3
votes
0answers
21 views

Nontrivial relations in rotation groups

Consider the subgroup $H$ of $SO(3)$ generated by rotations of order $5$ (i.e., rotations by $\frac{2\pi}5$) about the $x$ and $y$ axes. This group certainly isn't finite or discrete (as it's not ...
3
votes
0answers
65 views

Invariants under a transformation

Consider a $j=1,\,SU(2)$ representation (or fundamental $SO(3)$ representation). Suppose that $a_1, b_i, c_i$ with $i=1,2,3$ are vectors transforming under this representation i.e ...
2
votes
2answers
54 views

Dimension of $SO_n(\mathbb{R})$

Is there a simple proof that the dimension of $SO_n(\mathbb{R})$, a.k.a the group of rotations in $n$-dimensional space is $(n-1)n/2$? It would be great to see some proofs based only on the ...
6
votes
3answers
188 views

Prove where exp: Skew($3\times 3$) $\rightarrow SO(3)$ is local homeomorphism

The matrix exponential on skew-symmetric $3\times3$ matrices onto $SO(3)$ is not local homeomorphism everywhere. I have been instructed that one problem is with the spheres of radius $2n\pi$ ...
5
votes
1answer
107 views

Whether matrix exponential from skew-symmetric 3x3 matrices to SO(3) is local homeomorphism?

$SO(3)$ denotes 3x3 rotation matrices. This is Lie group, with corresponding Lie algebra being $\mathrm{Skew}_3$, the space of 3x3 skew-symmetric matrices. The link between them is the matrix ...
1
vote
1answer
26 views

Quaternion techniques for a geometric description of the composition of two rotations

Let $q \in S^3$. Therefore $q$ can be represented as $q=\cos(\alpha/2) + \sin(\alpha/2)u$ for some $\alpha \in \mathbb{R}$ and some $u \in S^3$ with it's real part zero. Recall that the quaternions ...
4
votes
1answer
91 views

A Neat Rotation Matrix Identity

Let $\mathbf{R}_i$ be $N$ rotation matrices that represent a rotation around axes $\mathbf{\omega}_i$ by an angle $|\mathbf{\omega}_i|$. Now say we know that the product of these matrices is unity, ...
1
vote
0answers
26 views

is there a polynomial-form minimal representation for SO(3)?

Is there a minimal local representation for $SO(3)$ such that if $(x_1,x_2,x_3)$ is the representation for some $R\in SO(3)$ then I can write the entries of the 3x3 rotation matrix for $R$ as a ...
3
votes
1answer
113 views

Relationship between two maps from $SU(2)$ to $SO(3)$

I have two maps from $SU(2) \to SO(3)$. For the first map, think of $SU(2)$ as the group of unit quaternions. Under this identification, we can give a map $f: SU(2) \to SO(3)$ given by $SU(2)$ ...
3
votes
1answer
141 views

Quaternions as group of rotation and scaling

It is very well known that unit quaternions are well suited to represent rotations in 3D. In particular, the group of unit quaternions forms a double cover of the special orthogonal group $SO(3)$. ...
3
votes
2answers
404 views

Jacobian of Reprojection Error

I am writing a program to find the transformation between two sets of 3D points extracted from a moving stereo camera. I am using an 'out of the box' Levenberg-Marquardt implementation to find this ...
1
vote
2answers
233 views

Minimization on the Lie Group SO(3)

Refering to a question previously asked Jacobian matrix of the Rodrigues' formula (exponential map). It was suggested in one of the answer to only calculate simpler Jacobian ...
0
votes
1answer
199 views

What transforms under SU(2) as a matrix under SO(3)?

A vector $\boldsymbol{r}$ in $\mathbb{R}^3$ transforms under rotation $\boldsymbol{A}$ to $\boldsymbol{r}'=\boldsymbol{Ar}$. It is equivalent to an SU(2) "rotation" as $$\left( ...
0
votes
1answer
46 views

$O(3)$ after identifying certain rotations

suppose i have $O(3)$ as a group and then proceed to identify rotations on the same axis. That is, assuming an element in the simple component is written as $$ e^{s_i I_i } $$ where $I_i$ are ...
9
votes
6answers
2k views

Jacobian matrix of the Rodrigues' formula (exponential map)

I am working an algorithm which is supposed to align a pair of images. The motion model, which describes the pose $p$ of an image (with respect to the second) in 3D space, is purely rotational. ...