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 exampleimport numpy as np# from matplotlib import pyplot as plt# pyplot imported for plotting graphsx = np.linspace(-4, 4, 9)# numpy.linspace creates an array of# 9 linearly placed elements between# -4 and 4, both inclusivey = np.linspace(-5, 5, 11)# The meshgrid function returns# two 2-dimensional arraysx_1, y_1 = np.meshgrid(x, y)print("x_1 = ")print(x_1)print("y_1 = ")print(y_1) |
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**2plt.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 indexingimport numpy as npx = np.linspace(-4, 4, 9)# numpy.linspace creates an array# of 9 linearly placed elements between# -4 and 4, both inclusivey = np.linspace(-5, 5, 11)# The meshgrid function returns# two 2-dimensional arraysx_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 respectivelyprint("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)) |
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).
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