E7.26A torus
The parametric description of a torus with radius \(c\) and tube radius \(a\) is
\[\begin{align*}
x &= (c + a\cos\theta) \cos\phi\\
y &= (c + a\cos\theta) \sin\phi\\
z &= a \sin\theta
\end{align*}\]
for \(\theta\) and \(\phi\) each between \(0\) and \(2\pi\). The code below outputs two views of a torus rendered as a surface plot.
torus
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
n = 100
theta = np.linspace(0, 2.0 * np.pi, n)
phi = np.linspace(0, 2.0 * np.pi, n)
theta, phi = np.meshgrid(theta, phi)
c, a = 2, 1
x = (c + a * np.cos(theta)) * np.cos(phi)
y = (c + a * np.cos(theta)) * np.sin(phi)
z = a * np.sin(theta)
fig, axes = plt.subplots(nrows=1, ncols=2, subplot_kw={"projection": "3d"})
ax1, ax2 = axes
ax1.set_zlim(-3, 3)
ax1.plot_surface(x, y, z, rstride=5, cstride=5, color="k", edgecolors="w")
ax1.view_init(36, 26)
ax2.set_zlim(-3, 3)
ax2.plot_surface(x, y, z, rstride=5, cstride=5, color="k", edgecolors="w")
ax2.view_init(0, 0)
ax2.set_xticks([])
plt.show()
- We need \(\theta\) and \(\phi\) to range over the interval \((0,2\pi)\) independently, so use a
meshgrid. - Note that we can use keywords such as
edgecolorsto style the polygon patches created byax.plot_surface. - Elevation angle above the \(xy\)-plane of \(36^\circ\), azimuthal angle in the \(xy\)-plane of \(26^\circ\).
