Matplotlib#

The module that we typically use to make plots in Python is called matplotlib. It is extremely powerful, and the examples page on its website contains many useful plots which you might want to adapt for your own purposes. You can import it like so:

import matplotlib.pyplot as plt

We also need to tell matplotlib how to display plots. The easiest option is the following, which tells matplotlib to show plots directly after the cell they are created in.

%matplotlib inline

Basic 2D plots#

For the first example, we will plot \(\sin\) on the interval \([-\pi, \frac{3\pi}{2}]\). The most convenient way of doing this involves numpy. We will first generate a sequence of \(x\)-coordinates and corresponding \(y\)-coordinates.

import numpy as np

x = np.arange(-np.pi, 3/2*np.pi, 0.01)   # x is an array of x-coords
y = np.sin(x)                            # y is a corresponding array of y-coords

Now that we have these coordinates, it is very simple to plot the function: just call plt.plot.

# The semicolon just suppresses a line of text above the plot.
# Try removing it to see the difference.
plt.plot(x, y);
../_images/18 Plotting_7_0.png

We often want to draw multiple lines on the same plot. As an example, we will plot sin against the first few terms of its MacLaurin expansion, \(\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} - O(x^7)\). To do so, we create a figure, which is like a blank canvas, and then plot the lines one by one.

# the argument to figure should be a unique identifier - it can be a number or a string
fig = plt.figure("sin-vs-series")

# Get the y-coords for the MacLaurin series.
y2 = x - x**3/6 + x**5/120

# We can't draw directly onto figures - we need to draw onto a "subplot" of the figure.
# The next line means "add a 1x1 grid of plots to the canvas, and give us the first plot"
ax = fig.add_subplot(1, 1, 1)

# Now we can plot onto ax
ax.plot(x, y)
ax.plot(x, y2);

# Note: we could have instead done ax.plot(x, y, x, y2).
../_images/18 Plotting_9_0.png

We should also add labels, a legend, and so on:

# A quicker way to produce figure and axes
fig, ax = plt.subplots()

# plot sin in red
ax.plot(x, y, color="red", label="sin")

# plot the series as a thick dashed blue line
ax.plot(x, y2, color="b", linestyle='--', linewidth=3, label="MacLaurin series")

# add labels to the axes
ax.set_xlabel("x")
ax.set_ylabel("y")

# add a legend - the labels are provided in the ax.plot commands above
ax.legend()

# add a title
ax.set_title("Comparison of sine with its three-term MacLaurin series");
../_images/18 Plotting_11_0.png

The plots you create with matplotlib are extremely customisable, and if you want to alter something about them you probably can - it’s just a matter of finding the right search terms to get the solution.

3D surface plots#

We can create 3-dimensional plots in a similar way to 2D plots. We need arrays of the \(x\)- and \(y\)-coordinates (the “grid” we are using), and also an array for the \(z\) coordinates.

%matplotlib notebook
# The previous line makes the 3D surface plot interactive.
# Change it to %matplotlib inline if you just want a static plot.

# To get 3D axes, we need to pass the subplot_kw argument below
fig, ax = plt.subplots(subplot_kw={"projection": "3d"}, figsize=(8, 8))

# Create the x and y coordinates
xp = np.arange(-5, 5, 0.01)
yp = np.arange(-5, 5, 0.01)

# Turn those coordinates into grids
X, Y = np.meshgrid(xp, yp)

# Compute the function on that grid
Z = np.sin(Y) * np.sin(0.5*X) * np.exp(0.1*X)

# Plot the surface. The "cmap" argument determines the "colour map"
ax.plot_surface(X, Y, Z, cmap="coolwarm");
ax.set_xlabel('x',fontsize=14)
ax.set_ylabel('y',fontsize=14)
ax.set_zlabel('f(x,y)',fontsize=14);

The plot that this produces is interactive - try clicking and dragging to rotate the surface. We have used the popular “coolwarm” colouring for the surface, but there are many other options for colour maps available - see https://matplotlib.org/stable/tutorials/colors/colormaps.html.

