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:
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 | import numpy as npimport matplotlib.pyplot as plt# Create a Möbius striptheta = np.linspace(0, 2 * np.pi, 100)w = np.linspace(-1, 1, 20)theta, w = np.meshgrid(theta, w)phi = 0.5 * thetar = 1 + w * np.cos(phi)x = np.cos(theta) * ry = np.sin(theta) * rz = w * np.sin(phi)# Plot the Möbius stripfig = 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 visualizationax.set_title('Möbius Strip in 3D')plt.show() |
Visualization Strategy:
- 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. Theedgecoloris set to ‘k’ (black) to distinguish between individual segments on the strip, andcoloris 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.
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