# Representation of 2D coordinate space with orientation

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?

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.

• The SE(2) form is very convenient when chaining (or composing, compounding) rotational+translational motions in the plane. Also useful when transforming the representation of points between coordinate frames. The 3x3 matrix is quite redundant so you may choose just to keep (x, y, theta) since converting between that form and the SE(2) is computationally simple. – Peter Corke Jan 23 '19 at 23:02