Numpy Meshgrid function

The numpy.meshgrid function is used to create a rectangular grid out of two given one-dimensional arrays representing the Cartesian indexing or Matrix indexing. Meshgrid function is somewhat inspired from MATLAB.
Consider the above figure with X-axis ranging from -4 to 4 and Y-axis ranging from -5 to 5. So there are a total of (9 * 11) = 99 points marked in the figure each with a X-coordinate and a Y-coordinate. For any line parallel to the X-axis, the X-coordinates of the marked points respectively are -4, -3, -2, -1, 0, 1, 2, 3, 4. On the other hand, for any line parallel to the Y-axis, the Y-coordinates of the marked points from bottom to top are -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. The numpy.meshgrid function returns two 2-Dimensional arrays representing the X and Y coordinates of all the points. 

Examples:

Input : x = [-4, -3, -2, -1, 0, 1, 2, 3, 4]
        y = [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5] 
Output :
x_1 = array([[-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.],
             [-4., -3., -2., -1.,  0.,  1.,  2.,  3.,  4.]])

y_1 = array([[-5., -5., -5., -5., -5., -5., -5., -5., -5.],
       [-4., -4., -4., -4., -4., -4., -4., -4., -4.],
       [-3., -3., -3., -3., -3., -3., -3., -3., -3.],
       [-2., -2., -2., -2., -2., -2., -2., -2., -2.],
       [-1., -1., -1., -1., -1., -1., -1., -1., -1.],
       [ 0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.,  0.],
       [ 1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.,  1.],
       [ 2.,  2.,  2.,  2.,  2.,  2.,  2.,  2.,  2.],
       [ 3.,  3.,  3.,  3.,  3.,  3.,  3.,  3.,  3.],
       [ 4.,  4.,  4.,  4.,  4.,  4.,  4.,  4.,  4.],
       [ 5.,  5.,  5.,  5.,  5.,  5.,  5.,  5.,  5.]])


Input : x = [0, 1, 2, 3, 4, 5]
        y = [2, 3, 4, 5, 6, 7, 8]

Output :
x_1 = array([[0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.],
             [0., 1., 2., 3., 4., 5.]])

y_1 = array([[2., 2., 2., 2., 2., 2.],
             [3., 3., 3., 3., 3., 3.],
             [4., 4., 4., 4., 4., 4.],
             [5., 5., 5., 5., 5., 5.],
             [6., 6., 6., 6., 6., 6.],
             [7., 7., 7., 7., 7., 7.],
             [8., 8., 8., 8., 8., 8.]]

Below is the code: 

Python3

# Sample code for generation of first example
import numpy as np
# from matplotlib import pyplot as plt
# pyplot imported for plotting graphs

x = np.linspace(-4, 4, 9)

# numpy.linspace creates an array of
# 9 linearly placed elements between
# -4 and 4, both inclusive
y = np.linspace(-5, 5, 11)

# The meshgrid function returns
# two 2-dimensional arrays
x_1, y_1 = np.meshgrid(x, y)

print("x_1 = ")
print(x_1)
print("y_1 = ")
print(y_1)

Output:

x_1 = 
[[-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]
 [-4. -3. -2. -1.  0.  1.  2.  3.  4.]]
y_1 = 
[[-5. -5. -5. -5. -5. -5. -5. -5. -5.]
 [-4. -4. -4. -4. -4. -4. -4. -4. -4.]
 [-3. -3. -3. -3. -3. -3. -3. -3. -3.]
 [-2. -2. -2. -2. -2. -2. -2. -2. -2.]
 [-1. -1. -1. -1. -1. -1. -1. -1. -1.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 1.  1.  1.  1.  1.  1.  1.  1.  1.]
 [ 2.  2.  2.  2.  2.  2.  2.  2.  2.]
 [ 3.  3.  3.  3.  3.  3.  3.  3.  3.]
 [ 4.  4.  4.  4.  4.  4.  4.  4.  4.]
 [ 5.  5.  5.  5.  5.  5.  5.  5.  5.]]

