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

[parton]

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.

 

[Dale]

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.

 

[parton]

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]

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.

 

[parton]

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"

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?

 

[Dale]

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