IG Math Space

By the end of this class session, students will be able to:

• Understand the purpose and use of the Matplotlib library for data visualization.
• Create 2D plots using plot(), polar(), and pie().
• Plot mathematical functions and famous curves using parametric and polar equations.
• Visualize 2D matrices using color maps with imshow().
• Create and use meshgrids for evaluating functions over 2D domains.
• Generate 3D surface, line, and wireframe plots using mplot3d.
• Apply visualization techniques to mathematical and scientific problems.

2.1 Introduction

Matplotlib is a Python package that produces publication-quality figures in a variety of hardcopy formats. It provides functions for matrix manipulation, generating plots, histograms, bar charts, error-charts, scatter-plots, and full control over line styles, font properties, and axes properties.

2.2 Importing Matplotlib

from pylab import *

Importing pylab makes plotting functions from pyplot and matrix manipulation functions from mlab available as local functions. It also imports NumPy automatically.

2.3 Basic Plotting

Example 1: Simple Plot

from pylab import *
data = [1, 2, 5]
plot(data)
show()

Example 2: Specifying Both Axes

from pylab import *
x = [1, 2, 5]
y = [4, 5, 6]
plot(x, y)
show()

Example 3: Customizing Plot Style

from pylab import *
x = [1, 2, 5]
y = [4, 5, 6]
plot(x, y, 'ro')        # red circles
xlabel('x-axis')
ylabel('y-axis')
axis([0, 6, 1, 7])      # [xmin, xmax, ymin, ymax]
show()

Example 4: Multiple Plots in Same Window

from pylab import *
t = arange(0.0, 5.0, 0.2)
plot(t, t**2, 'x')      # t² with x markers
plot(t, t**3, 'ro')     # t³ with red circles
show()

2.4 Multiple Plots (Subplots)

Example 5: Subplot

from pylab import *
subplot(2, 1, 1)
plot([1, 2, 3, 4])
subplot(2, 1, 2)
plot([4, 2, 3, 1])
show()

2.5 Polar Plots

Example 7: Circle Using Polar Plot

from pylab import *
th = linspace(0, 2*pi, 100)
r = 5 * ones(100)
polar(th, r)
show()

2.6 Pie Charts

Example 8: Pie Chart

from pylab import *
labels = 'Frogs', 'Hogs', 'Dogs', 'Logs'
fracs = [25, 25, 30, 20]
pie(fracs, labels=labels)
show()

3.1 Generating Input Data

• arange(start, stop, step) – third argument is stepsize
• linspace(start, stop, num_points) – total number of points specified
• logspace(start, stop, num_points) – logarithmic spacing

3.2 Sine Function and Friends

Example 9: Sine and Sin(x²)

from pylab import *
x = linspace(-pi, pi, 200)
y = sin(x)
y1 = sin(x*x)
plot(x, y)
plot(x, y1, 'r')
show()

3.3 Trouble with Circle

Example 10: Circle in Cartesian Coordinates

from pylab import *
a = 10.0
x = linspace(-a, a, 200)
yupper = sqrt(a**2 - x**2)
ylower = -sqrt(a**2 - x**2)
plot(x, yupper)
plot(x, ylower)
show()

3.4 Parametric Plots

Example 11: Circle Using Parametric Equations

from pylab import *
a = 10.0
th = linspace(0, 2*pi, 200)
x = a * cos(th)
y = a * sin(th)
plot(x, y)
show()

4.1 Astroid

Cartesian equation: x^(2/3) + y^(2/3) = a^(2/3)

Parametric equations: x = a·cos³(t), y = a·sin³(t)

Example 14: Astroid (Parametric)

from pylab import *
a = 2
t = linspace(-2*a, 2*a, 101)
x = a * cos(t)**3
y = a * sin(t)**3
plot(x, y)
show()

4.2 Ellipse

Example 15: Ellipse

from pylab import *
a = 2
b = 3
t = linspace(0, 2*pi, 100)
x = a * sin(t)
y = b * cos(t)
plot(x, y)
show()

4.3 Spirals of Archimedes and Fermat

Example 17: Archimedes Spiral

from pylab import *
a = 2
th = linspace(0, 10*pi, 200)
r = a * th
polar(th, r)
axis([0, 2*pi, 0, 70])
show()

Example 18: Fermat's Spiral

from pylab import *
a = 2
th = linspace(0, 10*pi, 200)
r = sqrt(a**2 * th)
polar(th, r)
polar(th, -r)
show()

4.4 Polar Rose

Example 19: Polar Rose

from pylab import *
k = 4
th = linspace(0, 10*pi, 1000)
r = cos(k*th)
polar(th, r)
show()

Example 22: Sum of Sines

from pylab import *
x = arange(-3*pi, 3*pi, 0.1)
y = arange(-3*pi, 3*pi, 0.1)
xx, yy = meshgrid(x, y)
z = sin(xx) + sin(yy)
imshow(z)
show()

5.1 Fractals — Julia Set

Example 24: Creating Meshgrid

from numpy import *
x = arange(0, 3, 1)
y = arange(0, 3, 1)
gx, gy = meshgrid(x, y)
print gx
print gy

Example 26: Surface Plot

from pylab import *
from mpl_toolkits.mplot3d import Axes3D

ax = Axes3D(figure())
x = arange(-3*pi, 3*pi, 0.1)
y = arange(-3*pi, 3*pi, 0.1)
xx, yy = meshgrid(x, y)
z = sin(xx) + sin(yy)

ax.plot_surface(xx, yy, z, cmap=cm.jet, cstride=1)
show()

Example 28: Sphere Wireframe

from pylab import *
from mpl_toolkits.mplot3d import Axes3D

ax = Axes3D(figure())
phi = linspace(0, 2*pi, 100)
theta = linspace(0, pi, 100)

x = 10 * outer(cos(phi), sin(theta))
y = 10 * outer(sin(phi), sin(theta))
z = 10 * outer(ones(size(phi)), cos(theta))

ax.plot_wireframe(x, y, z, rstride=2, cstride=2)
show()

1. What is Matplotlib? How do you import it for plotting?
2. Explain the difference between plot() and polar() with examples.
3. How do you create multiple plots in the same window using subplot()?
4. What is the format string in plot(x, y, 'ro')? What does each character mean?
5. Why is a circle difficult to plot in Cartesian coordinates? How do parametric equations solve this?
6. Write the parametric equations for the Astroid and Ellipse.
7. What is a meshgrid? How is it useful for 2D and 3D plotting?
8. Explain how imshow() is used to visualize a 2D matrix with colors.
9. What are fractals? Write the equation used for the Julia set.
10. Write a Python program to plot a surface using plot_surface().
11. How do you create a 3D wireframe plot? Give an example.
12. What is Mayavi? How does it differ from Matplotlib for 3D visualization?