TITLE: Visualization of a set
QUESTION [5 upvotes]: How can I imagine the set
  $$
M:=\left\{(x,y,z)\in\mathbb{R}^3:z=xy\right\}?
$$

Is there a program that can visualize that?

REPLY [3 votes]: You can notice that $z=xy=x_1^2-x_2^2$ for $x_1=\frac{x+y}{\sqrt{2}}$ and $x_2=\frac{x-y}{\sqrt{2}}$. Using $x_1$ and $x_2$ instead of $x$ and $y$ is simply rotating the coordinate system by $45^\circ$.
Now if $x_2=0$ then $z=x_1^2$ is a parabola. So we already know some of the points of the graph. Now simply imagine that on each point of this parabola you hang downward directed parabola. (Since for any fixed $x_1$ the function $z=x_1^2-x_2^2$ is again a parabola, but with a different orientation.)
In this way you get something looking like a saddle. This surface is called hyperbolic paraboloid.
On this picture the parabolas in both directions are nicely visible:

I've taken the picture from this site. I found it simply by searching for
hyperbolic paraboloid in Google Images.