python - 如何绘制随机平面

Question

我正在使用以下代码绘制通过原点的 3d 随机平面。

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

#Number of hyperplanes
n = 20
#Dimension of space
d = 3

plt3d = plt.figure().gca(projection='3d')
for i in xrange(n):
    #Create random point on unit sphere
    v = np.random.normal(size = d)
    v = v/np.sqrt(np.sum(v**2))
    # create x,y
    xx, yy = np.meshgrid(range(-5,5), range(-5,5))
    z = (-v[0] * xx - v[1] * yy)/v[2]
    # plot the surface
    plt3d.plot_surface(xx, yy, z, alpha = 0.5)
plt.show()

但是看图片我不相信他们是统一选择的。我做错了什么？

Answer 1

您的代码生成的平面具有随机分布的法线。它们只是看起来不像那样，因为 z 尺度比 x 和 y 尺度大得多。

您可以通过生成点来生成更好看的图像 均匀分布在平面上。为此，请根据以下参数对平面进行参数化 新坐标 (u, v)，然后在均匀间隔的网格上对平面进行采样 (u,v) 点。然后将这些 (u,v) 点转换为 (x,y,z) 空间中的点。

from __future__ import division
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
import math
import itertools as IT

def points_on_sphere(dim, N, norm=np.random.normal):
    """
    http://en.wikipedia.org/wiki/N-sphere#Generating_random_points
    """
    normal_deviates = norm(size=(N, dim))
    radius = np.sqrt((normal_deviates ** 2).sum(axis=0))
    points = normal_deviates / radius
    return points

# Number of hyperplanes
n = 10
# Dimension of space
d = 3

fig, ax = plt.subplots(subplot_kw=dict(projection='3d'))
points = points_on_sphere(n, d).T
uu, vv = np.meshgrid([-5, 5], [-5, 5], sparse=True)
colors = np.linspace(0, 1, len(points))
cmap = plt.get_cmap('jet')
for nhat, c in IT.izip(points, colors):
    u = (0, 1, 0) if np.allclose(nhat, (1, 0, 0)) else np.cross(nhat, (1, 0, 0))
    u /= math.sqrt((u ** 2).sum())
    v = np.cross(nhat, u)
    u = u[:, np.newaxis, np.newaxis]
    v = v[:, np.newaxis, np.newaxis]
    xx, yy, zz = u * uu + v * vv
    ax.plot_surface(xx, yy, zz, alpha=0.5, color=cmap(c))
ax.set_xlim3d([-5,5])
ax.set_ylim3d([-5,5])
ax.set_zlim3d([-5,5])        
plt.show()

或者，您可以使用 Till Hoffmann's pathpatch_2d_to_3d utility function 来避免复杂的数学运算:

for nhat, c in IT.izip(points, colors):
    p = patches.Rectangle((-2.5, -2.5), 5, 5, color=cmap(c), alpha=0.5)
    ax.add_patch(p)
    pathpatch_2d_to_3d(p, z=0, normal=nhat)

ax.set_xlim3d([-5,5])
ax.set_ylim3d([-5,5])
ax.set_zlim3d([-5,5])        
plt.show()