Parametrizing Continuous Paths on a Circle

I've been meaning to write this for some time now, partially as a follow up to my last post. I mentioned that winding numbers can be thought of as the homotopy classes for loops around a point. I also said that the fact winding numbers exist at all is not entirely obvious. Why? Well, how might we define what a winding number is? We might parametrize a continuous path @@\gamma: [0,1] \to \mathbb{R}^2\setminus\{\mathbf{0}\}@@ in polar coordinates, then define the winding number of that path to be @@\frac{1}{2\pi} \left(\theta(1) - \theta(0)\right)@@. That's the same approach the Wikipedia article uses, but it also points out that, in order for this to work, @@\theta: [0,1] \to \mathbb{R}@@ must be continuous. To me, it's clear we should be able to construct a continuous @@\theta@@ for every path, but not entirely obvious how exactly.

We'll start by "collapsing" the arbitrary path we're given down onto the unit circle. Essentially, we ignore the path's radius @@r@@ since it doesn't matter in calculating its winding number. Thus, we want to find a way to continuously (in @@\mathbb{R}@@) parametrize continuous paths on a circle (in @@\mathbb{R}/2\pi\mathbb{Z}@@).

Perhaps the most straightforward way to do this would be as follows. Take a continuous path on the unit circle @@\gamma@@ and consider @@f_0@@ the "natural" map from the circle to @@[-\pi, \pi]@@, shown below. Obviously, just taking @@f_0(\gamma)@@ would break continuity as soon as @@\gamma@@ "wraps around" the far side of the circle. There would be a discontinuous jump from @@\pi@@ to @@-\pi@@. To "patch" this, we can switch to using a different map at the point of discontinuity. For instance, at a point where @@\gamma@@ wraps around the circle and @@f_0@@ switches from positive to negative, we switch to using @@f_1 = f_0 + 2\pi@@. Similarly, if @@\gamma@@ wraps off the circle going down, switch to @@f_{-1} = f_0 - 2\pi@@. In general, if you are using @@f_n = f_0 + 2n\pi@@ as your current map and @@\gamma@@ "wraps around," switch to using @@f_{n+1}@@ if it wraps going up or @@f_{n-1}@@ if it wraps going down.

That approach kind of works. One drawback is that @@f_0@@ maps the far side of the circle to both @@\pi@@ and @@-\pi@@. We'd have to arbitrarily choose one then carefully define what exactly it means to wrap going up or down. We wouldn't be able to use the same definition in both cases. Perhaps more pressing is that this approach requires us to step through all the map transitions. It's very possible for there to be infinitely many such transitions. For example, consider a path that behaves somewhat like the function @@x\sin(x^{-1})@@, graphed below. A calculation that relies on stepping through all of the transitions would likely be ill-defined.

After some thinking, you might realize that we don't necessarily need to map one point on the circle to just one point in @@\mathbb{R}@@. We could instead "double cover" some parts of the unit circle, as shown below. I'll say that the ends of the cover are open – that we don't include @@\pi+D@@ and @@-(\pi+D)@@ – but I'm sure the following arguments would be very similar if I'd made it closed. This double cover is nice, but we still need create an @@f_0@@ to unambiguously map points on @@\gamma@@ to real numbers, and we resolve as follows. In order for the path to enter the "ambiguous zone," it must've at some prior time had @@f_0 = \pm(\pi-D)@@. If it entered through the positive side, use the positive "branch" of the map, otherwise use the negative branch.

That works in the absence of map transitions, and with careful definitions, we can get it to work in their presence as well. We define a map transition upward as a point where @@\gamma@@ exits the ambiguous zone with @@f_0@@ arbitrarily close to @@\pi+D@@, and a map transition downward when it exits arbitrarily close to @@-(\pi+D)@@. Clearly, these are the only two ways for a continuous path to exit the ambiguous zone while introducing a discontinuity into @@f_0@@. Moreover, on exit we attain @@f_0 = \pm(\pi-D)@@, which allows the definition from the last paragraph to work through transitions.

Again, we keep a counter in @@n@@ and use @@f_n = f_0 + 2n\pi@@, incrementing @@n@@ on a transition upward and decrementing on a transition downward. With the new definition for map transitions, we are guaranteed to have only finitely many of them. Why? Suppose we had infinitely many transitions, and thus a point about which map transitions are "dense." Consider a point just after a transition, where @@f_0(\gamma(t)) = \pm(\pi-D)@@. In order for another transition to happen, we must have at some time @@s@@ in the future @@f_0(\gamma(s)) = \mp(\pi-D)@@, exactly @@2D@@ away on the circle. However, this can never happen infinitely often since @@\gamma@@ is continuous.

Thus, we can, in a well-defined manner, create a continuous function in @@\mathbb{R}@@ to parametrize a continuous path on a circle. At least, I think we can. I used IVT and other properties of continuous functions without explicitly mentioning them in the above argument. Indeed, I may have glossed over too much and, for instance, accidentally ignored a condition or an edge case. Nonetheless, I'm fairly sure this approach can be made into a more formal proof.

Here more than ever, I feel that I'm missing a lot of machinery. For instance, my idea of covering the unit circle seems at least superficially similar to the idea of an atlas, though I can't say for sure because I don't yet know much about it. Again, I look forward to learning higher math, though I don't know if I'll ever be able to.