Generate Square Mesh, Given an Unordered X, Y and Z Vectors

Question

I have three arrays of the same length in Matlab, X, Y and Z. X(i), Y(i) and Z(i) forms a pair of 3D coordinates.

The issue now is, how to use these three arrays to generate square meshes, as shown below:

I got the image from mesh plot in Matlab documentation. So obviously mesh command is not what I want because it plots the meshes itself in the Matlab program, whereas I need the mesh elements ( along with the coordinates) so that I can plot them out myself in other program, such as C#.

In other words, I am looking for the mathematical algorithm to generate the meshes that allows mesh command to plot the below looking graph.

Edit: I realized that my question wasn't clear after a good night sleep. So here's more detail. I generate x and y vector by using this command [x,y]=meshgrid[rangex, rangy], and then I define a vector z with the function z(x,y). I would have to return a list of square elements ( as shown in the figure below) along with their corresponding x,y coordinates. So basically I just want to replot the following graph with those data.

Any ideas?

Answer 1

What you basically have are 3 matrices:

% define x_range and y_range as you wish
[x, y] - meshgrid(x_range,y_range)

z = some_function_of_x_and_y

Now you have to reshape these three matrices into row vectors:

sizes = size(x)   
x_row = reshape(x, sizes(1) * sizes(2), 1)
y_row = reshape(y, sizes(1) * sizes(2), 1)
z_row = reshape(z, sizes(1) * sizes(2), 1)

and another one of indices:

indeces = [1:length(x_row)]'

and now your list is:

result = [indeces x_row y_row z_row]

For instance:

x_range = [1,2,3];
y_range = [1,2,3];

>> [x,y] = meshgrid(x_range, y_range)

x =

     1     2     3
     1     2     3
     1     2     3


y =

     1     1     1
     2     2     2
     3     3     3

>> z = x+y

z =

     2     3     4
     3     4     5
     4     5     6

>> x_row = reshape(x, sizes(1) * sizes(2), 1);
>> y_row = reshape(y, sizes(1) * sizes(2), 1);
>> z_row = reshape(z, sizes(1) * sizes(2), 1);

>> indeces = [1:length(x_row)]';

>> result = [indeces x_row y_row z_row]

result =

     1     1     1     2
     2     1     2     3
     3     1     3     4
     4     2     1     3
     5     2     2     4
     6     2     3     5
     7     3     1     4
     8     3     2     5
     9     3     3     6

Now result holds the indeces in the first column, and (x,y,z) in the rest of the columns. You should be able to extract what you want from there.

Answer 2

    x = [1 2 3]; 
    y = [11 22 33]';
    [X, Y] = meshgrid(x,y)
X =

     1     2     3
     1     2     3
     1     2     3


Y =

    11    11    11
    22    22    22
    33    33    33

Answer 3

There are two possibilities here. If the points actually already forms a regular lattice in the (x,y) plane, and all you need to do is unscramble which points go where, then a sort will solve your problem. Specifically, use sortrows on the (x,y) pairs, and then a reshape applied to z will get the array into the proper shape. Something roughly like this...

[xy,tags] = sortrows([x(:),y(:)]);
z = reshape(z(tags),[n,m]);

However, if your data is scattered, then you need to use either an interpolation, or a surface fit. GRIDDATA will solve the interpolation problem, although it will only interpolate as far as the boundaries of your data.

GRIDFIT, a tool found on the file exchange, will solve the problem of fitting what is essentially a low order spline surface to your data.