# Plot both sets and I want to highlight the intersection of A and B.

I've two problems:

Given are the two sets
$$A = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid x_{0}^{2} = \vec{x} \, ^{2}, x_{0} \geq 0 \right \rbrace$$
and
$$B = \left \lbrace (x_{0}, x_{1}, x_{2}, x_{3}) \in \mathbb{R}^{4} \mid (k_{0} - x_{0})^{2} = (\vec{k} - \vec{x})^{2}, x_{0} \leq k_{0} \right \rbrace$$

where $$\vec{x} = (x_{1}, x_{2}, x_{3})$$

and $$k = (k_{0}, k_{1}, k_{2}, k_{3})$$ should be an arbitrary point (i.e. free of choice, but fix) with $$k_{0} > 0$$. For example: $$k = (k_{0}, 0, 0, 0)$$

Now I want to plot both sets and I want to highlight the intersection of A and B.

How do I do that??? Has someone any idea? I've some basics in Maple and Mathematica, but plotting is not one my strengths.

## Answers and Replies

Dale
Mentor
2020 Award

How do you plan to display a four-dimensional set on a two-dimensional monitor? Once you have that figured out then programming it should be straightforward, but I have no idea how you plan to do that.

I'm sorry, I did a mistake. I need a 3-dimensional plot of the two sets

$$A = \left \lbrace (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} \mid x_{0}^{2} = \vec{x} \, ^{2}, x_{0} \geq 0 \right \rbrace$$
and
$$B = \left \lbrace (x_{0}, x_{1}, x_{2}) \in \mathbb{R}^{3} \mid (k_{0} - x_{0})^{2} = (\vec{k} - \vec{x})^{2}, x_{0} \leq k_{0} \right \rbrace$$

where $$: \vec{x} = (x_{1}, x_{2})$$

and I want to highlight their intersection somehow.

Dale
Mentor
2020 Award

You can directly use ContourPlot3D in Mathematica to generate the 3D plot of A and B. I don't know about any way to highlight a region directly. I would guess that you will need to simultaneously solve A and B and then probably use ParametricPlot3D to plot it with some sort of plot style that will make it visible and then Show it together with the ContourPlot3D.

Btw, A and B are cones, so I would assume that their intersections will be conic sections like ellipses etc.

Thanks, for you help.

But I have another question neglecting both sets defined above. How can I plot something like this:

http://www.theory.caltech.edu/people/patricia/gifs/glcaus1.gif" [Broken]

In my case both cones should overlap (so we have to shift one of the cones) and I'd like to highlight the intersection of both. Is that possible and how???

Last edited by a moderator:
Dale
Mentor
2020 Award

That is the same as the first question. The sets A and B are cones.