================ by Jawad Haider

Chpt 4 - Visualization with Matplotlib¶

Example: Visualizing a Möbius strip¶

Example: Visualizing a Möbius strip

Example: Visualizing a Möbius strip¶

A Möbius strip is similar to a strip of paper glued into a loop with a half-twist. Topo‐ logically, it’s quite interesting because despite appearances it has only a single side! Here we will visualize such an object using Matplotlib’s three-dimensional tools. The key to creating the Möbius strip is to think about its parameterization: it’s a two-dimensional strip, so we need two intrinsic dimensions. Let’s call them θ, which ranges from 0 to 2π around the loop, and w which ranges from –1 to 1 across the width of the strip:

from mpl_toolkits import mplot3d

%matplotlib notebook
import numpy as np
import matplotlib.pyplot as plt

theta=np.linspace(0,2*np.pi, 30)
w=np.linspace(-0.25,0.25,8)
w,theta=np.meshgrid(w,theta)

Now from this parameterization, we must determine the (x, y, z) positions of the embedded strip. Thinking about it, we might realize that there are two rotations happening: one is the position of the loop about its center (what we’ve called θ), while the other is the twist‐ ing of the strip about its axis (we’ll call this ϕ). For a Möbius strip, we must have the strip make half a twist during a full loop, or Δϕ = Δθ/2.

phi = 0.5*theta

Now we use our recollection of trigonometry to derive the three-dimensional embed‐ ding. We’ll define r, the distance of each point from the center, and use this to find the embedded x, y, z coordinates:

# radius in x-y plane
r=1+w*np.cos(phi)

x=np.ravel(r*np.cos(theta))
y=np.ravel(r*np.sin(theta))
z=np.ravel(w*np.sin(theta))

Finally, to plot the object, we must make sure the triangulation is correct. The best way to do this is to define the triangulation within the underlying parameterization, and then let Matplotlib project this triangulation into the three-dimensional space of the Möbius strip. This can be accomplished as follows

# triangulate in the underlying parametrization
from matplotlib.tri import Triangulation
tri=Triangulation(np.ravel(w), np.ravel(theta))

ax=plt.axes(projection='3d')
ax.plot_trisurf(x,y,z, triangles=tri.triangles, cmap='viridis', linewidth=0.2)

ax.set_xlim(-1,1);ax.set_ylim(-1,1);ax.set_zlim(-1,1);

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Combining all of these techniques, it is possible to create and display a wide variety of three-dimensional objects and patterns in Matplotlib.