Visualization of a set

Question

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?

Answer 1

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.

Answer 2

In http://www.wolframalpha.com/ put "Plot z=xy" and see the result!

Answer 3

Having taught a multivariable calc class I usually tell my students to try to picture "slices" of the function. What I mean by this is choose a variable. In this case $x$ is a good choice. Then assume $x=1$, do the same for $x=2$ and so on. In each case you get a line and as you do this for all $x$ and "put them together" you get the picture you want.

Answer 4

The final shape is your set. I did it by Maple 16 as follows:

[> with(plots):
[> P := animate(plot3d, [k*x*y, x = -200 .. 200, y = -200 .. 200], k = 0 .. 100, frames = 60, axes = boxed):
[> display([P], insequence = true);