我有一组x和y坐标，相对于物体在不断旋转的圆形环境(10转/分钟)中的运动.如何区分物体的运动与环境的运动？
我try 了极坐标、速度和运动载体，得到了结果，但我想知道是否有人知道正确的路由. 主要是，我根据距旋转中心的距离对每个点进行反旋，转换为极坐标，创建一系列反旋位置.然后我计算每个original_points[idx]和derotated_points[idx+1]之间的运动载体.然后，我将第一个原始点作为新数组的起点，并向其添加第一个载体，存储新位置.这将是添加第二个载体的起点，等等……
看起来很合理，但我想知道是否还有其他方法.
这是一个非常好的领域，也是一个关于矩阵变换的好问题.如果您想了解这一点，请阅读本课程:https://graphics.stanford.edu/courses/cs248-01/
现在，让我们开始创建一些示例数据:
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
rpm = 10
omega = rpm * 2 * np.pi / 60
time = np.linspace(0, 60, 60)
speed = 0.1
x_linear = speed * time
y_linear = np.zeros_like(time)
x_rotated = x_linear * np.cos(omega * time) - y_linear * np.sin(omega * time)
y_rotated = x_linear * np.sin(omega * time) + y_linear * np.cos(omega * time)
df = pd.DataFrame({
'time': time,
'x': x_rotated,
'y': y_rotated
})
print(df)
这里，我假设rpm = 10，角速度(单位为rad(s))被给出为omega = rpm * 2 * np.pi / 60，物体的速度为speed = 0.1单位/秒.
这给你
time x y
0 0.000000 0.000000 0.000000e+00
1 1.016949 0.049276 8.895896e-02
2 2.033898 -0.107882 1.724206e-01
3 3.050847 -0.304652 -1.623727e-02
4 4.067797 -0.177888 -3.658219e-01
5 5.084746 0.292265 -4.160862e-01
6 6.101695 0.606713 6.485704e-02
7 7.118644 0.276790 6.558492e-01
8 8.135593 -0.502393 6.399064e-01
9 9.152542 -0.903602 -1.455835e-01
10 10.169492 -0.344989 -9.566443e-01
11 11.186441 0.736640 -8.418588e-01
12 12.203390 1.192763 2.579585e-01
13 13.220339 0.381661 1.265744e+00
...
56 56.949153 -5.686844 3.030958e-01
57 57.966102 -3.074643 -4.913986e+00
58 58.983051 2.858029 -5.159620e+00
59 60.000000 6.000000 -5.732833e-14
现在，要go 旋转
def derotate_coordinates(df, omega):
df['x_derotated'] = df['x']*np.cos(-omega*df['time']) - df['y']*np.sin(-omega*df['time'])
df['y_derotated'] = df['x']*np.sin(-omega*df['time']) + df['y']*np.cos(-omega*df['time'])
return df
比照适用的
df_derotated = derotate_coordinates(df.copy(), omega)
print(df_derotated)
会给你
time x y x_derotated y_derotated
0 0.000000 0.000000 0.000000e+00 0.000000 0.000000e+00
1 1.016949 0.049276 8.895896e-02 0.101695 0.000000e+00
2 2.033898 -0.107882 1.724206e-01 0.203390 -1.387779e-17
3 3.050847 -0.304652 -1.623727e-02 0.305085 0.000000e+00
4 4.067797 -0.177888 -3.658219e-01 0.406780 0.000000e+00
5 5.084746 0.292265 -4.160862e-01 0.508475 0.000000e+00
6 6.101695 0.606713 6.485704e-02 0.610169 0.000000e+00
7 7.118644 0.276790 6.558492e-01 0.711864 0.000000e+00
8 8.135593 -0.502393 6.399064e-01 0.813559 0.000000e+00
9 9.152542 -0.903602 -1.455835e-01 0.915254 2.775558e-17
10 10.169492 -0.344989 -9.566443e-01 1.016949 -5.551115e-17
11 11.186441 0.736640 -8.418588e-01 1.118644 0.000000e+00
12 12.203390 1.192763 2.579585e-01 1.220339 0.000000e+00
13 13.220339 0.381661 1.265744e+00 1.322034 -5.551115e-17
...
56 56.949153 -5.686844 3.030958e-01 5.694915 0.000000e+00
57 57.966102 -3.074643 -4.913986e+00 5.796610 0.000000e+00
58 58.983051 2.858029 -5.159620e+00 5.898305 -4.440892e-16
59 60.000000 6.000000 -5.732833e-14 6.000000 0.000000e+00
如果您想将其可视化
plt.figure(figsize=(12, 6))
plt.subplot(1, 2, 1)
plt.plot(df['x'], df['y'], 'ro-')
plt.title('Path in Rotating Frame')
plt.xlabel('X')
plt.ylabel('Y')
plt.axis('equal')
plt.subplot(1, 2, 2)
plt.plot(df_derotated['x_derotated'], df_derotated['y_derotated'], 'bo-')
plt.title('Path After Derotation')
plt.xlabel('X')
plt.ylabel('Y')
plt.axis('equal')
plt.show()
这给
更新:在3D中，仅仅因为这是一个迷人的话题
在这种情况下，让我们定义一个旋转螺旋并在极坐标下工作:
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D
t = np.linspace(0, 4 * np.pi, 100)
x = t * np.sin(t)
y = t * np.cos(t)
z = t
original_points = np.vstack([x, y, z])
def rotation_matrix_y(theta):
cos_theta, sin_theta = np.cos(theta), np.sin(theta)
return np.array([
[cos_theta, 0, sin_theta],
[0, 1, 0],
[-sin_theta, 0, cos_theta]
])
def rotation_matrix_z(theta):
cos_theta, sin_theta = np.cos(theta), np.sin(theta)
return np.array([
[cos_theta, -sin_theta, 0],
[sin_theta, cos_theta, 0],
[0, 0, 1]
])
theta_z = np.radians(45)
theta_y = np.radians(30)
rot_matrix_z = rotation_matrix_z(theta_z)
rot_matrix_y = rotation_matrix_y(theta_y)
combined_rot_matrix = rot_matrix_y @ rot_matrix_z
rotated_points = combined_rot_matrix @ original_points
inverse_combined_rot_matrix = np.transpose(combined_rot_matrix)
derotated_points = inverse_combined_rot_matrix @ rotated_points
fig = plt.figure(figsize=(18, 6))
ax1 = fig.add_subplot(131, projection='3d')
ax1.plot(*original_points, 'r')
ax1.set_title('Original Spiral')
ax1.set_xlim([-20, 20])
ax1.set_ylim([-20, 20])
ax1.set_zlim([0, 40])
ax2 = fig.add_subplot(132, projection='3d')
ax2.plot(*rotated_points, 'b')
ax2.set_title('Rotated Spiral')
ax2.set_xlim([-20, 20])
ax2.set_ylim([-20, 20])
ax2.set_zlim([0, 40])
ax3 = fig.add_subplot(133, projection='3d')
ax3.plot(*derotated_points, 'g')
ax3.set_title('Derotated Spiral')
ax3.set_xlim([-20, 20])
ax3.set_ylim([-20, 20])
ax3.set_zlim([0, 40])
plt.show()
这给 you
