numpy.meshgrid explanation

In [214]: nx, ny = (3, 2)
In [215]: x = np.linspace(0, 1, nx)
In [216]: x
Out[216]: array([ 0. ,  0.5,  1. ])
In [217]: y = np.linspace(0, 1, ny)
In [218]: y
Out[218]: array([ 0.,  1.])

Using unpacking to better see the 2 arrays produced by meshgrid:

In [225]: X,Y = np.meshgrid(x, y)
In [226]: X
Out[226]: 
array([[ 0. ,  0.5,  1. ],
       [ 0. ,  0.5,  1. ]])
In [227]: Y
Out[227]: 
array([[ 0.,  0.,  0.],
       [ 1.,  1.,  1.]])

and for the sparse version. Notice that X1 looks like one row of X (but 2d). and Y1 like one column of Y.

In [228]: X1,Y1 = np.meshgrid(x, y, sparse=True)
In [229]: X1
Out[229]: array([[ 0. ,  0.5,  1. ]])
In [230]: Y1
Out[230]: 
array([[ 0.],
       [ 1.]])

When used in calculations like plus and times, both forms behave the same. That's because of numpy's broadcasting.

In [231]: X+Y
Out[231]: 
array([[ 0. ,  0.5,  1. ],
       [ 1. ,  1.5,  2. ]])
In [232]: X1+Y1
Out[232]: 
array([[ 0. ,  0.5,  1. ],
       [ 1. ,  1.5,  2. ]])

The shapes might also help:

In [235]: X.shape, Y.shape
Out[235]: ((2, 3), (2, 3))
In [236]: X1.shape, Y1.shape
Out[236]: ((1, 3), (2, 1))

The X and Y have more values than are actually needed for most uses. But usually there isn't much of penalty for using them instead the sparse versions.

Your linear spaced vectors x and y defined by linspace use 3 and 2 points respectively.

These linear spaced vectors are then used by the meshgrid function to create a 2D linear spaced point cloud. This will be a grid of points for each of the x and y coordinates. The size of this point cloud will be 3 x 2.

The output of the function meshgrid creates an indexing matrix that holds in each cell the x and y coordinates for each point of your space.

This is created as follows:

# dummy
def meshgrid_custom(x,y):
xv = np.zeros((len(x),len(y)))
yv = np.zeros((len(x),len(y)))

for i,ix in zip(range(len(x)),x):
    for j,jy in zip(range(len(y)),y):
        xv[i,j] = ix
        yv[i,j] = jy

return xv.T, yv.T

So, for example the point at the location (1,1) has the coordinates:

x = xv_1[1,1] = 0.5
y = yv_1[1,1] = 1.0