\mathbb{R}^3$ we would have $(x,y,\pm \sqrt{x^2+y^2})= (x,y,z)$ which is exactly what was give. 
I think what you are seeing is correct, though: $\pm \sqrt{x^2+y^2}$ is the projection onto the $x$, $y$ plane for a more particular solution. One of the most misleading things is that teachers often confuse this statement: $C = x^2 + y^2$ with this statement $z = x^2 + y^2$. The difference being that in one case $z$ changes, and in the other $C$ is constant. When $C$ is a constant we can go ahead and solve for $y$ giving $y = \pm\sqrt{C - x^2}$ and when we define 
$$f(x) = y = \sqrt{C - x^2}$$ $$g(x) =-\sqrt{C - x^2} $$ we get the whole projection, at a particular slice of the cone. 
This is very different than solving for $z$ in which case you are defining a function of two variables! 
Actually, the generalized projection onto the $x$, $y$ plane when we bound $z$ is a filled in circle. You can think about this by computing one projection for some $C$ then a little bit larger, and you get this concentric circle pattern. 