Contour plots#

Basic contours#

We can also visualise 3D surfaces using contour plots.

%matplotlib inline

fig, ax = plt.subplots(figsize=(6, 6))

# Create the x and y coordinates
xp = np.arange(-5, 5, 0.01)
yp = np.arange(-5, 5, 0.01)

# Turn those coordinates into grids
X, Y = np.meshgrid(xp, yp)

# Compute the function on that grid
Z = np.sin(Y) * np.sin(0.5*X) * np.exp(0.1*X)

# Plot the contours
ax.contour(X, Y, Z);
../_images/18 Plotting_17_0.png

We can customise this plot by (for example) changing the number of contours used and adding a scale (important!)

%matplotlib inline

fig, ax = plt.subplots(figsize=(6, 6))

# force the aspect ratio of the plot to be 1:1
ax.set_aspect(1)

C = ax.contour(X, Y, Z, levels=10, cmap="coolwarm");
fig.colorbar(C)

ax.set_xlabel("x")
ax.set_ylabel("y");
../_images/18 Plotting_19_0.png

We could alternatively have the contour levels be presented on the contours:

%matplotlib inline

fig, ax = plt.subplots(figsize=(6, 6))
ax.set_aspect(1)

# "k" for blac"k"
C = ax.contour(X, Y, Z, levels=10, colors="k");
# add labels to the contours
ax.clabel(C)

ax.set_xlabel("x")
ax.set_ylabel("y");
../_images/18 Plotting_21_0.png

Note that when contours are drawn in black, a dashed line represents a negative value and a solid line a non-negative value.

Filled contours#

An alternative to using contour is contourf. This produces “filled” contour plots:

%matplotlib inline

fig, ax = plt.subplots(figsize=(6, 6))
ax.set_aspect(1)

C = ax.contourf(X, Y, Z, levels=15, cmap="coolwarm");
fig.colorbar(C)

ax.set_xlabel("x")
ax.set_ylabel("y");
../_images/18 Plotting_24_0.png

We can combine contourf and contour like so:

%matplotlib inline

fig, ax = plt.subplots(figsize=(6, 6))
ax.set_aspect(1)

# draw both a contour and a filled contour plot on ax
ax.contour(X, Y, Z, levels=15, colors="k")
C = ax.contourf(X, Y, Z, levels=15, cmap="coolwarm");
fig.colorbar(C)

ax.set_xlabel("x")
ax.set_ylabel("y");
../_images/18 Plotting_26_0.png

Plotting vector fields#

There are two main options available for plotting vector fields: quiver and streamplot. These are easiest to explain by example. We will consider the function $\(f(x, y) = \begin{pmatrix}x - 0.5y \\ \cos(x) \end{pmatrix}\)$ which could represent, for example, an autonomous system of ODEs.

# Define the vector function
def f(x, y):
    return y - 0.5*x, np.cos(x)

# Create the x and y coordinates
xp = np.arange(-5, 5, 0.5)
yp = np.arange(-5, 5, 0.5)

# Turn those coordinates into grids
X, Y = np.meshgrid(xp, yp)

# Compute the vector field.
# Note that U will contain the x-component and V will contain the y-component 
# of f(x, y) at each point.
U, V = f(X, Y)

fig, ax = plt.subplots(figsize=(8, 8))

ax.quiver(X, Y, U, V);

# We have neglected axes, a title, and so on - you can add them in the same
# way as for basic plots.
../_images/18 Plotting_29_0.png

You can see more examples of quiver plots here and here.

Note that it is quite hard to see what is going on in some areas. A streamplot might help:

fig, ax = plt.subplots(figsize=(8, 8))

ax.streamplot(X, Y, U, V);

# you should add labels!
../_images/18 Plotting_31_0.png

You can see more examples of stream plots here.

