Oriented Curves

In the early years, everybody must once have played with the Moebius strip. It's a really amazing thing, while the mathematical theory behind it is not of an entirely formal kind. The Klein bottle is much simpler, as it is a regular 2-dimensional surface in a 4-dimentional space, quite smooth and posing no conceptual problems. A surface with an edge is already a challenge, since there is no way to define the edge of a manifold from within, in its own terms; we need to get out, to relate the inside to the some outside, so that much will depend on choice of the embracing space and the mode of embedding. What holds for one case may not be applicable to another. That is why rigorous reasoning on singular spaces do not seem generally convincing, as there is always a feeling of an ad hoc theory stretched to the experiences to produce. Which, however, does not prevent science from being quite entertaining and instructive.

Can we get rid of singularities and switch to a simpler construct? Traditionally, topologists employ gluing; but, since the Moebius strip has a single edge, it is not evident how it could be glued. With yet another standard trick, regular contraction, one can reduce the edge down to a (puncture) point; the result is still singular, though it might be considered as simpler, in a sense.

Still, there is a different option. Note that the width of the Moebius strip does affect its topology. So, let us make the strip infinitely narrow thus making it into a closed curve. In this way, we get a (one-dimensional) manifold without edges; the outer peculiarities of the strip will become the inner structure of the curve. The Moebius strip can be produced gluing the ends of a regular ribbon with a half-turn on one of the ends. Now, we have to clarify what such a warp mean for a (spatial) curve to close.

Just visually, each point of any curve can be assigned with a (three-dimensional) orientation vector orthogonal to the direction along the curve; when we go from one point to another, the orientation vector will, in general, turn in its space. Gluing the ends of the curve with the same orientation, we get a regular space curve, which could be projected onto a plane so that the inner and outer regions of the projection would be clearly distinct, and the notions of inner and outer normal could be introduced. Gluing the ends with opposite orientation will produce the analog of the Moebius strip; in the projection on the plane, the outer normal will abruptly become inner after a full turn, and this behavior does not depend on the starting point. The apparent singularity is in no way related to the smoothness of the manifold: this is an artefact of the essentially nonlinear operation of projection and vector normalization.

Formally, there is an orientation system in each point of a spatial curve, one of the dimensions corresponding to the direction along the curve, another follows the transversal displacement (for a strip, this means the local motion from one edge to another; their vector product gives the position of the (three-dimensional) normal. Such an orientation system will generate the inner space of each point, to be distinguished form the outer, attached spaces (for instance with the axes along the local velocity and the radial acceleration); in general, the outer characteristics of the curve (like curvature and torsion) are not related to inner properties (the position and structure of the orientation frame).

Since the inner space does not depend on the outer, a displacement along the curve may, in addition to transversal orientation change, may also switch the very sense of the direction along the curve. In a three-dimensional embedding, this may produce an impression of singularity, cusp, retrograde motion. Still, embedding the same curve in four dimensions can preserve the uniform smoothness, so that the apparent irregularities could be explained by the choice of projection. There are many common-life examples. For instance, the mathematically smooth motion of a point pendulum shows up as a retrograde motion at the high ends of the trajectory; we also know that the motion of a distant airplane, when projected in the observation field, may produce weird curves that some people take for the maneuvers of a UFO. This is quite common in physics, when some nonlinear effects and singularities can be interpreted as the presence of hidden dimensions, up to the conjectures that the exustence of the light barrier (the impossibility of higher-than-light speeds) or the Schwarzschild singularity might hint to the higher dimensions of the physical space, where any movements are possible while the three-dimensional projection effectively crops the observable range.

The difference of the outer and inner spaces might be compared to the distinction of physical fields from geometry. The orientation of the inner space axes does not depend on the transforms of the outer coordinates (including mirror reflection)$ this is a formal expression of the independence of a physical system from the observer.

Displacement along the curve and the rotation of the normal in the plane orthogonal to the direction of motion can be characterized by two angle parameters: phase and orientation proper. If orientation gets modified by 2πk with the phase incremented by 2π, the curve corresponds to a regular band; with orientation shifted by 2π(k + 1/2), this is an analog of the Moebius strip (with the half-integer number of twists). In the latter case, the curve gets effectively split into two layers (or sheets), consecutively spanned during each 4π turn. One could normalize this extended loop: 4π → 2π; this will make the Moebius curve into a regular curve. This obviously corresponds to the operation of slicing the Moebius strip in the longitudinal direction (which is known to produce a twisted regular strip). In general, the dynamics of orientation rotation may differ for the two halves of the extended loop; this difference does not matter in the absence of constraints, as it can be eliminated by the redefining of the “time” variable.

In the general case, a phase shift by 2π will result in the orientation change by 2πq, where q is an arbitrary real number. For rational q, the curve will split into a finite number of layers; irrational factors will produce a kind of toroid. When the dynamics of orientation change depends on the phase, the spectrum (the density of curve turns around the layer surface) may exhibit quite nontrivial variations.

The rotation of orientation along a closed curve could be illustrated by a physical model with an orthogonal to the motion direction dipole in every point of the curve, so that the polarization of the next point depends on the previous. A finite number of layers then would correspond to a standing wave, while the non-periodic orientation changes describe a wave running along the curve. Beside some exotic interpretations (like treating the magnetic monopole as a Moebius strip), there are practically useful applications as well.

Oriented curves are hierarchical structures where each point unfolds into an inner space. The outer motion along the curve is naturally associated with a number of outer layers, with the curve (as a configuration space) becoming a stratified manifold. When it comes to the projection of such a construct onto a plane, it is not enough to merely establish correspondence between the point of the curve and the point of the plane: the inner and outer stratification too are to be reflected in the projection. This is possible in certain cases; for instance, the plane could be split in a number of sectors, each hosting a separate branch of the curve. More often, some features of the whole will be lost in projection. Similarly, higher-dimensional manifolds could be projected onto simpler spaces with a loss of important detail.

The levels of hierarchy essentially depend on each other. Consider a trivial mathematical illustration: let an oriented curve be represented by a narrow plane ellipse whose axes rotate as we move from one of its point to another; if, after a full span, the ellipse will take the original position, we deal with a regular curve.

An oriented curve is only apparently one-dimensional. In principle, all kinds of hierarchical structures could be unfolded in each point, while the transition from one point to another would mean folding one hierarchical structure and unfolding a different one (hierarchical conversion). In particular, the dimensionality of the inner space may change along the curve, with the same geometry of the embedding.

Jan 1985