Cycloid and Epicycloid Pattern using Python

 

import numpy as np

import matplotlib.pyplot as plt

r = 1

theta = np.linspace(0, 4 * np.pi, 1000)

x_cycloid = r * (theta - np.sin(theta))

y_cycloid = r * (1 - np.cos(theta))

R = 3

r_epicycloid = 1

x_epicycloid = (R + r_epicycloid) * np.cos(theta) - r_epicycloid * np.cos((R + r_epicycloid) / r_epicycloid * theta)

y_epicycloid = (R + r_epicycloid) * np.sin(theta) - r_epicycloid * np.sin((R + r_epicycloid) / r_epicycloid * theta)

plt.figure(figsize=(10, 5))

plt.subplot(1, 2, 1)

plt.plot(x_cycloid, y_cycloid, color='blue')

plt.title('Cycloid')

plt.axis('equal')

plt.subplot(1, 2, 2)

plt.plot(x_epicycloid, y_epicycloid, color='red')

plt.title('Epicycloid')

plt.axis('equal')

plt.tight_layout()

plt.show()

#source code --> clcoding.com 

Code Explanation:

1. Importing Libraries

import numpy as np

import matplotlib.pyplot as plt

NumPy: Used for creating numerical arrays and performing mathematical operations.

Matplotlib: Used for plotting the Cycloid and Epicycloid curves.

2. Define Parameters for the Cycloid

r = 1

theta = np.linspace(0, 4 * np.pi, 1000)

r = 1: Radius of the generating circle.

theta: Array of 1000 values equally spaced from 0 to 4π, representing the angle in radians.

This ensures a smooth plot for one complete oscillation.

3. Calculate the Cycloid Coordinates

x_cycloid = r * (theta - np.sin(theta))

y_cycloid = r * (1 - np.cos(theta))

Cycloid Equation:

x=r(t−sin(t)),y=r(1−cos(t))

Explanation:

x represents the horizontal motion of a point on the wheel as it rolls.

y represents the vertical motion of the point.

This forms a Cycloid — the path traced by a point on a rolling circle.

4. Define Parameters for the Epicycloid

R = 3

r_epicycloid = 1

R = 3: Radius of the stationary larger circle.

r_epicycloid = 1: Radius of the smaller rotating circle that rolls on the outside.

5. Calculate the Epicycloid Coordinates

x_epicycloid = (R + r_epicycloid) * np.cos(theta) - r_epicycloid * np.cos((R + r_epicycloid) / r_epicycloid * theta)

y_epicycloid = (R + r_epicycloid) * np.sin(theta) - r_epicycloid * np.sin((R + r_epicycloid) / r_epicycloid * theta)

Epicycloid Equation:

Explanation:

R is the radius of the fixed circle.

r is the radius of the rolling circle.

The trace of a point on the smaller circle as it rolls on the outer edge of the larger circle forms an Epicycloid.

6. Plot the Cycloid and Epicycloid

plt.figure(figsize=(10, 5))

Creates a figure of size 10x5 for side-by-side comparison.

7. Plot Cycloid

plt.subplot(1, 2, 1)

plt.plot(x_cycloid, y_cycloid, color='blue')

plt.title('Cycloid')

plt.axis('equal')

plt.subplot(1, 2, 1) creates the left plot (Cycloid).

plt.plot() plots the blue Cycloid.

plt.axis('equal') ensures the aspect ratio is equal, so the curve is accurately represented.

8. Plot Epicycloid

plt.subplot(1, 2, 2)

plt.plot(x_epicycloid, y_epicycloid, color='red')

plt.title('Epicycloid')

plt.axis('equal')

plt.subplot(1, 2, 2) creates the right plot (Epicycloid).

plt.plot() plots the red Epicycloid.

9. Final Touch and Display

plt.tight_layout()

plt.show()

plt.tight_layout() adjusts subplot spacing to prevent overlapping.

plt.show() renders the plot on the screen.