3D Surface of Revolution Paraboloid using Python

import matplotlib.pyplot as plt

import numpy as np

from mpl_toolkits.mplot3d import Axes3D

def f(x):

    return x**2

x=np.linspace(-3,3,100)

theta=np.linspace(0,2*np.pi,100)

X,Theta=np.meshgrid(x,theta)

Y=f(X)*np.cos(Theta)

Z=f(X)*np.sin(Theta)

fig=plt.figure(figsize=(6,6))

ax=fig.add_subplot(111,projection='3d')

ax.plot_surface(X,Y,Z,cmap='inferno',edgecolor='none',alpha=0.7)

ax.set_title('3D Surface of Revolution')

ax.set_xlabel('X axis')

ax.set_ylabel('Y axis')

ax.set_zlabel('Z axis')

plt.show()

#source code --> clcoding.com 

Code Explanation:

1. Importing Libraries

import numpy as np

import matplotlib.pyplot as plt

from mpl_toolkits.mplot3d import Axes3D

numpy: A numerical library that provides efficient ways to work with arrays and perform mathematical operations.

 matplotlib.pyplot: A plotting library used for creating various types of graphs and plots. It is commonly used for 2D plotting.

 mpl_toolkits.mplot3d: This is a module from Matplotlib that provides tools for 3D plotting. Specifically, Axes3D is used to create 3D plots.

 2. Define the Function for the Paraboloid

def f(x):

    return x ** 2

f(x): This defines a function that calculates the square of the input x, creating a simple parabolic shape when plotted in 2D.

 3. Create the Data for the Plot

x = np.linspace(-3, 3, 100)  # Generate 100 equally spaced values between -3 and 3

theta = np.linspace(0, 2 * np.pi, 100)  # Generate 100 equally spaced values from 0 to 2π (for full revolution)

X, Theta = np.meshgrid(x, theta)  # Create a meshgrid from the x and theta values

x = np.linspace(-3, 3, 100): This generates 100 evenly spaced values between -3 and 3. These represent the x-coordinates for the paraboloid.

 theta = np.linspace(0, 2 * np.pi, 100): This generates 100 evenly spaced values between 0 and 2π, which represent angles for a full revolution around the Z-axis.

 X, Theta = np.meshgrid(x, theta): np.meshgrid creates a 2D grid of x and theta values. It returns two 2D arrays, X and Theta, that correspond to the coordinates of a grid in the 3D space.

 4. Calculate the Coordinates for the 3D Surface

Y = f(X) * np.cos(Theta)  # Parametric equation for the Y coordinate (scaled by cosine of theta)

Z = f(X) * np.sin(Theta)  # Parametric equation for the Z coordinate (scaled by sine of theta)

Y = f(X) * np.cos(Theta): This calculates the Y-coordinates of the surface. The f(X) part gives the value of the paraboloid function (square of X), and multiplying by cos(Theta) rotates the paraboloid in the Y-axis direction.

 Z = f(X) * np.sin(Theta): Similarly, this calculates the Z-coordinates of the surface. It also uses the parabolic function f(X) and rotates the result around the Z-axis using sin(Theta).

 5. Set Up the Plot

fig = plt.figure(figsize=(10, 8))  # Create a figure with a size of 10x8 inches

ax = fig.add_subplot(111, projection='3d')  # Add a 3D subplot to the figure

fig = plt.figure(figsize=(10, 8)): Initializes a new figure with a specified size of 10 inches by 8 inches.

 ax = fig.add_subplot(111, projection='3d'): Creates a 3D subplot in the figure (111 means a single subplot), with projection='3d' enabling 3D plotting capabilities.

 6. Plot the Surface

ax.plot_surface(X, Y, Z, cmap='inferno', edgecolor='none', alpha=0.7)  # Plot the 3D surface

ax.plot_surface(X, Y, Z, cmap='inferno', edgecolor='none', alpha=0.7):

 This creates the 3D surface plot using the X, Y, and Z coordinates calculated earlier.

 cmap='inferno': Specifies the colormap to use for coloring the surface. inferno is a popular colormap that ranges from dark purple to bright yellow.

 edgecolor='none': Ensures that there are no lines around the edges of the surface.

 alpha=0.7: Adjusts the transparency of the surface to 70% (where 1 is fully opaque and 0 is fully transparent).

 7. Label the Axes and Set the Title

ax.set_xlabel("X axis")  # Label the X axis

ax.set_ylabel("Y axis")  # Label the Y axis

ax.set_zlabel("Z axis")  # Label the Z axis

ax.set_title("3D Surface of Revolution (Paraboloid)")  # Set the title of the plot

ax.set_xlabel("X axis"), ax.set_ylabel("Y axis"), ax.set_zlabel("Z axis"): Labels the X, Y, and Z axes, making the plot easier to interpret.

 ax.set_title("3D Surface of Revolution (Paraboloid)"): Adds a title to the plot indicating that this is a surface of revolution, specifically a paraboloid.

 8. Display the Plot

plt.show()  # Show the plot

plt.show(): This line displays the figure. It renders the 3D surface plot on the screen.