Timeline for Dimension of tangent space to manifold of cross section slices
Current License: CC BY-SA 3.0
7 events
| when toggle format | what | by | license | comment | |
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| Mar 9, 2016 at 16:06 | comment | added | Peter | Following the suggestions of Sebastian Goette I have posted the question on math.stackexchange ( math.stackexchange.com/questions/1690221/…) | |
| Mar 9, 2016 at 15:45 | comment | added | Peter | For the particular case of rotations, function values can be approximated as $\hat{\Phi}(x) = \Phi(x) - \nabla\Phi(x)^T\cdot[x]_{\times}\cdot\omega$ with $[.]_{\times}$ the skew symmetric cross product matrix, and $\omega$ the vector of exponential rotation coordinates. The term $[x]_{\times}\cdot\omega$ determines a tangent vector field for the curves under the group action. Therefore function values along the slightly perturbed slice are approximated in the form $\hat{\Phi} = \Phi + \sum_{i=1}^3 \omega_i\mathcal{L}_i$. I would like to know if this process can be replicated for $SE(3)$? | |
| Mar 9, 2016 at 15:45 | comment | added | Peter | I consider the slice to be a two-dimensional linear submanifold $\mathcal{S}$ of $\mathcal{M}=\mathbb{R}^3$. The action of the transformation group ($SE(3)$) maps points $x$ of $\mathcal{S}$ along curves, corresponding to flows of vector fields (diffeomorphisms), "induced" by the generators of the group. My question is if approximations to the function values $\Phi$ at the transformed points (jet) can be consistently provided with respect to arbitrary infinitesimal actions of the group, using a linear combination of basis vectors with coefficients of the Lie-algebra. | |
| Mar 8, 2016 at 19:13 | review | Close votes | |||
| Mar 9, 2016 at 8:33 | |||||
| Mar 8, 2016 at 18:55 | comment | added | Sebastian Goette | Welcome to mathoverflow. What do we know about $\Omega^\Phi$? Do you mean by "slice" the set $\Omega_s=s^{-1}\cap\mathbb R^2\times\{0\}$? And by "manifold of slices" the subset of $SE(3)$ such that $\Omega_s\ne\emptyset$? Please clarify this, and think about posting this question on math.stackexchange, as this does not really look like a research level question. | |
| Mar 8, 2016 at 18:30 | review | First posts | |||
| Mar 8, 2016 at 18:55 | |||||
| Mar 8, 2016 at 18:27 | history | asked | Peter | CC BY-SA 3.0 |