Elements of SO(3): Listing All Matrices

[pivoxa15]

Can someone list all the elements in each matrix of SO(3)? As opposed to the general definition of A^tA=1 with det(A)=1.

In other words give the general form of all matrices in SO(3). This is usually done with SU(2) but haven't seen it with SO(3).

 

[George Jones]

Can someone list all the elements in each matrix of SO(3)? As opposed to the general definition of A^tA=1 with det(A)=1.

In other words give the general form of all matrices in SO(3). This is usually done with SU(2) but haven't seen it with SO(3).

Look for Euler angles in a text, or using Google.

 

[pivoxa15]

Had a look. They seem to describe matrices wrt different coordinates x,y,z.

Is there a unique one or is SO(3) the set of all these matrices (i.e. wrt all axes)?

 

[mathwonk]

What is SO(3)? the 3x3 matrices which preserve length and orientation?

then they are rotations, so are determined by an axis of rotation and an angle of rotation.

so in some coordinates they look like a 1 in the upper left corner and a 2x2 rotation matrix in the bottom right block, i.e. an element of SO(2).

but the only general way to describe them is the one you gave first, i.e. every row is of length one, the rows are all orthogonal, and they give a right hand orientation when taken together.

since there are really a lot of vectors of length one and anyone of them can be the first row, it is hard to give an explicit list of all these matrices.

i.e. the first row can be any vector on the unit sphere in R^3.

 

[pivoxa15]

then they are rotations, so are determined by an axis of rotation and an angle of rotation.

You think SO(3) would be all the matrices under 'Table of matrices' in http://en.wikipedia.org/wiki/Euler_angle as they do describe every rotation of the 2 sphere.

 

[Cexy]

Every element of SO(3) looks like:

$\begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & \sin \beta \\ 0 & -\sin \beta & \cos \beta \end{bmatrix} \begin{bmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$

for suitable alpha, beta and gamma (respectively they are rotations about the z axis, x-axis and z axis again, aka the Euler angles). You can multiply them out if you want to, but I doubt you'll learn much from it.

 

[pivoxa15]

Every element of SO(3) looks like:

$\begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & \sin \beta \\ 0 & -\sin \beta & \cos \beta \end{bmatrix} \begin{bmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$

for suitable alpha, beta and gamma (respectively they are rotations about the z axis, x-axis and z axis again, aka the Euler angles). You can multiply them out if you want to, but I doubt you'll learn much from it.

There are so many other combinations of rotations on a sphere though. They are also non commutative. There's got to be more?

 

[HallsofIvy]

Every element of SO(3) looks like:

$\begin{bmatrix} \cos \gamma & \sin \gamma & 0 \\ -\sin \gamma & \cos \gamma & 0 \\ 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 \\ 0 & \cos \beta & \sin \beta \\ 0 & -\sin \beta & \cos \beta \end{bmatrix} \begin{bmatrix} \cos \alpha & \sin \alpha & 0 \\ -\sin \alpha & \cos \alpha & 0 \\ 0 & 0 & 1 \end{bmatrix}$

for suitable alpha, beta and gamma (respectively they are rotations about the z axis, x-axis and z axis again, aka the Euler angles). You can multiply them out if you want to, but I doubt you'll learn much from it.

There are so many other combinations of rotations on a sphere though. They are also non commutative. There's got to be more?

Yes, it would be more correct to say that matrices in SO(3) are "generated" by those-
all rotations can be done by products of those matrices.

 

[jostpuur]

Members of SO(3) are a secret thing that are very difficult to find anywhere. So far, I haven't found them from anywhere else than from my own notes Here they are.

$$\theta=(\theta_1,\theta_2,\theta_3) = |\theta|(n_1,n_2,n_3)\in\mathbb{R}^3$$

$$\exp\Big(\left[\begin{array}{ccc} 0 & -\theta_3 & \theta_2 \\ \theta_3 & 0 & -\theta_1 \\ -\theta_2 & \theta_1 & 0 \\ \end{array}\right]}\Big) =\sum_{k=0}^{\infty}\frac{1}{k!} \left[\begin{array}{ccc} 0 & -\theta_3 & \theta_2 \\ \theta_3 & 0 & -\theta_1 \\ -\theta_2 & \theta_1 & 0 \\ \end{array}\right]^k$$

$$=\left[\begin{array}{ccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \\ \end{array}\right] \;+\;\left[\begin{array}{ccc} 0 & -n_3 & n_2 \\ n_3 & 0 & -n_1 \\ -n_2 & n_1 & 0 \\ \end{array}\right] \sum_{k=0}^{\infty} \frac{(-1)^k}{(2k+1)!}|\theta|^{2k+1} \;+\;\left[\begin{array}{ccc} n_1^2 -1 & n_1n_2 & n_1n_3 \\ n_1n_2 & n_2^2 - 1& n_2n_3 \\ n_1n_3 & n_2n_3 & n_3^2 - 1 \\ \end{array}\right] \sum_{k=1}^{\infty}\frac{(-1)^{k+1}}{(2k)!}|\theta|^{2k}$$

