Representation of 2D coordinate space with orientation

Question

I wanted to know what is the mathematical representation of 2D coordinate space with orientation in the same plane.

2D coordinate space is represented as $R^{2}$, how do I include orientation in the same? or is it included in it?

Answer 1

You need to resort to the Special Euclidean groups.

In particular, in your planar case, the group is $SE\left(2\right)$ and thus the representation is the following:

$ T=\left(\begin{matrix} R & v \\ \mathbf{0} & 1\end{matrix}\right), $

where $R \in \mathbf{R}^{2\times2}$ is the matrix accounting for the rotation, whereas $v \in \mathbf{R}^2$ is the translation part.

Therefore, if you want to roto-translate a point $p \in \mathbf{R}^2$, just do this:

$ p'=T \cdot \left(\begin{matrix} p \\ 1\end{matrix}\right) $

Finally, to invert this kind of transformations, have a look at this Q&A.