Curves in $SE(3)$ and their behaviours
Received:December 18, 2019  Revised:December 18, 2019
Key Word: the $\pi$-operator   the pre-image   the $i$-th order Darboux derivative   the $k$-th order covariant curvature.
Fund ProjectL:The National Natural Science Foundation of China (No. 61473059)
 Author Name Affiliation E-mail Zhong Hua HOU School of Mathematical Sciences, Dalian University of Technology zhhou@dlut.edu.cn Wei SHI School of Mathematical Sciences, Dalian University of Technology Wei WU School of Mathematical Sciences, Dalian University of Technology
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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.