Chapter 4, Lie Groups and Lie Algebra

nicolasrosa commented 3 years ago

Hello @gaoxiang12

I'm reading chapter 4 and I got a little confused about the definitions of $\phi(t).$

First, you said that this variable is a 3D vector, $\phi(t) \in \mathbb{R}^3$ , and that its skew-symmetric matrix is: $\mathbf{\dot{R}}(t)\mathbf{R}(t)^{T} = \phi(t)\hat{}$

Second, you said "We see that $\phi$ reflects the derivative of R, so it is called the Tangent Space near the origin of SO(3)". Are you refering to $\phi(t)$ or $\phi(t_0)$ ? The second one, right?

Third, you said that "The previously mentioned $\phi$ is actually a kind of Lie algebra.". I think this sentence is a little misleading since the presented definiton of Lie Algebra was $\mathfrak{g}=(\mathbb{V}, \mathbb{F}, [,])$ , and the $\phi$ presented so far is just a 3D vector.

I believe the notation for Lie Algebra $\mathfrak{so}(3)$ should be $\mathfrak{so}(3)=(\mathbb{R}^3, \mathbb{R}?, \Phi')$ , where $\Phi' = [\phi_1, \phi_2]=(\Phi_1 \Phi_2 - \Phi_2 \Phi_1)\check{}=(\phi_1\hat{} \phi_2\hat{} - \phi_2\hat{} \phi_1\hat{})\check{}$ is the Lie bracket/Arithmetic Operator. Am I right?

In the equation 4.13, how is possible the $\mathfrak{so}(3)$ also include $\mathbb{R}^{3x3}$ values, if the arithmetic operator contains the " $\check{}$ " (Skew-Symmetric Matrix back to Vector) ?

This way, I believe the equation 4.13 should be changed to $\mathbb{V}_{\mathfrak{so}(3)} = \{ \phi \in \mathbb{R}^3\}$ .

Considering everything said, analyzing a beforementioned sentence "you will find that they are exactly the correspondence between Lie Group and Lie Algebra". In our case: Lie Group: $\mathbf{R} \in SO(3)$ Lie Algebra: $\mathfrak{so}(3)=(\phi \in \mathbb{R}^3, \mathbb{R}?, \Phi')$

So, is the $\phi$ the AngleAxis $\theta\vec{\mathbf{n}}$ presented in the topic 3.3.1 Rotation Vector? $\vec{\mathbf{a}}$ presented in section 4.2.1 is the $\vec{\mathbf{n}}$ , right?

gaoxiang12 commented 3 years ago

Hi @nicolasrosa, in my opinion, the vector \phi to skew-symmetric matrix \Phi is a one-by-one relationship, so it would be ok to write so(3) either in vector or matrix form. Given an arbitrary 3D vector \phi we can find one (and only one) corresponding \Phi, and vise versa. I agree that write it in the matrix form is better for understanding and consistency.

There is a general definition of Lie groups and Lie algebra, and SO3/so3 is just a special instance of that. In some books, people may start from the generalized definition but we start from an example. In 4.1.2 I just want to naturally introduce the definition of Lie algebra so I use a somewhat less strict way by solving the ODE say '\phi(t) keeps equal to \phi (t_0) in a short time' and then give a solution to that.

The last sentence is right.

Sorry I don't know how to type latex in this github box. Please let me know if anything is still not clear.

nicolasrosa commented 3 years ago

Hello @gaoxiang12,

After getting used to the one-to-one relationship, I understood the relaxation in the notations. Thank you.

Obs.: I was using this website https://latex.codecogs.com/eqneditor/editor.php for typing the equations and later I just paste the equations here using the following code:

<img src="https://render.githubusercontent.com/render/math?math=MATH_LATEX_HERE">

gaoxiang12 commented 3 years ago

Yes exactly. It is just a basic property of the cross product.

gaoxiang12 / slambook-en

Chapter 4, Lie Groups and Lie Algebra #22