Projecting the 3-simplex onto the sphere, with planes of projection.

simplex-projection-2.py

import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
fig = plt.figure(figsize=(21, 18))
ax = fig.add_subplot(111, projection='3d')

u_c = np.linspace(0, np.pi / 2, 100)
v_c = np.linspace(0, np.pi / 2, 100)

x_c = 1 * np.outer(np.cos(u_c), 
                   np.sin(v_c))
y_c = 1 * np.outer(np.sin(u_c), 
                   np.sin(v_c))
z_c = 1 * np.outer(np.ones(np.size(u_c)),
                   np.cos(v_c))

ax.plot_surface(x_c, y_c, z_c,
                rstride=5, cstride=5,
                color='pink', linewidth=0.2, alpha=0.5, antialiased=True)

u = np.linspace(0, 1, 100)
v = np.linspace(0, 1, 100)

x_s = 1 * np.outer(1 - u,
                   v)
y_s = 1 * np.outer(u, 
                   v)
z_s = 1 * np.outer(np.ones(np.size(u)),
                   1 - v)

ax.plot_surface(x_s, y_s, z_s,
                rstride=5, cstride=5,
                color='darkblue', linewidth=1, alpha=0.8, antialiased=True)

z_l1 = 0.466666
y_l1 = u * 0.533333
x_l1 = (1 - u) * 0.533333

norms1 = (z_l1 * z_l1 + 
          y_l1 * y_l1 +
          x_l1 * x_l1) ** 0.5

ax.plot(x_l1, y_l1, z_l1, color='lightblue',
        linewidth=2)
ax.plot(x_l1 / norms1, y_l1 / norms1, 
        z_l1 / norms1, color='darkblue',
        linewidth=3)

z_l2 = 0.2
y_l2 = u * 0.8
x_l2 = (1 - u) * 0.8

norms2 = (z_l2 * z_l2 + 
          y_l2 * y_l2 +
          x_l2 * x_l2) ** 0.5

ax.plot(x_l2, y_l2, z_l2, color='lightblue',
        linewidth=2)
ax.plot(x_l2 / norms2, y_l2 / norms2, 
        z_l2 / norms2, color='darkblue',
        linewidth=3)

r = np.linspace(0, 1, 100)[None, :]
r_norms1 = r / norms1[:, None]

x_p1 = x_l1[:, None] * r_norms1
y_p1 = y_l1[:, None] * r_norms1
z_p1 = z_l1 * np.ones_like(y_l1)[:, None] * r_norms1

ax.plot_surface(x_p1, y_p1, z_p1,
                rstride=10, cstride=10,
                color='lightblue', linewidth=0.2, 
                alpha=0.25, antialiased=True)

r_norms2= r / norms2[:, None]

x_p2 = x_l2[:, None] * r_norms2
y_p2 = y_l2[:, None] * r_norms2
z_p2 = z_l2 * np.ones_like(y_l2)[:, None] * r_norms2

ax.plot_surface(x_p2, y_p2, z_p2,
                rstride=10, cstride=10,
                color='lightblue', linewidth=0.2, 
                alpha=0.25, antialiased=True)

ax.view_init(elev = 18, azim = -45)

plt.show()