The Lie group of orientation-preserving rigid motions of the plane — $\mathrm{SE}(2) = \mathbb{R}^2 \rtimes S^1$. When equipped with a sub-Riemannian metric (forward + steer, no sideways slip) it becomes the natural state space for the Dubins car, the kinematic bicycle, and Petitot’s contact model of primary visual cortex V1.