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Bk2_Ch18_立体几何_06可视化单位球体几何转换

《可视之美》

python

鸾尾花书

数据可视化

《可视之美》python鸾尾花书数据可视化

刀刀

发布于 2024-05-28

推荐镜像 :Basic Image:bohrium-notebook:2023-04-07

推荐机型 :c2_m4_cpu

可视化单位球体几何转换

单位球

©️ Copyright 2024 @ Authors
共享协议：本作品采用知识共享署名-非商业性使用-相同方式共享 4.0 国际许可协议进行许可。
作者信息：Notebook原地址为 https://github.com/Visualize-ML

代码

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Chapter 18

可视化单位球体几何转换

Book_2《可视之美》 | 鸢尾花书：从加减乘除到机器学习

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[1]

# 导入包

from mpl_toolkits.mplot3d import axes3d

import matplotlib.pyplot as plt

import numpy as np

import os

# 如果文件夹不存在，创建文件夹

if not os.path.isdir("Figures"):

os.makedirs("Figures")

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[2]

def plot_implicit(fn, X_plot, Y_plot, Z_plot, ax, bbox, filename):

# 等高线的起止范围

xmin, xmax, ymin, ymax, zmin, zmax = bbox*3

ax.set_proj_type('ortho')

# 绘制三条参考线

k = 1.5

ax.plot((xmin * k, xmax * k), (0, 0), (0, 0), 'k', lw = 0.1)

ax.plot((0, 0), (ymin * k, ymax * k), (0, 0), 'k', lw = 0.1)

ax.plot((0, 0), (0, 0), (zmin * k, zmax * k), 'k', lw = 0.1)

# 等高线的分辨率

A = np.linspace(xmin, xmax, 500)

# 产生网格数据

A1,A2 = np.meshgrid(A,A)

# 等高线的分割位置

B = np.linspace(xmin, xmax, 40)

# 绘制 XY 平面等高线

if X_plot == True:

for z in B:

X,Y = A1,A2

Z = fn(X,Y,z)

cset = ax.contour(X, Y, Z+z, [z],

zdir='z',

linewidths = 0.25,

colors = '#0066FF',

linestyles = 'solid')

# 绘制 XZ 平面等高线

if Y_plot == True:

for y in B:

X,Z = A1,A2

Y = fn(X,y,Z)

cset = ax.contour(X, Y+y, Z, [y],

zdir='y',

linewidths = 0.25,

colors = '#88DD66',

linestyles = 'solid')

# 绘制 YZ 平面等高线

if Z_plot == True:

for x in B:

Y,Z = A1,A2

X = fn(x,Y,Z)

cset = ax.contour(X+x, Y, Z, [x],

zdir='x',

linewidths = 0.25,

colors = '#FF6600',

linestyles = 'solid')

ax.quiver(0, 0, 0,

xmax, 0, 0,

length = 1,

color = 'r',

normalize=False,

arrow_length_ratio = .07,

linestyles = 'solid',

linewidths = 0.25)

ax.quiver(0, 0, 0,

0, ymax, 0,

length = 1,

color = 'g',

normalize=False,

arrow_length_ratio = .07,

linestyles = 'solid',

linewidths = 0.25)

ax.quiver(0, 0, 0,

0, 0, zmax,

length = 1,

color = 'b',

normalize=False,

arrow_length_ratio = .07,

linestyles = 'solid',

linewidths = 0.25)

ax.set_zlim(zmin * k,zmax * k)

ax.set_xlim(xmin * k,xmax * k)

ax.set_ylim(ymin * k,ymax * k)

ax.set_box_aspect([1,1,1])

ax.view_init(azim=60, elev=30)

ax.axis('off')

fig.savefig('Figures/' + filename + '.svg', format='svg')

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[3]

# 缩放 > 旋转

# xT @ x = 1

# A@x = z, A = V @ S

# x = inv(A) @ z

# zT @ inv(A @ AT) z = 1

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单位球

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[4]

Identity = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 1]])

A = Identity

def unit(x,y,z):

f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

return f[0,0] - 1

fig = plt.figure(figsize=(6,6))

ax = fig.add_subplot(111, projection='3d')

plot_implicit(unit, True, True, True, ax, (-4,4), '单位球')

