Curves in $SE(3)$ and Their Behaviours
Received:December 18, 2019  Revised:August 01, 2020
Key Words: $\pi$-operator   pre-image   $i$-th order Darboux derivative   $k$-th order covariant curvature
Fund Project:Supported by the National Natural Science Foundation of China (Grant No.61473059).
 Author Name Affiliation Zhonghua HOU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Wei SHI School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China Wei WU School of Mathematical Sciences, Dalian University of Technology, Liaoning 116024, P. R. China
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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.
Citation:
DOI:10.3770/j.issn:2095-2651.2021.02.007