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import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from matplotlib.animation import FuncAnimation
%matplotlib notebook
8.9. JNB LAB: Calculus Animations#
In this JNB, we will show how to create animations that illustrate basic calculus concepts. To create an animation, we use a for loop to create single frames of the animation ‘0.png’, ‘1.png’, ‘2.png’,…. and then use a block of code to combine the individual frames into a .gif animation.
8.9.1. Tangent Lines#
Let’s begin by showing how secants approach a tangent line. We will use the function \(f(x)=x^2\) and create a sequence of secants that approach the tangent line at point \(x=4\).
First, let’s define our function and specify the point of tangency.
f = lambda x : x**2 # Define the function f(x)
a=4 #x coordinate of point of tangency
Now let’s create the first frame (\(n=0\)) of the sequence of images \(\{n.png\}\) used to create the animation.
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n=0
#---set the figure specs
fig = plt.figure(figsize=(5,5))
plt.xlim(3,9)
plt.ylim(5,100)
#---plot the function f(x) in black---
x = np.linspace(3, 9, 100)
y=f(x)
plt.plot(x,y,color='k')
#---plot the tangent line in red---
y_tan= f(a)+2*a*(x-a)
plt.plot(x,y_tan,'r--')
#---plot the secant segment in blue---
b=a+5/(n+1)
x_sec=np.linspace(a, b, 50)
y_sec= f(a) + (f(b) - f(a)) / (b - a) * (x_sec - a)
plt.plot(x_sec,y_sec,color='b')
# Set the x-axis and y-axis labels
plt.xlabel('x')
plt.ylabel('y')
# Set the title
plt.title('Secant Line Animation')
# Set the legend
plt.legend(['function','tangent','secant'])
# Save the figure
plt.savefig(str(n)+'.png')
plt.show()
To create our animation, we will:
i) import a couple of functions used to create the animation
ii) specify the number of frames
iii) specify our function and point of tangency (assuming we haven’t done so previously)
iv) use a for loop to create the individual frames in the animation
v) combine the frames into an animation file “secant.gif” which is found in the same folder as this lab.
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# i) import functions
import matplotlib.animation as animation
from matplotlib.animation import FuncAnimation
%matplotlib notebook
# ii) specify the number of frames
frames=5
# iii) specify the function f(x) and poit of tangency
f = lambda x : x**2
a=4 #x coordinate of point of tangency
# iv) use a for loop to create the individual frames
for n in range(frames):
fig = plt.figure(figsize=(3,3))
plt.xlim(3,9)
plt.ylim(5,100)
#---plot the function f(x) in black---
x = np.linspace(3, 9, 100)
y=f(x)
plt.plot(x,y,color='k')
#---plot the tangent line in red---
y_tan= f(a)+2*a*(x-a)
plt.plot(x,y_tan,'r--')
#---plot the secant segment in blue---
b=a+5/(n+1)
x_sec=np.linspace(a, b, 50)
y_sec= f(a) + (f(b) - f(a)) / (b - a) * (x_sec - a)
plt.plot(x_sec,y_sec,color='b')
# Set the x-axis and y-axis labels
plt.xlabel('x')
plt.ylabel('y')
# Set the title
plt.title('Secant Line Animation')
# Set the legend
plt.legend(['function','tangent','secant'])
plt.savefig(str(n)+'.png')
plt.show()
# v) combine the frames into an animation file secant.gif
from PIL import Image
images = []
for n in range(frames):
exec('a'+str(n)+'=Image.open("'+str(n)+'.png")')
images.append(eval('a'+str(n)))
images[0].save('secant.gif',
save_all=True,
append_images=images[1:],
duration=1000,
loop=0)
Exercise#
Exercise
1.1 Create an animation with 3 frames which shows secants approaching the tangent line to \(y=\frac{1}{x}\) at \(x=1\).
8.9.2. Riemann Sums#
Just as the tangent line is obtained by a limiting process, the area under curve is obtainable as a limit of a Riemann Sum. We can create an animation which shows the approximating rectangles filling in the area under a curve. For example, we consider the area under \(y=1/x\) in the interval \(1\le x \le 5\).
