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numpy.meshgrid() Function – Generate Coordinate Matrices in NumPy

In Python, numpy.meshgrid() is a useful function from the NumPy library. It is particularly useful for generating 2-Dimensional coordinate grids (or matrices) from two 1-Dimensional arrays representing the x-axes and y-axes. The generated coordinate grids are used in many applications, such as plotting 3-D surfaces, numerical computing in partial differential equations, grid based simulation in physics, solving mathematical problems, and image processing.

In previous articles, we learned about NumPy Arrays and How to create NumPy Arrays using Lists or Tuples or using different methods like np.arange(), np.linspace() and np.logspace(). In this tutorial, we will explain the concept of meshgrid() in detail, demonstrate how to use it, and explore common applications of the function.

Table of Contents

numpy.meshgrid() – Generate Coordinate Matrices in NumPy

The numpy.meshgrid() function takes two 1-Dimensional arrays and returns two 2-Dimensional arrays (matrices) representing all the possible pairs of coordinates in the Cartesian plane. Each output array represents the grid coordinates of either the x or y dimension. In simpler terms, meshgrid() “spreads” one 1-D array along the x-axis and another along the y-axis to generate a full grid of coordinates.

Syntax => numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')

numpy.meshgrid(*xi, copy=True, sparse=False, indexing='xy')
  • *xi – Represent grid point for each dimension in 1-D arrays. In the 2-D grid, it represents arrays of x-values and y-values.
  • copy (optional) – The default value is True, which helps create a copy of the array. If False, changes are allowed to be made in place.
  • sparse (optional) – It produces Sparse matrices to save memory used in large grids, only if sparse is True.
  • indexing (optional) – Cartesian indexing ('xy', default), where x is first dimension and y is the second dimension. Matrix indexing ('ij'), here first dimensions correspond to row indices(i) and column indices(j).

Example Basic Syntax => X, Y = np.meshgrid(x, y)

  • x is a 1-D array representing the x-coordinates.
  • y is a 1-D array representing the y-coordinates.
  • X is a 2-D array containing the x-coordinates of the mesh grid.
  • Y is a 2-D array containing the y-coordinates of the mesh grid.

How numpy.meshgrid() Works

Let’s start by understanding how numpy.meshgrid() works with a simple example. Suppose we have two 1-D arrays x and y:

numpy.meshgrid() Function - How numpy.meshgrid() Works
numpy.meshgrid() Function – How numpy.meshgrid() Works
import numpy as np

x = np.array([-1, 0, 1, 2, 3]) # NumPy array for x coordinates
y = np.array([3, 4, 5])        # NumPy array for y coordinates
X, Y = np.meshgrid(x, y)   # X and Y resulting grid coordinates
print(X)
# Output => [[-1  0  1  2  3]
#            [-1  0  1  2  3]
#            [-1  0  1  2  3]]
print(Y)
# Output => [[3 3 3 3 3]
#            [4 4 4 4 4]
#            [5 5 5 5 5]]
  • The x array => [-1, 0, 1, 2, 3] and the y array => [3, 4, 5]
  • X is the 2-D array containing the x-coordinates, repeating each value of x for every value in y. => [[-1 0 1 2 3] [-1 0 1 2 3] [-1 0 1 2 3]]
  • Y is the 2-D array containing the y-coordinates, repeating each value of y for every value in x. => [[3 3 3 3 3] [4 4 4 4 4] [5 5 5 5 5]]
  • Together, X and Y form a grid of coordinates. X and Y are coordinate matrices, where each point (x,y) represents a pair of coordinates on the grid. We can visualize this 2-D grid as a set of points in the Cartesian plane.

