The formulation of the Möbius strip involves a parameterization in terms of two variables: \(\theta\), which is an angle that varies from \(0\) to \(2 \pi\), and \(w\), which is a distance from the centerline of the strip and varies between -1 and 1.
Given these parameters, the \((x, y, z)\) coordinates for points on the Möbius strip can be determined using the following equations:
import numpy as np import matplotlib.pyplot as plt # Create a Möbius strip theta = np.linspace(0, 2 * np.pi, 100) w = np.linspace(-1, 1, 20) theta, w = np.meshgrid(theta, w) phi = 0.5 * theta r = 1 + w * np.cos(phi) x = np.cos(theta) * r y = np.sin(theta) * r z = w * np.sin(phi) # Plot the Möbius strip fig = plt.figure() ax = fig.add_subplot(111, projection='3d') ax.plot_surface(x, y, z, edgecolor='k', color='c') ax.view_init(50, 30) # Adjust view angle for better visualization ax.set_title('Möbius Strip in 3D') plt.show()
- Parameter Space Definition: We begin by defining our parameter space for θ and w. The variable θ spans the entire 360-degree range of the strip, while w varies between -1 and 1 to account for the width of the strip.
- Meshgrid Creation: Using NumPy’s
meshgridfunction, we create a grid of points in the (θ, w) parameter space. Each combination of (θ, w) corresponds to a unique point on the Möbius strip.
- Position Calculation: For every combination of (θ, w), we compute the x, y, and z coordinates using the above equations.
- 3D Plotting: Using
Matplotlib’s 3D plotting capabilities, we then plot the computed (x, y, z) coordinates as a surface. The
edgecoloris set to ‘k’ (black) to distinguish between individual segments on the strip, and
coloris set to ‘c’ (cyan) for the surface color.
- View Angle Adjustment: The line
ax.view_init(50, 30)adjusts the view angle of the 3D plot, making it easier to perceive the Möbius strip’s twist.