Create a rotation matrix from 2 normals

Question

Good Evening,

I want to find a way how I can could create a rotation matrix with just 2 normal vectors. One is the origin vector (0,1,0) and one is the normal where I want to move the points to .

So in theory I than have to multiply my r-matrix by every point.

So this picture represents my issue:

I have also googled, I also have found something but I think this isnt what I realy want.

EDIT: This is also ment for 3D space, the picture is just for better understanding.

Answer 1

Good Morning.

Suppose you want to write the rotation which map a vector U to a vector V. Then W=U^V (cross product) is the axe of rotation and is an invariant. Let M be the associated matrix.

We have finally:

                              (V,W,V^W) = M.(U,W,U^W)

Now let write the code :

from pylab import cross,dot,inv

def rot(U,V):
    W=cross(U,V)
    A=np.array([U,W,cross(U,W)]).T
    B=np.array([V,W,cross(V,W)]).T
    return dot(B,inv(A))

An example :

In [2]: U = np.array([4, 3, 8])
Out[3]: V = np.array([1, 3, 4])

In [6]: M=rot(U,V)
In [7]: dot(M,U)
Out[7]: array([ 1., 3., 4.])

In [9]: W=cross(U,V)

In [10]: allclose(W,dot(M,W))
Out[10]: True    

Note that U and V need not to be unit vectors, just not parallel. The transformation is a rotation if the norms are equal.