计算旋转矩阵以对齐 3D 空间中的两个向量? [英] Calculate rotation matrix to align two vectors in 3D space?

问题描述

我有两个独立的 3D 数据点向量代表曲线，我正在使用 matplotlib 在 3D 图中将它们绘制为散点数据.

两个向量都从原点开始，并且都是单位长度.这些曲线彼此相似，但是，两条曲线之间通常存在旋转(出于测试目的，我实际上使用了一条曲线并对其应用了旋转矩阵以创建第二条曲线).

我想对齐两条曲线，使它们在 3D 中对齐，例如旋转曲线 b，使其起点和终点与曲线 a 对齐.我一直试图通过从第一个点中减去最后一个点来做到这一点，以获得表示从每条曲线的起点到终点的直线的方向向量，将它们转换为单位向量，然后计算交叉和点积和使用此答案中概述的方法(

解决方案

问题出在这里:

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

np.square(k) 对矩阵的每个元素进行平方.你想要 np.matmul(k,k) 或 k @ k ，它是自乘矩阵.

我还会实施该答案的评论中提到的附带情况(尤其是 s=0)，否则您最终会在很多情况下出错.

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.

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).

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.

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

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)

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  ]]

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

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.        ]]

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.

解决方案

Problem is here:

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

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.

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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