A Roto-translation invariance

A.1 Rotations in 2 dimensions In 2-dimensional settings, there exists a single scalar angular position, the yaw angle θ. In order to perform the transformation, we have to express the angular positions in a format suitable for linear transformations; we do so by transforming them to rotation matrices, perform a matrix multiplication, and then transform the angular positions back to angle format. In 2 dimensions, we use eq. After the rotation, we can convert them back to angle format using the 2-argument arc-tangent function: θ = atan2(sin θ, cos θ) (14) Simplified rotations In 2 dimensions, the computations can be simplified since rotations commute. In practice, we use the 2-argument arc-tangent function atan2(y, x) to compute θ.