Timeline for 3D similarities and quaternions?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Sep 20, 2020 at 6:53 | history | edited | YCor |
edited tags
|
|
| Jun 8, 2020 at 15:04 | vote | accept | Goulifet | ||
| Jun 8, 2020 at 9:45 | answer | added | wlad | timeline score: 3 | |
| Jun 8, 2020 at 9:21 | comment | added | wlad | @BertramArnold Shouldn't that be $(a,b,x)\circ v = avb + x$ | |
| Jun 8, 2020 at 9:09 | comment | added | wlad | In terms of matrices, the group of similarity transformations is a subgroup of the group of affine transformations, the latter of which has a matrix representation | |
| Jun 8, 2020 at 9:05 | comment | added | Bertram Arnold | In dimension $4$, any orthogonal transformation is of the form $v\mapsto a\cdot v\cdot b$ with $a,b\in \mathbb S^3\subset H^\times$. This defines the double covering $SU(2)\times SU(2)\to SO(4)$. This generalizes immediately to similarities, the group of which is $\mathbb (H^\times \mathbb H^\times / \big((\lambda,1)\sim (1,\lambda)\big)\ltimes \mathbb H$ acting by $(a,b,x)\circ v = avb + v$. To get to $SO(3)$, restrict to those transformations preserving $\mathbb R^\perp\subset \mathbb H$, which is equivalent to $x\in \mathbb R^\perp,ab\in\mathbb R$, i.e. $b$ is proportional to $\overline a$. | |
| Jun 8, 2020 at 9:04 | comment | added | wlad | What about $v \mapsto qvq^* + u$ where $v$ is a vector quaternion, $u$ is a vector quaternion, and $q$ is a quaternion | |
| Jun 8, 2020 at 8:16 | history | edited | YCor | CC BY-SA 4.0 |
formatting
|
| Jun 8, 2020 at 2:22 | comment | added | Theo Johnson-Freyd | This question is, more or less, what led Hamilton to discover the quaternions in the first place. | |
| Jun 8, 2020 at 1:46 | comment | added | Steven Stadnicki | One problem is that the rotation action of the quaternions isn't given by $q\mapsto(f_q(z): z\mapsto qz)$ as it is with complex numbers representing 2d rotations, but rather $q\mapsto(f_q(z): z\mapsto q^{-1}zq)$ (possibly with the specific conjugation inverted depending on your conventions). | |
| Jun 8, 2020 at 1:27 | history | asked | Goulifet | CC BY-SA 4.0 |