Autorotation

Abstract

A continuous autorotation vector field along a framed space curve is defined, which describes the rotational progression of the frame. We obtain an exact integral for the length of the autorotation vector. This invokes the infinitesimal rotation vector of the frame progression and the unit vector field for the corresponding autorotation vector field. For closed curves we define an autorotation number whose integer value depends on the starting point of the curve. Upon curve deformations, the autorotation number is either constant, or can make a jump of (multiples of) plus-minus two, which corresponds to a change in rotation of multiples of 4π. The autorotation number is therefore not topologically conserved under all transformations. We discuss this within the context of generalised inflection points and of frame revisit points. The results may be applicable to physical systems such as polymers, proteins, and DNA. Finally, turbulence is discussed in the light of autorotation, as is the Philippine wine dance, the Dirac belt trick, and the 4π cycle of the flying snake.

Importance and Effectiveness of Representing the Shapes of Cosserat Rods and Framed Curves as Paths in the Special Euclidean Algebra

“…Nevertheless, we have shown that the geometry of the base curve is not a necessary starting point to define the shape of a rod, since the strain fields u i and v i , i = 1, 2, 3, are the only degrees of freedom needed to characterize that shape. In summary, it seems more appropriate to explicitly use rod theory when dealing with mechanical models (as exemplified by the works of Domokos [62] and Domokos & Healey [63], and many others cited above) and framed curves when focusing on geometry (as exemplified by the works of Starostin & van der Heijden [64], Bohr & Markvorsen [65], da Silva [66], and Honda & Takahashi [67]).…”