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from matplotlib.animation import FuncAnimation
%matplotlib notebook
f = lambda x : 1/x
x=np.arange(1,6,.0001)
y=f(x)
frames=10
for n in np.arange(0,frames,1):
plt.figure(figsize=(10,7))
plt.gca().set_xticks(np.arange(1,7,1))
plt.gca().set_yticks([.1,.2,.3,.4,.5,.6,.7,.8,.9,1])
plt.plot(x,y,'k',markersize=1)
x_left = np.linspace(1,6,n+2)
x_left=x_left[:-1]
plt.bar(x_left,1/x_left,width=5/(n+1),alpha=1-.9*(1/(n+1)),align='edge',color='r',edgecolor='k')
plt.title('Riemann Sums')
plt.xlabel('x')
plt.ylabel('y')
plt.ylim((0,1.1))
plt.text(3,1,"Area Under Curve = ln(5) = 1.609...")
Rn=0
for k in np.arange(0,n+1,1):
Rn=Rn+(5/(n+1))*(1/x_left[k])
plt.text(4,.9, "R"+str(n+1)+"="+str(Rn))
#plt.grid()
plt.savefig(str(n)+'.png')
plt.show()
#----- Save the Animation as riemannsums.gif
from PIL import Image
images = []
for n in range(frames):
exec('a'+str(n)+'=Image.open("'+str(n)+'.png")')
images.append(eval('a'+str(n)))
images[0].save('riemannsums.gif',
save_all=True,
append_images=images[1:],
duration=800,
loop=0)
Exercise#
Exercise
2.1 Create an animation for a Riemann sum approximation of \(\int_0^{\pi/2}\cos(x)\,dx\).
8.9.3. Parametric Curves#
JNBs can be used to create .gif animations of parametric curves. For example, the code below is used to animate a cycloid curve traced by a point on the rim of a wheel.
Show code cell source
import numpy as np
import pandas as pd
import matplotlib.pyplot as plt
import matplotlib.animation as animation
from matplotlib.animation import FuncAnimation
%matplotlib notebook
frames=50
for n in range(frames):
fig = plt.figure(figsize=(15, 2))
plt.gca().axis([-np.pi, 2*np.pi, 0, 1.2])
#---Plot the Cycloid--------
t = np.arange(0, 2*np.pi, 0.001)
xt=t/2-.5*np.sin(t)
yt=.5-.5*np.cos(t)
plt.gca().plot(xt, yt,color='k')
#----Plot the Wheel-----------
circle = plt.Circle((np.pi*n/frames, 0.5), 0.5,color='r',fill=False)
plt.gca().add_artist(circle)
#Plot the point on the rim of the wheel
plt.gca().plot(np.pi*n/frames -.5*np.sin(2*np.pi*n/frames), .5-.5*np.cos(2*np.pi*n/frames), 'bo')
plt.savefig(str(n)+'.png')
plt.close()
#----- Save the Animation as cycloid.gif
from PIL import Image
images = []
for n in range(frames):
exec('a'+str(n)+'=Image.open("'+str(n)+'.png")')
images.append(eval('a'+str(n)))
images[0].save('cycloid.gif',
save_all=True,
append_images=images[1:],
duration=300,
loop=0)
Exercise#
Exercise
3.1 A hypocycloid curve is formed by a point on a circle rotating along the inside rim of another circle. The parametric equations are
\(x(t) = (a-b) \cos t + b \cos(\frac{a-b}{b}t)\)
\(y(t) = (a-b) \sin t - b \sin(\frac{a-b}{b}t)\)
Create an animation of a hypocycloid curve for the case \(a=1\), \(b=1/4\). Your animation should resemble the one shown below.
