# Visualizing certain sets

I'm not sure if this question is appropriate for this site, so I appologize in advance if it's not.

For the purpose of a certain project I am working on, I would like to have the following sets plotted in 3D: $A = \{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_1 \geq x_2, x_3 = 0\}$

$B = \{(x_1,x_2,x_3) \in \mathbb{R}^3 : x_1 = x_2 \geq x_3 \geq 0\}$ and

$C = \{x \in \mathbb{R}^3 : x_1 \geq x_2 = x_3 \geq 0\}$.

Note that these are not subspaces. I do not know any sort of mathematical software, but if someone could let me know how I could plot these sets in 3D in a (preferably free) program then I would appreciate it.

You see, I imagine that plotting 3 different hyperplanes would be easy, but the fact that these sets are not hyperplanes (they are triangle shaped, I believe) makes it harder. I would also like to find the angles in each triangle.

I just read about gnu-plot. Maybe that could help?

Thanks.

I think you're best off trying to understand what these sets are, then you can use any 3D graphing tool (e.g. GNU Plot, MATLAB, Python with matplotlib, etc) to plot them. Here are some tips:

• Each part of your set definitions defines a plane. The set is restricted to the plane defined by the equality and bounded by the planes defined by the inequality.
• Each inequality bounds the set to one half of the plane or the other. You can plot the line represented by the intersection of the "equality plane" and each "inequality plane" to identify the boundary of your set on the plane.
• You can easily determine which parts of the plane are inside or outside your set based on picking points on the plane and testing them against the inequalities.

The basic idea is to find the equality plane and then the boundaries of the set on this plane. You can then plot the set (or part of it since it's an infinite triangular region in these cases) based on this information. Going through this process for your set $C$ gives the following figures which were made using lines and patches in MATLAB. Python is free and has similar capabilities using matplotlib (e.g. http://matplotlib.org/examples/mplot3d/polys3d_demo.html).

Plot $x_2=x_3$:

Plot the set boundaries on this plane (red dashed lines) which are just the inequality planes intersected with the equality plane:

Figure out the relevant region (e.g. by picking points and testing the inequalities) and plot just that region, this is your set: