Created
June 29, 2021 13:57
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Spherical triangle sampling according to [Arvo 95]
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| #!/bin/env python3 | |
| from mpl_toolkits import mplot3d | |
| import numpy as np | |
| import matplotlib.pyplot as plt | |
| import math | |
| def vnorm(v): | |
| return v / np.linalg.norm(v) | |
| def draw_unit_sphere(ax): | |
| u = np.linspace(0, 2 * np.pi, 20) | |
| v = np.linspace(0, np.pi, 20) | |
| x = np.outer(np.cos(u), np.sin(v)) | |
| y = np.outer(np.sin(u), np.sin(v)) | |
| z = np.outer(np.ones(np.size(u)), np.cos(v)) | |
| ax.plot_surface(x, y, z, rstride=2, cstride=2, alpha=0.3) | |
| fig = plt.figure() | |
| ax = plt.axes(projection='3d') | |
| # Triangle | |
| A0 = np.array([3, 1, 2]) | |
| B0 = np.array([2, 3, 1]) | |
| C0 = np.array([1, 1, 2]) | |
| # Projected points on unit sphere | |
| A = A0 / np.linalg.norm(A0) | |
| B = B0 / np.linalg.norm(B0) | |
| C = C0 / np.linalg.norm(C0) | |
| draw_unit_sphere(ax) | |
| # Draw triangle & projected points | |
| ax.scatter3D([A[0], A0[0]], [A[1], A0[1]], [A[2], A0[2]], label='A') | |
| ax.scatter3D([B[0], B0[0]], [B[1], B0[1]], [B[2], B0[2]], label='B') | |
| ax.scatter3D([C[0], C0[0]], [C[1], C0[1]], [C[2], C0[2]], label='C') | |
| ax.plot([A0[0], 0], [A0[1], 0], [A0[2], 0]) | |
| ax.plot([B0[0], 0], [B0[1], 0], [B0[2], 0]) | |
| ax.plot([C0[0], 0], [C0[1], 0], [C0[2], 0]) | |
| #tri = mplot3d.art3d.Poly3DCollection([A0, B0, C0]) | |
| #tri.set_edgecolor('k') | |
| #ax.add_collection3d(tri) | |
| nA = vnorm(np.cross(B0 - A0, A0)) | |
| nB = vnorm(np.cross(C0 - B0, B0)) | |
| nC = vnorm(np.cross(A0 - C0, C0)) | |
| ax.plot( | |
| [A0[0], A0[0] + nA[0]], | |
| [A0[1], A0[1] + nA[1]], | |
| [A0[2], A0[2] + nA[2]]) | |
| ax.plot( | |
| [B0[0], B0[0] + nB[0]], | |
| [B0[1], B0[1] + nB[1]], | |
| [B0[2], B0[2] + nB[2]]) | |
| ax.plot( | |
| [C0[0], C0[0] + nC[0]], | |
| [C0[1], C0[1] + nC[1]], | |
| [C0[2], C0[2] + nC[2]]) | |
| dot_alpha = np.dot(nC, nA) | |
| dot_beta = np.dot(nA, nB) | |
| dot_gamma = np.dot(nB, nC) | |
| print("dot_alpha = {}".format(dot_alpha)) | |
| print("dot_beta = {}".format(dot_beta)) | |
| print("dot_gamma = {}".format(dot_gamma)) | |
| cos_alpha = -(dot_alpha) | |
| cos_beta = -(dot_beta) | |
| cos_gamma = -(dot_gamma) | |
| print("cos(α) = {}".format(cos_alpha)) | |
| print("cos(β) = {}".format(cos_beta)) | |
| print("cos(γ) = {}".format(cos_gamma)) | |
| alpha = math.acos(cos_alpha) | |
| beta = math.acos(cos_beta) | |
| gamma = math.acos(cos_gamma) | |
| print("α + β + γ = {} + {} + {} = {}".format(alpha, beta, gamma, alpha + beta + gamma)) | |
| sin_alpha = math.sin(alpha) | |
| sin_beta = math.sin(beta) | |
| cos_c = (cos_gamma + cos_beta * cos_alpha) / (sin_beta * sin_alpha) | |
| print("cos_c = {}".format(cos_c)) | |
| spherical_area = alpha + beta + gamma - np.pi | |
| print("spherical area: {}".format(spherical_area)) | |
| def ortho_component(x, y): | |
| return vnorm(x - np.dot(x, y) * y); | |
| def sample(e1, e2): | |
| small_area = e1 * spherical_area | |
| s = math.sin(small_area - alpha) | |
| t = math.cos(small_area - alpha) | |
| u = t - cos_alpha | |
| v = s + sin_alpha * cos_c | |
| q = ((v * t - u * s) * cos_alpha - v) / ((v * s + u * t) * sin_alpha) | |
| c_hat = q * A + math.sqrt(1 - q * q) * ortho_component(C, A) | |
| z = 1 - e2 * (1 - np.dot(c_hat, B)); | |
| return z * B + math.sqrt(1 - z * z) * ortho_component(c_hat, B) | |
| def plot_samples(ax): | |
| n = 10 | |
| x = np.empty((n + 1) * (n + 1)) | |
| y = np.empty((n + 1) * (n + 1)) | |
| z = np.empty((n + 1) * (n + 1)) | |
| for i in range(0, n + 1): | |
| for j in range(0, n + 1): | |
| d = sample(i / n, j / n) | |
| x[i * (n + 1) + j] = d[0] | |
| y[i * (n + 1) + j] = d[1] | |
| z[i * (n + 1) + j] = d[2] | |
| ax.scatter3D(x, y, z) | |
| if spherical_area > 0 and spherical_area < 2 * np.pi: | |
| plot_samples(ax) | |
| plt.legend() | |
| plt.show() |
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