A 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.

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

n = 100

theta = np.linspace(0, 2.*np.pi, n)
phi = np.linspace(0, 2.*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 = plt.figure()
ax1 = fig.add_subplot(121, projection='3d')
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 = fig.add_subplot(122, projection='3d')
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 \texttt{meshgrid}.

  • Note that we can use keywords such as edgecolors to style the polygon patches created by ax.plot_surface.

  • Elevation angle above the $xy$-plane of $36^\circ$, azimuthal angle in the $xy$-plane of $26^\circ$.

A toroidal surface