$$=\left[\begin{array}{ccc} n_1^2(1-\cos|\theta|) + \cos|\theta| & n_1n_2(1-\cos|\theta|) - n_3\sin|\theta| & n_1n_3(1-\cos|\theta|) + n_2\sin|\theta| \\ n_1n_2(1-\cos|\theta|) + n_3\sin|\theta| & n_2^2(1-\cos|\theta|) + \cos|\theta| & n_2n_3(1-\cos|\theta|) - n_1\sin|\theta| \\ n_1n_3(1-\cos|\theta|) - n_2\sin|\theta| & n_2n_3(1-\cos|\theta|) + n_1\sin|\theta| & n_3^2(1-\cos|\theta|) + \cos|\theta| \\ \end{array}\right]$$

It is also possible to interpret the operator

$$e^{\theta\times}$$

as an member of SO(3). First convince yourself with some geometric arguments, that the mapping

$$x\mapsto x + ((x\cdot n)n - x)(1-\cos|\theta|) + (n\times x)\sin|\theta|$$

is the rotation of x around the angle theta, and then verify that the series

$$e^{\theta\times}x = x \;+\; \theta\times x \;+\; \frac{1}{2}\theta\times(\theta\times x) \;+\; \frac{1}{3!}\theta\times(\theta\times(\theta\times x)) \;+\; \cdots$$

converges towards this. This is the same thing as the matrix calculation, in fact.

 

[jostpuur]

Here's explanation why this

$$x\mapsto x + ((x\cdot n)n - x)(1-\cos|\theta|) + (n\times x)\sin|\theta|$$

is the rotation. If you have two dimensional space spanned by $e_1$ and $e_2$, then the rotation of a vector $x=|x|e_1$ is given by

$$x\mapsto |x|\cos(\theta) e_1 + |x|\sin(\theta) e_2.$$

Suppose then that we have $x,\theta\in\mathbb{R}^3$. First write x as

$$x=(x\cdot n)n + (x - (x\cdot n)n).$$

The first term is the projection onto the subspace spanned by theta, and the second is perpendicular to it. Now the component $(x\cdot n)n$ remains unchanged in the rotation, while the vectors

$$x-(x\cdot n)n\quad\quad(\propto e_1)$$

and

$$n\times(x-(x\cdot n)n) = n\times x\quad\quad(\propto e_2)$$

span the two dimensional subspace in which the rotation occurs. So the rotated vector is

$$(x\cdot n)n + (x-(x\cdot n)n)\cos|\theta| + (n\times x)\sin|\theta|.$$

Little algebra then brings to my previous expression for this.

 

[ObsessiveMathsFreak]

Every rotation in three dimensions can be expressed as the composition of two reflections through planes. If u1 is the normal vector to a plane, then to reflect any vector v through that plane, use the matrix $$R_u=\left(I-2*uu^T\right)$$.(Prove this works).

A reflection is the composition of two such matrices i.e.
$$R_{u_1 u_2}=R_{u_1} R_{u_2}$$

Perhaps this point of view may help.

 

[pivoxa15]

Members of SO(3) are a secret thing that are very difficult to find anywhere. So far, I haven't found them from anywhere else than from my own notes Here they are.

Are you suggesting you worked all of that out yourself?

 

[jostpuur]

Are you suggesting you worked all of that out yourself?

Depends on what you mean by "all". I had seen the definition of $\mathfrak{so}(n)$, and also the equation $\exp(\mathfrak{so}(n))=SO(n)$, so it is not a very great discovery to discover the question "what happens when I calculate $\exp(X)$ for arbitrary $X\in\mathfrak{so}(3)$?", but on the other hand, it can take surprisingly lot of time to even succeed in that...

It is probably easy for you to believe, that I had not seen the operator $e^{\theta\times}$, or the explicit formula of the rotated vector, anywhere, because you have not seen them either in any pedagogical texts. So I did, to some extent, discover them myself.

In the end, I don't feel like working this out myself. I feel like I have merely studied this out of very unreasonable pedagogical texts, that leave lot of working out for the reader.

 

[jostpuur]

My comment that it would be difficult to find the members of SO(3) in literature wasn't very correct, because of course the Euler angels are one way of representing the members of SO(3) too. I meant that it is difficult to find the explicit mapping $\mathbb{R}^3\to SO(3)$, that maps angle vector theta into the corresponding rotation matrix. IMO Euler angels are not very elegant compared to this Lie algebra approach.