# Rotating one 3-vector to another

I have written an algorithm for solving the following problem: Given two 3-vectors, say: $a,b$, find rotation of $a$ so that its orientation matches $b$.

However, I am not sure if the following algorithm works in all cases:

1) Find axis and angle using cross product and dot product:

$$\mathbf{x}=\frac{a\times b}{||a\times b||}\\ \theta=\cos^{-1}(\frac{a\cdot b}{||a||\cdot ||b||})$$

3) Find rotation matrix using exponential map:

$$\mathbf{R}=e^{\mathbf{A}\theta} =\mathbf{I}+\sin(\theta)\cdot \mathbf{A}+\left(1-\cos(\theta)\right) \cdot \mathbf{A}^{2}$$

where $\mathbf{A}$ is a skew-symmetric matrix corresponding to $\mathbf{x}$:

$$\mathbf{A}=[\mathbf{x}]_{\times}=\begin{bmatrix}0 & -\mathbf{x}_{3} & \mathbf{x}_{2} \\ \mathbf{x}_{3} & 0 & -\mathbf{x}_{1} \\ -\mathbf{x}_{2} & \mathbf{x}_{1} & 0\end{bmatrix}$$

Notes:

The axis is computed using cross product as this gives vector perpendicular to both $a$ and $b$. Only direction of the axis is important, hence it is divided by its magnitude. However, I am not sure if $\mathbf{x}$ will always have the proper direction (the result can be $-\mathbf{x}$ instead of $\mathbf{x}$?).

The rotation matrix is computed using Rodrigues' rotation formula.

Finally, the vector $\mathbf{R}a$ should have same direction as $b$.

I have tested this numerically and it seems working, but I would like to be sure the formulas work for any two $a,b$.

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This is the right general approach, but the corner case $\|a\times b\| \approx 0$ must be handled.

If $\theta < \epsilon,$ $R=I$.

If $\pi-\theta < \epsilon$, you can choose for $\mathbf{x}$ any vector orthogonal to $\mathbf{a}$, for instance $\mathbf{x} = \frac{\mathbf{a} \times e_i}{\|\mathbf{a}\times e_i\|}$, where $i$ is the index of the component of $\mathbf{a}$ with least magnitude.

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I was aware of the first case (very small angle), but you have shown the angle approaching 180° is also important to handle. Thanks. –  Libor Feb 2 '13 at 21:53
Libor, user7530: Can anyone explain the origin of rotation matrix formula? I've no idea about rotation matrix and skew matrix. I'd also like to know what happens at 0 and 180 to handle the cases separately? I also have no idea about orthogonal vector. I can understand till finding the angle between vectors and the axis of rotation. –  cegprakash Jun 9 '14 at 23:17
@cegprakash Have you read the article on Wikipedia about the Rodrigues Rotation Formula? What you ask is a rather broad topic, you may want to try asking specific, new questions. –  user7530 Jun 9 '14 at 23:40

Let $F_n(x) = x - {2 \over n . n} n (n \ . \ x)$ be the transformation that reflects $x$ through the plane that is perpendicular to $n$. Composing two reflections gives a rotation: if the angle from $a$ to $b$ is $\phi$ then $F_b(F_a(x))$ rotates $x$ on the plane spanned by $a$ and $b$ by $2 \phi$.

Given the normalized vectors $\hat a=a/|a|$, $\hat b=b/|b|$, $\hat c=c/|c|$, where $c = \hat a + \hat b$, the angle from $\hat a$ to $\hat c$ is $\phi$, half the angle from $\hat a$ to $\hat b$.

In fact, $F_{\hat c}(F_{\hat a}(x))$ rotates $x$ by $\phi$ on the plane spanned by $a$ and $b$. This is valid even if $a$ and $b$ are parallel. This avoids computing a cross product, inverse cosine, sine and cosine, or division by a magnitude that can be arbitrarily close to zero.

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On second thought, this still isn't valid if the two vectors point in opposite directions. Other than expecting the user to explicitly provide a bisecting vector, reflect on (t-a) and then (t-b) where t is a coordinate vector corresponding to the smallest component of the vector a. This is detailed in this paper: cs.brown.edu/~jfh/papers/Moller-EBA-1999/paper.pdf –  Centrinia Mar 18 at 12:12

I have a simpler method comes from Erigen's "Mechanics of Continua". Here R is rotational matrix that rotate vector "a" align with vector "b"

Matlab Code:

%%%%%% Rotate vector a align with vector b%%%%%%%%%%
syms ax ay az bx by bz k real

a=[ax ay az]'

au=a./sqrt(ax^2+ay^2+az^2)

b=[bx by bz]'

bu=b./sqrt(bx^2+by^2+bz^2)

R=[bu(1)*au(1) bu(1)*au(2) bu(1)*au(3);

bu(2)*au(1) bu(2)*au(2) bu(2)*au(3)

bu(3)*au(1) bu(3)*au(2) bu(3)*au(3)]

% You can verify it by type

c=R*a

cu=c./sqrt(c(1)^2+c(2)^2+c(3)^2)

simple(bu-cu)

%the result is zero means c(a after rotation) and b are aligned with each other.

simple(sqrt(c(1)^2+c(2)^2+c(3)^2)-sqrt(c(1)^2+c(2)^2+c(3)^2))

%the result is zero means c(a after rotation) and a are of the same length

%%%%%%%End%%%%%%%%%%%%

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Thanks. Could you please point me to the theory? The matrix seems to be simply $R=ab^{T}$ where $a$ and $b$ are normalized vectors. It is not clear to me why $R^{T}=R^{-1}$ (i.e. $R^{T}R=I$) in that case. The book is unfortunately hard to obtain and too much hassle just to get this information... –  Libor Jul 26 '14 at 14:14

I'm not clear on why you have a factor of $A^2$ in your expression for $R$. In particular, wikipedia lists the matrix form for the Rodrigues formula as

$$R = I \cos \theta + A \sin \theta + (1-\cos \theta) x x^T$$

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Note that for $k$ a unit vector, $A^2 = kk^T - I$. –  user7530 Feb 2 '13 at 21:40
function [ R ] = RotAtoB( a,b )
x = [a(2)*b(3) - b(2)*a(3);a(3)*b(1) - b(3)*a(1);a(1)*b(2) - b(1)*a(2)];
x = x/norm(x);
theta = acos(a'*b/(norm(a)*norm(b)));
A = [0    -x(3)  x(2)
x(3)   0   -x(1)
-x(2)  x(1)   0  ];
R = eye(3) + sin(theta)*A + (1-cos(theta))*A^2;
end


Calculating the powers of matrix : Nth power of a square matrix and the Binet Formula for Fibonacci sequence

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What does the link have to do with this? –  Vladhagen Nov 6 '14 at 5:01