Timeline for Representation of Lie algebra $\operatorname{SE}(2)$
Current License: CC BY-SA 4.0
18 events
| when toggle format | what | by | license | comment | |
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| Aug 17, 2021 at 14:37 | comment | added | Chivul | Let us continue this discussion in chat. | |
| Aug 17, 2021 at 11:03 | comment | added | Ben McKay | The definition of $\partial_x$ is that it is the operator of partial derivative with respect to the $x$ variable, acting on the space of smooth functions with compact support as usual in calculus, and then extended uniquely to square integrable functions by density of smooth functions with compact support. The same for the other operators. You might look at Reed and Simon, or other functional analysis books, for the density argument and further discussion. | |
| Aug 17, 2021 at 9:30 | history | edited | Chivul | CC BY-SA 4.0 |
added 75 characters in body
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| Aug 17, 2021 at 9:28 | comment | added | Chivul | @BenMcKay: ah, I'm sorry. I will edit it. Thank you. | |
| Aug 17, 2021 at 9:25 | comment | added | Ben McKay | @Vichakh: The map $R$ is acting on an infinite dimensional space of functions, not on the plane, so $R$ is not $\mathcal{A}$. You might define $R$ for the benefit of mathoverflow users. | |
| Aug 17, 2021 at 9:25 | comment | added | Chivul | Thank you all. I have already answered my question myself in a way a little different from Chik67's below. | |
| Aug 17, 2021 at 9:24 | comment | added | Chivul | @BenMcKay: The action $R$ is given by $\mathcal{A}$ included in my question. | |
| Aug 17, 2021 at 9:21 | comment | added | Ben McKay | @Vichakh: I was not suggesting you clarify the definition of $SE(2)$, but the definition of $R$; on which vector space does it act, and how? This is defined in the paper, but it would be nice if your question were self-contained, so we don't have to read that paper. | |
| Aug 17, 2021 at 4:37 | answer | added | Chik67 | timeline score: 3 | |
| Aug 17, 2021 at 4:11 | comment | added | Chivul | Thank you @DeaneYang. For the purpose of the article, the author wants to use differentiation. Unfortunately, this is the first time I've seen this representation. | |
| Aug 17, 2021 at 4:10 | comment | added | Deane Yang | Here's one way to look at it. Represent each point $z \in \mathbb{C}$ as a column matrix $$Z = \begin{bmatrix} 1 \\ z \end{bmatrix}$$. Given a translation $v \in \mathbb{C}$ and rotation $e^{i\theta}$, consider the matrix $$ M = \begin{bmatrix} 1 & 0 \\ v & e^{i\theta} \end{bmatrix}$$ Then the rigid motion is given simply by $$ Z \mapsto MZ$$ | |
| Aug 17, 2021 at 3:20 | comment | added | Chivul | Thank you for your help, @LSpice <3 | |
| Aug 17, 2021 at 2:52 | comment | added | LSpice |
Since the automated sizing of $\mathrm{SE}\left(2\right)$ \left(2\right) doesn't seem necessary here, and creates weird spacing between the name of the group and the parenthesis, I edited it to $\operatorname{SE}(2)$ \operatorname{SE}(2), which looks better to me. I hope that was all right.
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| Aug 17, 2021 at 2:52 | history | edited | LSpice | CC BY-SA 4.0 |
PDF -> abs; deleted 'thanks'
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| Aug 17, 2021 at 2:34 | history | edited | Chivul | CC BY-SA 4.0 |
added 86 characters in body
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| Aug 17, 2021 at 2:28 | comment | added | Chivul | Thank you. $R_{\left(\gamma,\theta\right)}$ is an action of $SE\left(2\right)$ with the translation $\gamma$ and the rotation $\theta$. | |
| Aug 16, 2021 at 11:43 | comment | added | Ben McKay | The question needs to include the definition of this $R_{\gamma,\theta}$, as defined in the paper. | |
| Aug 16, 2021 at 10:30 | history | asked | Chivul | CC BY-SA 4.0 |