Geometric characterization of the rotation centers of a particle in a flow

Abstract

We provide a geometrical characterization of the instantaneous rotation centers $\overrightarrow{O}\left( p,t\right) $ of a particle in a flow $\mathcal{F}$ over time $t$. Specifically, we will prove that: a) at a specific instant $t$, the point $\overrightarrow{O}\left( p,t\right) $ is the center of curvature at the vertex of the parabola which best fits the path-particle line $\gamma\left( t\right) $ on its Darboux plane at $p$, and b) over time $t$, the geometrical locus of $\overrightarrow{O}\left(p,t\right) $ is the line of striction of the principal normal surface generated by $\gamma\left( t\right) $.