/tmp/ipykernel_1031/944982215.py:7: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

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[5]

alpha = np.deg2rad(45)

R_x = np.array([[1, 0, 0],

[0, np.cos(alpha), -np.sin(alpha)],

[0, np.sin(alpha), np.cos(alpha)]])

S_z = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 3]])

# A@x = z, A = V @ S

A = S_z

def scale_z_rotate_x(x,y,z):

f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

return f[0,0] - 1

fig = plt.figure(figsize=(6,6))

ax = fig.add_subplot(111, projection='3d')

plot_implicit(scale_z_rotate_x, True, True, True, ax, (-4,4), '缩放')

/tmp/ipykernel_1031/1555199623.py:15: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

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[6]

S_z = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 3]])

alpha = np.deg2rad(30)

R_x = np.array([[1, 0, 0],

[0, np.cos(alpha), -np.sin(alpha)],

[0, np.sin(alpha), np.cos(alpha)]])

A = R_x @ S_z

def scale_z_rotate_x(x,y,z):

f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

return f[0,0] - 1

fig = plt.figure(figsize=(6,6))

ax = fig.add_subplot(111, projection='3d')

plot_implicit(scale_z_rotate_x, True, True, True, ax, (-4,4), '缩放 --- x旋转')

/tmp/ipykernel_1031/1429942179.py:15: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

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[7]

S_z = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 3]])

alpha = np.deg2rad(30)

R_x = np.array([[1, 0, 0],

[0, np.cos(alpha), -np.sin(alpha)],

[0, np.sin(alpha), np.cos(alpha)]])

beta = np.deg2rad(45)

# 从 y 正方向看

R_y = np.array([[np.cos(beta), 0, np.sin(beta)],

[0, 1, 0],

[-np.sin(beta),0, np.cos(beta)]])

A = R_y @ R_x @ S_z

def scale_z_rotate_x_y(x,y,z):

f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

return f[0,0] - 1

fig = plt.figure(figsize=(6,6))

ax = fig.add_subplot(111, projection='3d')

plot_implicit(scale_z_rotate_x_y, True, True, True, ax, (-4,4), '缩放 --- xy旋转')

/tmp/ipykernel_1031/871422812.py:23: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

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[8]

S_z = np.array([[1, 0, 0],

[0, 1, 0],

[0, 0, 3]])

alpha = np.deg2rad(30)

R_x = np.array([[1, 0, 0],

[0, np.cos(alpha), -np.sin(alpha)],

[0, np.sin(alpha), np.cos(alpha)]])

beta = np.deg2rad(45)

# 从 y 正方向看

R_y = np.array([[np.cos(beta), 0, np.sin(beta)],

[0, 1, 0],

[-np.sin(beta),0, np.cos(beta)]])

gamma = np.deg2rad(60)

R_z = np.array([[np.cos(gamma), -np.sin(gamma), 0],

[np.sin(gamma), np.cos(gamma), 0],

[0, 0, 1]])

A = R_z @ R_y @ R_x @ S_z

def scale_z_rotate_x_y_z(x,y,z):

f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

return f[0,0] - 1

fig = plt.figure(figsize=(6,6))

ax = fig.add_subplot(111, projection='3d')

plot_implicit(scale_z_rotate_x_y_z, True, True, True, ax, (-4,4), '缩放 --- xyz旋转')

/tmp/ipykernel_1031/780611931.py:28: VisibleDeprecationWarning: Creating an ndarray from ragged nested sequences (which is a list-or-tuple of lists-or-tuples-or ndarrays with different lengths or shapes) is deprecated. If you meant to do this, you must specify 'dtype=object' when creating the ndarray.
  f = [[x, y, z]] @ np.linalg.pinv(A @ A.T) @ [[x], [y], [z]]

代码

文本

《可视之美》

python

鸾尾花书

数据可视化

《可视之美》python鸾尾花书数据可视化

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Book2《可视之美》 | 鸢尾花书：从加减乘除到机器学习

刀刀

更新于 2024-06-03

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