Note: This article has also featured on geeksforgeeks.org
Matplotlib is a pretty extensive library which supports Animations of graphs as well.
The animation tools center around the matplotlib.animation base class, which provides a framework around which the animation functionality is built. The main interfaces are TimedAnimation and FuncAnimation and out of the two, FuncAnimation is the most convenient one to use.
Installation:
 Matplotlib: Refer to MATPLOTLIB Tutorial Series  Part 1
 Numpy: You can install numpy module using following pip command:
pip install numpy
 FFMPEG: It is required only for saving the animation as a video. The executable can be downloaded from here.
Example 1
# importing required modules import matplotlib.pyplot as plt import matplotlib.animation as animation import numpy as np # create a figure, axis and plot element fig = plt.figure() ax = plt.axes(xlim=(50, 50), ylim=(50, 50)) line, = ax.plot([], [], lw=2) # initialization function def init(): # creating an empty plot/frame line.set_data([], []) return line, # lists to store x and y axis points xdata, ydata = [], [] # animation function def animate(i): # t is a parameter t = 0.1*i # x, y values to be plotted x = t*np.sin(t) y = t*np.cos(t) # appending new points to x, y axes points list xdata.append(x) ydata.append(y) # set/update the x and y axes data line.set_data(xdata, ydata) # return line object return line, # setting a title for the plot plt.title('A growing coil!') # hiding the axis details plt.axis('off') # call the animator anim = animation.FuncAnimation(fig, animate, init_func=init, frames=500, interval=20, blit=True) # save the animation as mp4 video file anim.save('animated_coil.mp4', writer = 'ffmpeg', fps = 30) # show the plot plt.show()
Here is how the output animation looks like:
Now, let us try to understand the code in pieces:

fig = plt.figure() ax = plt.axes(xlim=(50, 50), ylim=(50, 50)) line, = ax.plot([], [], lw=2)
Here, we first create a figure, i.e a top level container for all our subplots.
Then we create an axes element ax which acts as a subplot. The range/limit for x and y axis are also defined while creating the axes element.
Finally, we create the plot element, named as line . Initially, the x and y axis points have been defined as empty lists and linewidth (lw) has been set as 2. 
def init(): line.set_data([], []) return line,
Now, we declare a initialization function, init . This function is called by animator to create the first frame.

def animate(i): # t is a parameter t = 0.1*i # x, y values to be plotted x = t*np.sin(t) y = t*np.cos(t) # appending new points to x, y axes points list xdata.append(x) ydata.append(y) # set/update the x and y axes data line.set_data(xdata, ydata) # return line object return line,
This is the most important function of above program. animate() function is called again and again by the animator to create each frame. The number of times this function will be called is determined by number of frames, which is passed as frames argument to animator.
animate() function takes the index of ith frame as argument.t = 0.1*i
Here, we cleverly use the index of current frame as a parameter!
x = t*np.sin(t) y = t*np.cos(t)
Now, since we have the parameter t, we can easily plot any parametric equation. For example, here, we are plotting a spiral using its parametric equation.
line.set_data(xdata, ydata) return line,
Finally, we use set_data() function to set x and y data and then return plot object, line .

anim = animation.FuncAnimation(fig, animate, init_func=init, frames=500, interval=20, blit=True)
Now, we create the FuncAnimation object, anim . It takes various arguments explained below:
fig : figure to be plotted.
animate : the function to be called repeatedly for each frame.
init_func : function used to draw a clear frame. It is called once before the first frame.
frames : number of frames. (Note: frames can also be an iterable or generator.)
interval : duration between frames ( in milliseconds)
blit : setting blit=True means that only those parts will be drawn, which have changed. 
anim.save('animated_coil.mp4', writer = 'ffmpeg', fps = 30)
Now, we save the animator object as a video file using save() function.
You will need a movie writer for saving the animation video.
In this example, we have used FFMPEG movie writer. So, writer is set as ‘ffmpeg’.
fps stands for frame per second.
Example 2
This example shows how you can make a rotating curve by applying some simple mathematics!
# importing required modules import matplotlib.pyplot as plt import matplotlib.animation as animation import numpy as np # create a figure, axis and plot element fig = plt.figure() ax = plt.axes(xlim=(25, 25), ylim=(25, 25)) line, = ax.plot([], [], lw=2) # initialization function def init(): # creating an empty plot/frame line.set_data([], []) return line, # set of points for a star (could be any curve) p = np.arange(0, 4*np.pi, 0.1) x = 12*np.cos(p) + 8*np.cos(1.5*p) y = 12*np.sin(p)  8*np.sin(1.5*p) # animation function def animate(i): # t is a parameter t = 0.1*i # x, y values to be plotted X = x*np.cos(t)  y*np.sin(t) Y = y*np.cos(t) + x*np.sin(t) # set/update the x and y axes data line.set_data(X, Y) # return line object return line, # setting a title for the plot plt.title('A rotating star!') # hiding the axis details plt.axis('off') # call the animator anim = animation.FuncAnimation(fig, animate, init_func=init, frames=100, interval=100, blit=True) # save the animation as mp4 video file anim.save('basic_animation.mp4', writer = 'ffmpeg', fps = 10) # show the plot plt.show()
Here is how the output of above program looks like:
Here, we have used some simple mathematics to rotate a given curve.
 The star shape is obtained by putting k = 2.5 and 0 < t < 4π in the parametric equation given below:
The same has been applied here:p = np.arange(0, 4*np.pi, 0.1) x = 12*np.cos(p) + 8*np.cos(1.5*p) y = 12*np.sin(p)  8*np.sin(1.5*p)
You can study more about Hypotrochoids here.
 Now, in each frame, we rotate the star curve using concept of rotation in complex numbers. Let x, y be two ordinates. Then after rotation by angle θ, the new ordinates are:
The same has been applied here:X = x*np.cos(t)  y*np.sin(t) Y = y*np.cos(t) + x*np.sin(t)
You can study more about rotation concept here.
All in all, animations are a great tool to create amazing stuff and many more things can be created using them.
So, this was how animated plots can be generated and saved using Matplotlib.
For more examples, refer: animation Examples .