向量的旋转 (Python)

Question

我使用以下代码通过两个 2D 旋转在 3D 中旋转矢量：

注意：L 是

np.array([11.231303753070549, 9.27144871768164, 18.085790226916288])

下图中以蓝色显示的预定义向量。

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

def angle_between(p1, p2):
    ang1 = np.arctan2(*p1[::-1])
    ang2 = np.arctan2(*p2[::-1])
    return ((ang1 - ang2) % (2 * np.pi))

L = np.vstack([L,np.zeros(3)])
line_xy = [0.,1.]
line_L = [L[0,0],L[0,1]]
a = angle_between(line_xy, line_L)

def rotation(vector,theta):    
        v1_new = (vector[0]*np.cos(theta)) - (vector[1]*np.sin(theta))
        v2_new = (vector[1]*np.cos(theta)) + (vector[0]*np.sin(theta))        
        z_trans = [v1_new,v2_new,vector[2]]
        line_yz= [0.,1.]
        theta2 = angle_between(line_yz, [z_trans[1],z_trans[2]])
        v1_new = (z_trans[0]*np.cos(theta2)) - (z_trans[1]*np.sin(theta2))
        v2_new = (z_trans[1]*np.cos(theta2)) + (z_trans[0]*np.sin(theta2))
        y_trans = np.array([z_trans[0],v1_new,v2_new])        
        return z_trans,y_trans

L2,L3 = rotation(L[0,:],a)

L2 = np.vstack([L2,np.zeros(3)])
L3 = np.vstack([L3,np.zeros(3)])

fig = plt.figure()
ax = fig.add_subplot(111, projection='3d')
#ax.scatter(x1*1000,y1*1000,z1*1000,c ='r',zorder=2)
ax.plot(L[:,0],L[:,1],L[:,2],color='b',zorder=1)
line = np.array([[0,0,0],[0,0,15]])
ax.plot(line[:,0],line[:,1],line[:,2],color = 'g')
ax.set_xlabel('X Kpc')
ax.set_ylabel('Y Kpc')
ax.set_zlabel('Z Kpc')

ax.plot(L2[:,0],L2[:,1],L2[:,2],color='g')
ax.plot(L3[:,0],L3[:,1],L3[:,2],color='y')

我在这里做的是计算 x=0, y=1 之间的角度（即 line_xy 部分），然后使用旋转函数的第一部分围绕 z 轴旋转它：

v1_new = (vector[0]*np.cos(theta)) - (vector[1]*np.sin(theta))
v2_new = (vector[1]*np.cos(theta)) + (vector[0]*np.sin(theta))        
z_trans = [v1_new,v2_new,vector[2]]

然后重复该过程，但这次使用旋转函数的第二部分绕 x 轴旋转：

line_yz= [0.,1.]
theta2 = angle_between(line_yz, [z_trans[1],z_trans[2]])
v1_new = (z_trans[0]*np.cos(theta2)) - (z_trans[1]*np.sin(theta2))
v2_new = (z_trans[1]*np.cos(theta2)) + (z_trans[0]*np.sin(theta2))
y_trans = np.array([z_trans[0],v1_new,v2_new]) 

旋转是通过标准的 2D 旋转方程完成的：

x' = x cos(theta) - y sin(theta) y' = y cos(theta) + x sin(theta)

但是由于某种原因，第二次旋转后，线（黄色）与绿线（旋转此向量的原始目标）不对齐。

我试过检查弧度和度数的角度，但它似乎只适用于弧度。

当检查角度 theta2 时，它出现在 35 度左右，这看起来似乎是合理的。

Answer 1

我对你的问题不是很清楚，但希望这会有所帮助。

如果您想围绕特定轴旋转 3D 矢量，请利用矩阵变换而不是元素方式（就像您在上面写的那样）。下面是围绕任何轴旋转 3-D 矢量的代码：

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

def unit_vector(vector):
    """ Returns the unit vector of the vector."""
    return vector / np.linalg.norm(vector)

def angle_between(v1, v2):
    """Finds angle between two vectors"""
    v1_u = unit_vector(v1)
    v2_u = unit_vector(v2)
    return np.arccos(np.clip(np.dot(v1_u, v2_u), -1.0, 1.0))

def x_rotation(vector,theta):
    """Rotates 3-D vector around x-axis"""
    R = np.array([[1,0,0],[0,np.cos(theta),-np.sin(theta)],[0, np.sin(theta), np.cos(theta)]])
    return np.dot(R,vector)

def y_rotation(vector,theta):
    """Rotates 3-D vector around y-axis"""
    R = np.array([[np.cos(theta),0,np.sin(theta)],[0,1,0],[-np.sin(theta), 0, np.cos(theta)]])
    return np.dot(R,vector)

def z_rotation(vector,theta):
    """Rotates 3-D vector around z-axis"""
    R = np.array([[np.cos(theta), -np.sin(theta),0],[np.sin(theta), np.cos(theta),0],[0,0,1]])
    return np.dot(R,vector)

将原始蓝色矢量旋转 45 度 (pi/2)

L_predef = np.array([11.231303753070549, 9.27144871768164, 18.085790226916288]) #blue vector
new_vect = z_rotation(L_predef, np.pi/2.0)

fig = plt.figure()
ax = fig.gca(projection='3d')
ax.plot(np.linspace(0,L_predef[0]),np.linspace(0,L_predef[1]),np.linspace(0,L_predef[2]))
ax.plot(np.linspace(0,new_vect[0]),np.linspace(0,new_vect[1]),np.linspace(0,new_vect[2]))

plt.show()