The output of coordinates by meshgrid can also be used for plotting functions within the given coordinate range. 
An Ellipse: 

*** QuickLaTeX cannot compile formula:
 

*** Error message:
Error: Nothing to show, formula is empty

Python3

ellipse = xx * 2 + 4 * yy**2
plt.contourf(x_1, y_1, ellipse, cmap = 'jet')

plt.colorbar()
plt.show()

Output: 

Random Data: 

Python3

random_data = np.random.random((11, 9))
plt.contourf(x_1, y_1, random_data, cmap = 'jet')

plt.colorbar()
plt.show()

Output: 

A Sine function: 

*** QuickLaTeX cannot compile formula:
 

*** Error message:
Error: Nothing to show, formula is empty

Python3

sine = (np.sin(x_1**2 + y_1**2))/(x_1**2 + y_1**2)
plt.contourf(x_1, y_1, sine, cmap = 'jet')

plt.colorbar()
plt.show()

Output: 

We observe that x_1 is a row repeated matrix whereas y_1 is a column repeated matrix. One row of x_1 and one column of y_1 is enough to determine the positions of all the points as the other values will get repeated over and over. So we can edit above code as follows: 
x_1, y_1 = np.meshgrid(x, y, sparse = True)
This will produce the following output: 

x_1 = [[-4. -3. -2. -1.  0.  1.  2.  3.  4.]]
y_1 = [[-5.]
 [-4.]
 [-3.]
 [-2.]
 [-1.]
 [ 0.]
 [ 1.]
 [ 2.]
 [ 3.]
 [ 4.]
 [ 5.]]

The shape of x_1 changed from (11, 9) to (1, 9) and that of y_1 changed from (11, 9) to (11, 1) 
The indexing of Matrix is however different. Actually, it is the exact opposite of Cartesian indexing. 

For the matrix shown above, for a given row Y-coordinate increases as 0, 1, 2, 3 from left to right whereas for a given column X-coordinate increases from top to bottom as 0, 1, 2. 
The two 2-dimensional arrays returned from Matrix indexing will be the transpose of the arrays generated by the previous program. The following code can be used for obtaining Matrix indexing: 

Python3

# Sample code for generation of Matrix indexing
import numpy as np


x = np.linspace(-4, 4, 9)
# numpy.linspace creates an array
# of 9 linearly placed elements between
# -4 and 4, both inclusive
y = np.linspace(-5, 5, 11)

# The meshgrid function returns
# two 2-dimensional arrays
x_1, y_1 = np.meshgrid(x, y)


x_2, y_2 = np.meshgrid(x, y, indexing = 'ij')

# The following 2 lines check if x_2 and y_2 are the
# transposes of x_1 and y_1 respectively
print("x_2 = ")
print(x_2)
print("y_2 = ")
print(y_2)

# np.all is Boolean and operator;
# returns true if all holds true.
print(np.all(x_2 == x_1.T))
print(np.all(y_2 == y_1.T))

Output:

x_2 = 
[[-4. -4. -4. -4. -4. -4. -4. -4. -4. -4. -4.]
 [-3. -3. -3. -3. -3. -3. -3. -3. -3. -3. -3.]
 [-2. -2. -2. -2. -2. -2. -2. -2. -2. -2. -2.]
 [-1. -1. -1. -1. -1. -1. -1. -1. -1. -1. -1.]
 [ 0.  0.  0.  0.  0.  0.  0.  0.  0.  0.  0.]
 [ 1.  1.  1.  1.  1.  1.  1.  1.  1.  1.  1.]
 [ 2.  2.  2.  2.  2.  2.  2.  2.  2.  2.  2.]
 [ 3.  3.  3.  3.  3.  3.  3.  3.  3.  3.  3.]
 [ 4.  4.  4.  4.  4.  4.  4.  4.  4.  4.  4.]]
y_2 = 
[[-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]
 [-5. -4. -3. -2. -1.  0.  1.  2.  3.  4.  5.]]

True

True

The sparse = True can also be added in the meshgrid function of Matrix indexing. In this case, the shape of x_2 will change from (9, 11) to (9, 1) and that of y_2 will change from (9, 11) to (1, 11).