Special Euclidean Group (SE3)

The upper left corner () is the rotation matrix, the right side is the translation vector (), the lower-left corner is 0 vector (), and the lower right corner is 1 (). This set of transform matrix is known as the Special Euclidean Group.

\mathrm{SE}(3) = \left\{ \mathbf{T} = \left[ {\begin{array}{*{20}{c}} \mathbf{R} & \mathbf{t} \\ {{\mathbf{0}^T}} & 1 \end{array}} \right] \in \mathbb{R}^{4 \times 4} | \mathbf{R} \in \mathrm{SO}(3), \mathbf{t} \in \mathbb{R}^3\right\} \end{equation}$$ This is used for [[notes/Spatial Algebra|Euclidean Transformation]]s, and enables us to compose transforms together. SO(n) is a subgroup of SE(n). ### Inverse of Euclidean Transforms There's something magical here: $$(T_{a,b})^{-1} =T_{b,a}$$ Why does this happen? I am still confused, it seems like concidence. - AHHH, yes I actually learned this, see [[notes/Linear Transformation#inverse-linear-transformations|Inverse Linear Transformation]], where we saw that $[L^{-1}] = [L]^{-1}$ as long as the matrix $[L]$ is invertible. [[notes/Spatial Algebra|Euclidean Transform]]s are linear transformations - This doesn't work with [[notes/Affine Transformation|Affine Transform]] because they are not [[notes/Linear Transformation|Linear Transformation]]s! I asked [[notes/ChatGPT|ChatGPT]]: Here's why the inverse of a transformation matrix in SE(3) effectively "inverts" the expression in the frame: 1. **Translations**: When you take the inverse of the matrix, the translation component changes sign, effectively moving the origin back to its original position. 2. **Rotations**: The upper left 3x3 submatrix is orthogonal. The inverse of an orthogonal matrix is simply its transpose, so taking the inverse of the rotation submatrix effectively reverses the rotation. By combining the effects of translation reversal and rotation reversal, the inverse of the SE(3) matrix effectively undoes the original transformation, bringing you back to the original frame of reference. ### Related - [[notes/Special Orthogonal Group|Special Orthogonal Group]]