计算旋转矩阵以在3D空间中对齐两个向量? [英] Calculate rotation matrix to align two vectors in 3D space?

问题描述

我有两个分别代表曲线的3D数据点矢量，我正在使用matplotlib在3D图中将它们绘制为散点数据.

I have two separate vectors of 3D data points that represent curves and I'm plotting these as scatter data in a 3D plot with matplotlib.

两个向量都从原点开始，并且两个向量均为单位长度.这些曲线彼此相似，但是，两条曲线之间通常存在一个旋转(出于测试目的，我实际上是在使用一条曲线并将旋转矩阵应用于第二条曲线).

Both the vectors start at the origin, and both are of unit length. The curves are similar to each other, however, there is typically a rotation between the two curves (for test purposes, I've actually being using one curve and applying a rotation matrix to it to create the second curve).

我想对齐两条曲线，以便它们以3D对齐，例如旋转曲线b，使其起点和终点与曲线a对齐.我一直在尝试通过从第一点减去最后一点来获得一个方向矢量，该方向矢量代表从每条曲线的起点到终点的直线，将它们转换为单位矢量，然后计算叉积和点积，使用此答案中概述的方法( https://math.stackexchange.com/a/476311/357495 )计算旋转矩阵.

I want to align the two curves so that they line up in 3D e.g. rotate curve b, so that its start and end points line up with curve a. I've been trying to do this by subtracting the final point from the first, to get a direction vector representing the straight line from the start to the end of each curve, converting these to unit vectors and then calculating the cross and dot products and using the methodology outlined in this answer (https://math.stackexchange.com/a/476311/357495) to calculate a rotation matrix.

但是，当我这样做时，计算出的旋转矩阵是错误的，我不确定为什么吗?

However, when I do this, the calculated rotation matrix is wrong and I'm not sure why?

我的代码在下面(我正在使用Python 2.7):

My code is below (I'm using Python 2.7):

# curve_1, curve_2 are arrays of 3D points, of the same length (both start at the origin) 

curve_vec_1 = (curve_1[0] - curve_1[-1]).reshape(3,1)
curve_vec_2 = (curve_2[index][0] - curve_2[index][-1]).reshape(3,1)
a,b = (curve_vec_1/ np.linalg.norm(curve_vec_1)).reshape(3), (curve_vec_2/ np.linalg.norm(curve_vec_2)).reshape(3)
v = np.cross(a,b)
c = np.dot(a,b)
s = np.linalg.norm(v)
I = np.identity(3)
vXStr = '{} {} {}; {} {} {}; {} {} {}'.format(0, -v[2], v[1], v[2], 0, -v[0], -v[1], v[0], 0)
k = np.matrix(vXStr)
r = I + k + np.square(k) * ((1 -c)/(s**2))

for i in xrange(item.shape[0]):
    item[i] = (np.dot(r, item[i]).reshape(3,1)).reshape(3)

在我的测试案例中，曲线2只是曲线1，并应用了以下旋转矩阵:

In my test case, curve 2 is simply curve 1 with the following rotation matrix applied:

[[1  0       0    ]
[ 0  0.5     0.866]
[ 0  -0.866  0.5  ]]

(绕x轴旋转60度).

(just a 60 degree rotation around the x axis).

由我的代码计算出的用于再次对齐两个向量的旋转矩阵为:

The rotation matrix computed by my code to align the two vectors again is:

[[ 1.         -0.32264329  0.27572962]  
 [ 0.53984249  1.         -0.35320293]
 [-0.20753816  0.64292975  1.        ]]

两条原始曲线(分别为a和b分别为蓝色和绿色)的方向矢量的图以及通过计算的旋转矩阵(红色)转换的b的结果如下.我正在尝试计算旋转矩阵，以使绿色矢量与蓝色对齐.

The plot of the direction vectors for the two original curves (a and b in blue and green respectively) and the result of b transformed with the computed rotation matrix (red) is below. I'm trying to compute the rotation matrix to align the green vector to the blue.

推荐答案

问题在这里:

r = I + k + np.square(k) * ((1 -c)/(s**2))

np.square(k)将矩阵的每个元素平方.您需要np.matmul(k,k)或k @ k，这是矩阵自身相乘的结果.

np.square(k) squares each element of the matrix. You want np.matmul(k,k) or k @ k which is the matrix multiplied by itself.

我还将实现该答案的注释中提到的附带情况(尤其是s=0)，否则在很多情况下您都会出错.

I'd also implement the side cases (especially s=0) mentioned in the comments of that answer or you will end up with errors for quite a few cases.

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