# SE(n)

The set of all transformations in $$n$$ dimensional Euclidean space which preserve the Euclidean distance between all points once applied.

The case $$n=3$$ is used to represent rigid body transformations. Namely, let $$T = \begin{bmatrix} R & p\\ \textbf{0} & 1 \end{bmatrix}$$ where $$R$$ is from SO(n) with $$n=3$$, $$p \in \mathbb{R}^3$$, $$\textbf{0}$$ is a row vector of zeros and 1 is simply the scalar. All possible matrices $$T$$ define $$SE(3)$$, and is closed under matrix multiplication. In the context of robotics they are called homogeneous transforms, and are often used to:

• Represent a configuration $$T_{sb} = \begin{bmatrix}R_{sb} & p\\ \textbf{0} & 1 \end{bmatrix}$$
• Change reference frames $$T_{sc} = T_{sb}T_{bc}$$.
• Displace frames and vectors, where points $$\mathbf{p} \in \mathbb{R}^3$$ are first converted into $\begin{bmatrix} \mathbf{p} 1 \end{bmatrix}$

## References

Lynch, Kevin M., and Frank C. Park. 2017. Modern Robotics: Mechanics, Planning, and Control. 1st ed. USA: Cambridge University Press.

Created: 2022-03-13 Sun 21:44

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