Parametrized curves: whence ``regularity''?

The first oddity one runs across in the study of the differential geometry of curves is the requirement that such a curve be regular. We shall describe the basic idea behind this little word and its significance.

Reparametrization and Arclength

Given all of the difficulties surrounding the parametrization of a curve, it would be good to have some geometrically meaningful way to do it. We want to have some way to choose a parametrization that actually has something to do with the shape of the curve. Fortunately, there is the parametrization by arclength. In this case, we use the distance along the curve from some fixed point as the parameter.

Differential Geometry of Curves: Why Parametrizations?

In differential geometry, we make a definite choice to study parametrized curves. The reasons for this are simple, but important. This choice also has some consequences for what kinds of things we will be able to handle. Here I want to discuss this basic idea.

Spherical Curves

When does a curve in space lie on a sphere? That is, is there a way to tell if some given curve actually lies on a sphere, without being given that information in advance?

My students asked this nice question, whose answer shows off the basic techniques of classical differential geometry very clearly. So, I figure it is helpful to share the answer, and also the way we found it in class.

The Tangent Spherical Image and another interpretation of Curvature

There is another slick little interpretation of the curvature. This one is both easier and harder to see: easier in that the argument for it is much shorter and doesn't use quite such an intricate argument; harder in that it lies so close to the surface it is just harder to notice.

As usual, we start with some $C^2$ regular curve $\gamma$ in $\mathbb{R}^3$, and we assume that it is parametrized by arclength with arclength parameter $s$. At this point, it can seem a trivial observation, but the important thing is that the tangent vector $T(s)$ has length which is always equal to $1$.

Now we step back and consider the assignment $\alpha: s \to T(s)$ as a function on its own terms, and let $\gamma$ fade into the background for a minute. For each input $s$, $T(s)$ is a vector in $\mathbb{R}^3$. Usually, one visualizes each of these vectors as attached to the corresponding points $\gamma(s)$, but we will just take them all based at the origin of $\mathbb{R}^3$. Because $||T(s)|| = 1$, the curve $\alpha$ lies on the unit sphere!

This curve $\alpha$ is called the (tangent) spherical image of $\gamma$.

An Interpretation of Curvature: Osculating Plane and Circle

To be very clear, this post follows Spivak Vol 2 pretty closely.
Suppose that we have some curve in space, and we have chosen a point $P$ on it. We are concerned with the following three questions:

1. There is no guarantee that the curve lies in a plane, but if there were a plane through $P$ which came closest to containing the curve, how would you find it?
2. Can we find a circle which serves as a good approximation to the curve near $P$?
3. Is there a simple and reasonable interpretation of the curvature of the curve at $P$?

Of course, the answers are all "yes." The plane is called the osculating plane, the circle is the osculating circle, and the curvature gets an interpretation from the size of the osculating circle. The goal of this post is to show that these objects exist and have real geometric significance.

Sage Cell Test

At some point I want to include some more pictures and examples here. This is supposed to be about geometry, so there really should be some pictures. Let's test out the handy Sage Cell Server to see if it can help us.

Did that work?