How to generate a 4 D plot.

Dear Friends,

We have four variables: ,,, with conditions ,. We want to plot this function

with assumption that .

How can we generate the plot for F?

[Walter Roberson]

You have 4 independent values and 1 dependent value, so your plot would have to be 5 dimensional. Did you have some ideas on how you wanted to encode the 5 different dimensions ?

[Rajkumar Verma]

In this problem,I want the generate this graph for all pairs [(x,y), ()] which satisfy the conditions and . Here are connected and are connected.

Answer 1

[Walter Roberson]

vals = 0 : 0.01 : 1;
[p1, p2] = ndgrid(vals);
mask = p1 + p2 <= 1;
p1 = p1(mask);
p2 = p2(mask);

[idx1, idx2] = ndgrid(1:length(p1));
V = @(M) M(:);
x = V(p1(idx1));
y = V(p2(idx1));
u = V(p1(idx2));
v = V(p2(idx2));

Now x, y, u, v are each column vectors, and for any given index, x(K), y(K), u(K), v(K) is such that x(K)+y(K)<=1 and u(K)+v(K)<=1 . This gives you 26532801 (over 26 million) points.

You can now evaluate F in vectorized form. You can then do corrections for the cases where x and u are both 0 or y and v are both 0 or 1-x-y is 0 together with 2 - x - u - u - v being 0.

This will leave you with 26532801 x, y, u, v, F coordinate tuples.

With the data all calculated, you now have to decide what a 5 dimension plot should look like. Did you have some ideas on how you want to represent 5 dimensional data on a 2 dimensional display ?

[Rajkumar Verma]

I have no idea how can we display this data.

[Rik]

You could make 10 2D plots.

[Rajkumar Verma]

How ?

[Rik]

Put the data in an ND array and use a loop to plot pairs in subplot() axes.

Answer 2

[John D'Errico]

You want to plot some function of 4 variables. So essentially, you have a 5-dimensional plot you want to see. Not gonna happen. Your monitor is TWO dimensional. We plot 3-d plots as needed, because your brain understands how to visualize a 3-d thing from a 2-d scene. You rotate it around, etc, then gaining an understanding of the object.

But you have 5 dimensions of interest. Not 2 or 3. Not even 4 (where there is one trick you can do with an iso-surface, sort of a contour plot in 3-d.) You can also play with Chernoff faces, but I have never personally found them to be of much value.

https://en.wikipedia.org/wiki/Chernoff_face

So unless you have bought one of those new holodeck monitors, available from Starfleet Command for a princely sum of federation credits (surely you can guess what it costs to ship stuff from the other side of the galaxy) you cannot visualize 5-dimensional stuff on a 2-d monitor.

I'm sorry, but we all live in sphereland. (Read the book! At least, start with Flatland, truly a fun book to read.) 5-dimensions is out of view for us common 3-d people, who can see only 3 dimensions.

https://en.wikipedia.org/wiki/Flatland