For closed smooth curves in Euclidean 3-space the most general type of tangential symmetry described above will be too general to lead to conclusions on the shape of the curve. Hence we start immediately with the notions of parallelism and self-parallelism [FR]:
The
for all
Hence the existence of a parallel mate for
we get for the total normal twist of
where there is no need to parametrize
A n is the number of
periods until the trace of the end point of the vector will provide a closed
curve.
This gives a fairly clear picture of space curves admitting this kind
of symmetries. But there are other advantages related to the visual perception
of these curves. There is an obvious one, coming from the kinematic interpretation
of the parallel transfer in the normal bundle of a curve. Consider a normal
frame as a rigid two-dimensional figure, moving without acceleration freely
along the curve, with the constraint to remain in the normal plane forever.
Then the motion will be described by the parallel transfer in the normal
bundle of the curve. Hence, considering the intersection of a tangentially
symmetric curve, having n-gon, this motion
of the frame along the central curve will preserve this figure after one
period, though a permutation of the vertices may have happened.
Interpreting the original curve as a figure located on a tube around the central curve, it carries information for the visualization of the center curve as follows: The tube may be taken as a more solid image of the center curve. The twist of the center curve may be visualized by drawing families of curves on the surface of this tube. The most appropriate curves for this will be those obtained by parallel transfer in the normal bundle of the center curve, and they will be closed curves only, if they exhibit a tangential symmetry. Clearly, then the surface of the torus bounding the tube can be foliated by curves with this kind of tangential symmetry. Furthermore, if another profile than a disk should be taken to thicken the center curve to a solid body, this only will be possible if the tangential symmetry of the original curve is respected by this profile. This can be observed in many images where closed space curves are displayed.
The more restricted form of tangential symmetry which is given by the
notion of transnormality has been studied by M.C. Irwin [I], and several
geometric results have been obtained for them in [W4]. Within the current
context the only interesting result is, that in this case the symmetry
group can be There also are a lot of considerations concerning tangential symmetries of curves in higher-dimensional spaces. In particular, the case of Minkowski 4-space is of special interest for Relativity. But this is beyond the goal of this presentation. |