Annotations#

We often want to mark points on a plot. The simplest way of doing this is using the scatter function (usually used to draw scatter plots). In the example below, we mark the visible stationary points on the stream plot from above. The same technique will work for any of the plots above (though for a 3D plot, you will need to provide 3 coordinates for the annotation).

from numpy import pi

def f(x, y):
    return y - 0.5*x, np.cos(x)

xp = np.arange(-5, 5, 0.5)
yp = np.arange(-5, 5, 0.5)
X, Y = np.meshgrid(xp, yp)
U, V = f(X, Y)

fig, ax = plt.subplots(figsize=(8, 8))

ax.streamplot(X, Y, U, V, color="k");

# Mark the stable stationary points.
# First, separate them into lists of x-coords and corresponding y-coords.
stable_x = [-3/2 * pi, pi/2]
stable_y = [-3/4 * pi, pi/4]

# Now plot them as blue stars.
# The zorder argument controls the "depth" of the markers, i.e. whether they appear on top of
# or below the stream plot.
ax.scatter(stable_x, stable_y, marker="*", color="b", s=130, zorder=2);

# Now plot the unstable stationary point as a red dot
ax.scatter(-pi/2, -pi/4, marker="o", color="r", s=130, zorder=2)

# We can also add text, though this will look cluttered on the stream plot.
# The first two arguments are the x, y coordinates of where to put the text.
ax.text(-pi/2 - 0.5, -pi/4 + 0.3, "unstable!", fontsize=20, color="r", zorder=2);
../_images/18 Plotting_35_0.png

Better than adding cluttered text to the stream plot would be to have the markers included in a legend, like so:

fig, ax = plt.subplots(figsize=(8, 8))

ax.streamplot(X, Y, U, V, color="k");

stable_x = [-3/2 * pi, pi/2]
stable_y = [-3/4 * pi, pi/4]

# This time we add a label so that this will appear in the legend.
ax.scatter(stable_x, stable_y, marker="*", color="b", s=130, zorder=2, label="Stable");
ax.scatter(-pi/2, -pi/4, marker="o", color="r", s=130, zorder=2, label="Unstable")

# To place the legend outside of the plot, use "loc" and "bbox_to_anchor" like so:
ax.legend(loc='upper left', bbox_to_anchor=(1, 1));

# This still needs a title and axes labels!
../_images/18 Plotting_37_0.png

Saving plots#

You might want to export a plot, so that you can use it in another application. The easiest way to do this is using savefig.

%matplotlib inline

fig, ax = plt.subplots()

ax.plot(x, y, color="red", label="sin")
ax.plot(x, y2, color="b", linestyle='--', linewidth=3, label="MacLaurin series")

ax.set_xlabel("x")
ax.set_ylabel("y")
ax.legend()
ax.set_title("Comparison of sine with its three-term MacLaurin series");

plt.savefig("images/sin-vs-maclaurin.png")
../_images/18 Plotting_39_0.png

You should now have a file sin-vs-maclaurin.png in the same folder as this notebook.

Animated plots#

If you have a time-dependent plot, it might be fun to animate it. The basic idea is to provide a function which “redraws” the plot for each frame of the animation.

%%capture
# suppress output from this cell

# the two new imports we need
from matplotlib import animation
from IPython.display import HTML

fig, ax = plt.subplots(figsize=(8, 8))

xp = np.arange(-5, 5, 0.01)
yp = np.arange(-5, 5, 0.01)
X, Y = np.meshgrid(xp, yp)
theta = np.arctan2(X, Y)
R = np.sqrt(X**2 + Y**2)

# animate(i) should draw the i-th frame of the animation
def animate(i):
    # the function Z depends on i, the frame number (think of this as time)
    Z = theta**2 * np.sin(i/20 * R)
    ax.clear()
    ax.contourf(X, Y, Z);
    ax.set_xlabel('x',fontsize=14)
    ax.set_ylabel('y',fontsize=14)
    ax.set_title("$\\theta^2 \\sin(\\frac{tR}{50})$, $t = %.2f$"%(i/20),fontsize=14);

anim = animation.FuncAnimation(fig, animate,
                               frames=50, interval=50, blit=False)

# This cell may take a short while

HTML(anim.to_html5_video())    # display the animation

# anim.save('animation.mp4')  # save the animation as mp4