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()
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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 $26^\circ$.
