0
votes
0answers
12 views

Rotations of form $R_z(a)R_y(b)R_x(c)R_z(-a)R_y(-b)R_x(-c)$

Any proper rotation (in three dimensions) can be expressed using the Tait-Bryan (sometimes called improper Euler) angles in the form $$ R(\phi,\theta,\psi)=R_z(\phi)R_y(\theta)R_x(\psi) $$ where ...
1
vote
0answers
36 views

Lie bracket as defining element for transformations

Why is it precisely the Lie bracket that encodes the information about a given transformation? A Lie algebra is defined by its commutator. Using the exponential map one ends up with a given ...
1
vote
1answer
113 views

Jacobian of exponential mapping in SO3/SE3

Following this post Jacobian matrix of the Rodrigues' formula (exponential map) What if I really need the Jacobian of the exponential mapping function in $\omega \neq 0$? Basically, I want to ...
2
votes
0answers
59 views

Gradient of a real-valued function on SO(3)

I have struggling with a problem of evaluating the gradient of a cost function on the Lie group of rotations: SO(3). The cost is the following: \begin{equation} ...
3
votes
0answers
38 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
68 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
60 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
209 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
178 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
35 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
97 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
32 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 ...
4
votes
1answer
135 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
157 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
579 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
303 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
221 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
48 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 ...
10
votes
6answers
3k 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. ...