Rotate from one vector to another vector in high dimensional space

algebra

Problem statement

In an n-dimensional space, given 2 vectors $x$ and $y$ with angle $\theta$, how to compute the rotation matrix that rotates $x$ to $y$ along the 2d subspace generated by $x$ and $y$ ?

Idea

In high dimensional space, the rotation happens:

• along the 2D plane $P$ generated by $x$ and $y$
• around the complemented (n-2)-dimensional subspace $Q$, which means this stays the same

Thus, the idea is

1. Project onto $P$
2. Rotate in $P$
3. Project onto Q
4. Sum the projections onto Q with the rotated vector in $P$

Here are the steps:

• In the 2D plane generated by $x$ and $y$:
  • find 2 orthonormal vectors from $x$ and $y$ to represent that space $$\begin{array}{rl}u& \leftarrow \frac{x}{||x||}\\ v& \leftarrow y-(y\cdot u)u\\ v& \leftarrow \frac{v}{||v||}\end{array}$$
  • the projection in to this space is given by $P=u{u}^T+v{v}^T$
  • map $P$ to ${\mathbb{R}}^2$ and do the rotation there
    • $u$ and $v$ are 2 bases of a plane and $[uv]$ is the ${\mathbb{R}}^2$ mapping to that plane. ($\cdot [uv]$)
    • do the rotation with angle $\theta$ in that space ($\cdot R_{\theta }$) $$R_{\theta }=\left[\begin{array}{cc}cos(\theta )& -sin(\theta )\\ sin(\theta )& cos(\theta )\end{array}\right]$$ where $cos(\theta )=\frac{x\cdot y}{||x||||y||}$
  • then map back to n-dimensional ($\cdot [uv{]}^T$)
• In complemented (n - 2)-dimensional subspace:
  • Which is the complemented (n-2)-dimensional subspace of the 2D plane mentioned above
  • The projection on to this can be computed by subtracting the projection matrix onto the space generated by $x$ and $y$ from the identity matrix $Q=I-P$
• The final rotation matrix is ^e160cf $$\begin{array}{rlr}R& =I-(u{u}^T+v{v}^T)+[uv]R_{\theta }[uv{]}^T& (1)\\ & =I+(u{v}^T-v{u}^T)sin(\theta )+(u{u}^T+v{v}^T)(cos(\theta )-1)& (2)\end{array}$$

Implementation

import numpy as np
 
# number of dimensions
n = 17
 
# 2 random vectors
x = np.random.rand(17)
y = np.random.rand(17)
 
# compute 2 orthonormal vectors to describe the plane
u = x / np.linalg.norm(x)
v = y - (y @ u) * u
v = v / np.linalg.norm(v)
 
# compute the rotation matrix (counter-clockwise)
cos_theta = x @ y / np.linalg.norm(x) / np.linalg.norm(y)
sin_theta = np.sqrt(1 -  cos_theta ** 2)
R_theta = [
	 [cos_theta, -sin_theta], 
	 [sin_theta, cos_theta]
]
 
# the final transformation
uv = np.column_stack([u, v])
u = u.reshape(-1, u.shape[-1])
v = v.reshape(-1, v.shape[-1])
R = np.eye(n) - (np.outer(u, u) + np.outer(v, v)) + uv @ R_theta @ uv.T

References

• linear algebra - Finding the rotation matrix in n-dimensions - Mathematics Stack Exchange