Special Euclidean group:
$\rm SE(3)=SO(3)\rtimes \mathbb{R}^3$
How to explain this expression of $\rm SE(3)$, about rigid body workspace.
2026-05-15 18:21:55.1778869315
What's the meaning of semidirect product?
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Let us represent the map $x\mapsto Ax+b$ by the pair $(A,b)$. Then composing the maps corresponding to $(A_1,b_1)$ with the one corresponding to $(A_2,b_2)$, you get the map corresponding to $(A_1A_2,b_1+A_1b_2)$. This is the way how multiplication in a semi-direct product works, whereas in a direct product, you would simple have component-wise multiplication, i.e. $(A_1,b_1)(A_2,b_2)=(A_1A_2,b_1+b_2)$.