Create numpy.meshgrid() using linspace() Function

import numpy as np

x = np.linspace(-3, 3, 7, dtype=int)   # NumPy array for x coordinates
y = np.linspace(-5, 4, 10, dtype=int) # NumPy array for y coordinates
X, Y = np.meshgrid(x, y)   # X and Y resulting grid coordinates
print(X)
# [[-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]
#  [-3 -2 -1  0  1  2  3]]

print(Y)
# [[-5 -5 -5 -5 -5 -5 -5]
#  [-4 -4 -4 -4 -4 -4 -4]
#  [-3 -3 -3 -3 -3 -3 -3]
#  [-2 -2 -2 -2 -2 -2 -2]
#  [-1 -1 -1 -1 -1 -1 -1]
#  [ 0  0  0  0  0  0  0]
#  [ 1  1  1  1  1  1  1]
#  [ 2  2  2  2  2  2  2]
#  [ 3  3  3  3  3  3  3]
#  [ 4  4  4  4  4  4  4]]
  • np.linspace(-3, 3, 7, dtype=int) creates x array => [-3, -2, -1, 0, 1, 2, 3]
  • np.linspace(-5, 4, 10, dtype=int) creates y array => [-5, -4, -3, -2, -1, 0, 1, 2, 3, 4]
  • X is the 2-D array containing the x-coordinates, repeating each value of x for every value in y.
  • Y is the 2-D array containing the y-coordinates, repeating each value of y for every value in x.
  • Together, X and Y form a grid of coordinates. X and Y are coordinate matrices, where each point (x,y) represents a pair of coordinates on the grid. We can visualize this 2-D grid as a set of points in the Cartesian plane.

Create numpy.meshgrid() with sparse=True

Create numpy.meshgrid() with sparse=True
Create numpy.meshgrid() with sparse=True
import numpy as np

x = np.linspace(-2, 3, 6, dtype=int)  # NumPy array for x coordinates
y = np.linspace(-3, 4, 8, dtype=int) # NumPy array for y coordinates
X, Y = np.meshgrid(x, y, sparse=True)   # X and Y resulting grid coordinates
print(X)
# [[-2 -1  0  1  2  3]]

print(Y)
# [[-3]
#  [-2]
#  [-1]
#  [ 0]
#  [ 1]
#  [ 2]
#  [ 3]
#  [ 4]]
  • np.linspace(-2, 3, 6, dtype=int) creates x array => [-2, -1, 0, 1, 2, 3]
  • np.linspace(-3, 4, 8, dtype=int) creates y array => [-3, -2, -1, 0, 1, 2, 3, 4]
  • X is the 2-D array containing the x-coordinates, repeating each value of x for every value in y. => [[-2 -1 0 1 2 3]]
  • Y is the 2-D array containing the y-coordinates, repeating each value of y for every value in x.
  • Together, X and Y form a grid of coordinates. X and Y are coordinate matrices, where each point (x,y) represents a pair of coordinates on the grid. We can visualize this 2-D grid as a set of points in the Cartesian plane.

Create numpy.meshgrid() with Matrix Indexing

Create numpy.meshgrid() with Matrix Indexing
Create numpy.meshgrid() with Matrix Indexing
import numpy as np

x = np.linspace(-2, 3, 6, dtype=int)  # NumPy array for x coordinates
y = np.linspace(-3, 4, 8, dtype=int) # NumPy array for y coordinates
X_ij, Y_ij = np.meshgrid(x, y, indexing="ij")   # X_ij and Y_ij resulting grid coordinates
print(X_ij)
# [[-2 -2 -2 -2 -2 -2 -2 -2]
#  [-1 -1 -1 -1 -1 -1 -1 -1]
#  [ 0  0  0  0  0  0  0  0]
#  [ 1  1  1  1  1  1  1  1]
#  [ 2  2  2  2  2  2  2  2]
#  [ 3  3  3  3  3  3  3  3]]

print(Y_ij)
# [[-3 -2 -1  0  1  2  3  4]
#  [-3 -2 -1  0  1  2  3  4]
#  [-3 -2 -1  0  1  2  3  4]
#  [-3 -2 -1  0  1  2  3  4]
#  [-3 -2 -1  0  1  2  3  4]
#  [-3 -2 -1  0  1  2  3  4]]
  • np.linspace(-2, 3, 6, dtype=int) creates x array => [-2, -1, 0, 1, 2, 3]
  • np.linspace(-3, 4, 8, dtype=int) creates y array => [-3, -2, -1, 0, 1, 2, 3, 4]
  • X_ij is the 2-D array containing the x-coordinates, repeating each value of x for every value in y. => [[-2 -1 0 1 2 3]]
  • Y_ij is the 2-D array containing the y-coordinates, repeating each value of y for every value in x.
  • Together, X and Y form a grid of coordinates. X_ij and Y_ij are coordinate matrices, where each point (x,y) represents a pair of coordinates on the grid. We can visualize this 2-D grid as a set of points in the Matrix.

