Curves in $SE(3)$ and Their Behaviours

Abstract:

      Consider $SE(3)$ as a submanifold of the Euclidean space $E^{12}$. At first, the intrinsic and extrinsic geometries of $SE(3)$ are studied. Then the Frenet typed equations for curves in $SE(3)$ are derived. The relationship between the $k$-th order covariant curvatures of a curve in $SE(3)$ and the parametric equation of its pre-image in $se(3)$ is established. Finally, some kinds of curves in $SE(3)$ with vanish $5$-th order covariant curvature are constructed. The behaviours of these curves are exhibited.