Numpy Meshgrid function (original) (raw)

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 below 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:

import numpy as np

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

y = np.linspace( - 5 , 5 , 11 )

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: [Tex]$$x_1^2+4y_1^2 = 0 [/Tex]

ellipse = xx * 2 + 4 * yy * * 2

plt.contourf(x_1, y_1, ellipse, cmap = 'jet' )

plt.colorbar()

plt.show()

Output: Random Data:

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: [Tex]$$\dfrac{\sin(x_1^2+y_1^2)}{x_1^2+y_1^2}[/Tex]

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:

import numpy as np

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

y = np.linspace( - 5 , 5 , 11 )

x_1, y_1 = np.meshgrid(x, y)

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

print ( "x_2 = " )

print (x_2)

print ( "y_2 = " )

print (y_2)

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).