= Translational connection for a convex body rolling on a plane
{title2=$\mathcal A=dc_{\parallel}+(\Omega\times r)_{\parallel}$}
For a smooth strictly convex <rigid body> with a smooth unique contact point, a configuration consists of its orientation $R\in SO(3)$ and horizontal position $c_{\parallel}\in\mathbb R^2$; its height is fixed by contact. Translation makes this a <principal bundle> over $SO(3)$ with group $\mathbb R^2$. If $r(R)$ points from its reference point to contact and $\Omega$ is its spatial <angular velocity>, <rolling without slipping> requires $\dot c+\Omega\times r=0$. The displayed translation-valued <principal connection> vanishes on precisely these velocities. It is translation-invariant and equals the identity on vertical translation vectors. A prescribed orientation curve consequently has a unique horizontal lift from a given initial position. No-slip alone leaves spin free and does not specify a connection over the plane with orientation as its fiber; an additional no-twist constraint is needed for that familiar spherical model.
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