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?
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Sign up to join this communityYou 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.