Using numpy.meshgrid() to Plot 3-D surface

One common use case for meshgrid() is in 3-D surface plotting. For a function of two variables $z = f(x, y)$, we can use meshgrid() to create a grid of $(x, y)$ values, then evaluate $f(x, y)$ at each grid point to get the corresponding $z$-values.

Let’s plot the surface of the function $z = \sin(x^{2} + y^{2})$ over a 2-D grid.

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-5, 5, 20) # NumPy array for x coordinates
y = np.linspace(-5, 5, 20) # NumPy array for y coordinates

X, Y = np.meshgrid(x, y) # Create the meshgrid
Z = np.sin(X**2 + Y**2) # Compute Z values based on the function

# Create a 3D plot
fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
ax.plot_surface(X, Y, Z, cmap='viridis')
plt.show() # Show the plot
Using numpy.meshgrid() to Plot 3-D surface - Function sin(x^2 + y^2) over a 2-D grid
Using numpy.meshgrid() to Plot 3-D surface – Function sin(x^2 + y^2) over a 2-D grid
  • x and y are defined as evenly spaced values between -5 and 5.
  • np.meshgrid(x, y) generates the 2-D grid of points.
  • Z = np.sin(X**2 + Y**2) computes the value of the function at each grid point.
  • The plot_surface() method from matplotlib is used to create a 3-D surface plot.

Applications of numpy.meshgrid() Function

Here are a few common use cases for numpy.meshgrid().

Evaluating Functions on a Grid

We can use numpy.meshgrid() to evaluate any mathematical function over a 2-D grid. For example, to evaluate the function $f(x, y) = x^{2} + y^{2}$ over a grid of values:

x = np.linspace(-3, 3, 7)
y = np.linspace(-2, 2, 5)
X, Y = np.meshgrid(x, y)
Z = X**2 + Y**2
print(Z)
# [[13.  8.  5.  4.  5.  8. 13.]
#  [10.  5.  2.  1.  2.  5. 10.]
#  [ 9.  4.  1.  0.  1.  4.  9.]
#  [10.  5.  2.  1.  2.  5. 10.]
#  [13.  8.  5.  4.  5.  8. 13.]]

Vector Fields

We can visualize vector fields (e.g., magnetic fields, fluid dynamics) using meshgrid() to generate a grid of points where we can compute the vector components at each point.

Image Processing

In image processing, the meshgrid() function can be used to generate coordinates corresponding to each pixel of an image. These coordinates can be used to manipulate image data, perform transformations, or apply filters.

Contour Plots

We can create contour plots (2-D projections of 3-D surfaces) using meshgrid().

import numpy as np
import matplotlib.pyplot as plt

x = np.linspace(-5, 5, 100)
y = np.linspace(-5, 5, 100)
X, Y = np.meshgrid(x, y)
Z = np.sin(X**2 + Y**2)
plt.contour(X, Y, Z)
plt.title("Contour plot of z = sin(x^2 + y^2)")
plt.show()
Contour plots (2-D projections of 3-D surfaces) using meshgrid()
Contour plots (2-D projections of 3-D surfaces) using meshgrid()

Conclusion

The numpy.meshgrid() function is a powerful tool for generating coordinate matrices that simplify the creation of multidimensional grids for various numerical computations and visualizations. By understanding how meshgrid() works and how it integrates with functions like linspace(), you can create finely tuned coordinate grids for 2-D or 3-D plotting and simulations.

The ability to generate sparse grids and utilize matrix indexing further enhances its flexibility, allowing for optimized performance in larger datasets. Whether for surface plotting, contour mapping, or solving complex mathematical problems, numpy.meshgrid() proves to be an essential function for working with multi-dimensional arrays in Python, particularly in scientific computing and data visualization.

Code snippets and programs related to numpy.meshgrid() Function and other topics covered in this tutorial, can be accessed from GitHub Repository. This GitHub repository all contains programs related to other topics in NumPy tutorial.

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