5. SymPy with Graphs

5.1. Introduction

In this exercise, we will use Python’s SymPy library to create and visualize mathematical expressions using both 2D and 3D plots.

5.2. Code Example: 2D Plots

We will use the SymPy library to define symbolic expressions and visualize them using 2D plots.

Define the Symbolic Variable

from sympy import Symbol
from sympy.plotting import plot

# Define the symbolic variable
x = Symbol('x')

# Display the symbolic variable
print(x)

Plot the Expressions

# Plot the quadratic function x^2 with a custom color
plot(x**2, line_color='fuchsia')

# Plot the quadratic function x^2 with a different color code
plot(x**2, line_color='#e30052')

# Plot the quadratic function x^2 with the default color
plot(x**2)

# Plot the sine and cosine functions over a specific interval
plot(sin(x), cos(x), (x, -pi, pi))

# Plot multiple functions with different intervals and a custom title
plot((sin(x), (x, -2*pi, 2*pi)), (cos(x), (x, -pi, pi)),
     line_color='green', title='Graph Example with SymPy')

Note

Use these plots to explore the behavior of functions within specified intervals.

Converting Plots to PNG

In some cases, especially when working in a web environment like this interactive code editor, we need to convert plots into images (such as PNG) for them to be displayed. This is because the editor may not support direct rendering of SymPy plots in their native format. By converting the plots into images and encoding them as base64, we can embed them into HTML for visualization within the notebook or a web page.

Alternatively, if you’re working locally on your own machine, you can directly display plots without converting them to images by using the show() method provided by SymPy’s plotting module. This method will render the plot in a new window or within your Jupyter Notebook if you’re using one.

# Directly plot the expression without conversion when working locally
plot(sin(x), cos(x), (x, -pi, pi), show=True)

5.3. Code Example: 3D Plots

We can also create 3D surface plots using SymPy.

3D Surface Plot

from sympy.plotting import plot3d
from sympy import Symbol

# Define symbolic variables for 3D plotting
x = Symbol('x')
y = Symbol('y')

# Display the symbolic variables
print(x, y)

# Plot a 3D surface for the expression x * y
plot3d(x * y, (x, -10, 10), (y, -10, 10))

# Plot multiple 3D surfaces
plot3d(x * y, x / y, (x, -5, 5), (y, -5, 5))

# Plot surfaces with more complex expressions
plot3d((x**2 + y**2, (x, -5, 5), (y, -5, 5)),
       (x * y, (x, -3, 3), (y, -3, 3)))

3D Parametric Plots

from sympy.plotting import plot3d_parametric_line
from sympy import cos, sin

# Plot a 3D parametric line
plot3d_parametric_line(cos(x), sin(x), x, (x, -5, 5))

# Plot a 3D parametric surface
from sympy.plotting import plot3d_parametric_surface
u, v = symbols('u v')
plot3d_parametric_surface(cos(u + v), sin(u - v), u - v,
                          (u, -5, 5), (v, -5, 5))

Implicit Plots

from sympy import plot_implicit, Eq, And
from sympy import symbols

# Define the symbolic variables
x, y = symbols('x y')

# Plot an implicit equation
p1 = plot_implicit(Eq(x**2 + y**2, 5),
                   (x, -5, 5), (y, -2, 2),
                   adaptive=False, points=400)

# Plot a region defined by an inequality
p2 = plot_implicit(y > x**2)

# Plot using boolean conjunctions
p3 = plot_implicit(And(y > x, y > -x))

Note

Experiment with these plots to understand how SymPy handles symbolic math and visualization.

5.4. Interactive Code Editor

To experiment with the code interactively, use the provided interactive code blocks below. Run all the code blocks to see the results and explore different functionalities.

from sympy import Symbol, sin, cos, pi from sympy.plotting import plot from io import BytesIO import base64 x = Symbol('x') p = plot(sin(x), cos(x), (x, -pi, pi), show=False) # You can directly display the plot using plot() when working locally # Convert plot to PNG buffer = BytesIO() p.save(buffer) buffer.seek(0) img = buffer.getvalue() # Encode to base64 img_base64 = base64.b64encode(img).decode('utf-8') # Create HTML img tag img_tag = f'<img src="data:image/png;base64,{img_base64}">' # Display using PyScript's HTML class from pyscript import HTML print("The plot is displayed below in the field:") display(plt, "plot_area") # Replace with plt.show() if running locally

Note

Ensure you run all the code blocks provided to see the complete results and understand the functionalities demonstrated.

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