The Geometry of the Frenet-Serret Equations for Space Curves

The geometry of a parameterized curve in space can be understood by following the motion of its Frenet framing. We have previously discussed the way the framing $\{T,N,B\}$ is defined at teach point of a unit-speed curve $s\mapsto \alpha (s)$, and derived the basic equations:

$$\begin{array}{ccccc}{T}^{\prime }& =& & \kappa N& \\ N^{\prime }& =& -\kappa T& & +\tau B\\ B^{\prime }& =& & -\tau N& \end{array}$$

These are the Frenet-Serret Equations. We would like to explore now what these equations mean and how they help us interpret the meaning of curvature, $\kappa$, and torsion, $\tau$.

This post has no picture in it. This is intentional. To get the most out of this description, you should follow along and embody the motion for yourself.

To get "oriented" (That's a joke, son), use your right hand to make a little moving frame for yourself: first make a fist. Then stick your index finger straight out and your thumb straight up. These two fingers will make a right angle. Then uncurl your middle finger so that it points out from your palm at a 90 degree angle. The three digits sticking out will be the vectors of your frame: the index finger is $T$; the middle finger is $N$; the thumb is $B$.

The important thing to remember is that your hand can move, turn, and spin, but it cannot change shape as it does so. The framing you are trying to represent is a rigid one: the fingers represent unit vectors that stay at right angles to one another.

First, note that the tangent vector $T$ tells us about the direction of travel of the curve. Way back from multivariable calculus, you know that this is the direction of the tangent line to the curve, which is the line that the curve is closest to at the given point. Tracking the motion of $T$ tells us how to how the curve changes direction, that is, how it curves. This is the job of the curvature $\kappa$ and the normal $N$. $$T^{\prime }=\kappa N$$ The normal $N$ tells us the direction in which $T$ is changing, and $\kappa$ tells us how fast. For a stronger geometric interpretation, consider that $N$ points toward the center of the osculating circle, and $\kappa$ is the reciprocal of the radius of that circle. In a way, the curvature measures the how badly your curve fails to be a straight line.

If we think in terms of the physical model (put your fingers back in place!), as your hand travels along the curve, you should turn it so that your index finger turns toward your middle finger. The curvature tells you how fast to do that.

Now, the tangent vector $T$ and the normal vector $N$ span out the osculating plane. This is the plane which the curve is closest to lying in. (If your curve is a planar curve, $T$ and $N$ will be a basis for the plane containing the curve.) The next thing to do is to measure how badly the curve fails to lie in the osculating plane.

Now as we move along the curve, the osculating plane will change and move around, too. It is a bit tricky to measure this motion directly, but we can use its normal vector as a proxy. The binormal vector $B=T\times N$ is a normal vector to the osculating plane, and its rate of change $B^{\prime }$ measures how the osculating plane moves around. Note that the equation $$B^{\prime }=-\tau N$$ tells us that B moves in the direction $-N$ with speed $\tau$.

In terms of our physical model (fingers!), as your hand travels along the curve you should also rotate so your thumb turns away from your middle finger with speed $\tau$. If you do this correctly, positive values of $\tau$ will mean that you follow the curve upward in space (toward your head) as you curl around to the left, and negative values of $\tau$ will mean that you follow the curve downward in space (toward your feet) as you curl around to the left.

Okay, that last paragraph is only really true as long as your middle finger stays to the left side of your index finger. If you turn your hand over, things get funny. Once you are comfortable, you should be able to work out what happens if you meet a curve that is bending to the right when you meet it. (Hint: start with your thumb pointing down.)

Now, if you stare at it long enough, you can see why the last of the Frenet-Serret equations is forced on us. $$N^{\prime }=-\kappa T+\tau B$$ If your hand is moving in the way indicated above (turning index finger toward middle finger while rotating thumb around axis of index finger away from middle finger), then your middle finger is doing a more complicated motion. But that complicated motion is just the combination of two motions forced on it: one, turning away from your index finger with rate $\kappa$ and rotating toward your thumb with rate $\tau$. This is the only way to keep the frame a rigid object while doing the motions described above.

Study Plan: Curve Shortening Flow

For a while now I have wanted to dig into the details of Gage-Grayson-Hamilton work on the "curve shortening flow." I have several reasons for this: 

1. If I am to call myself a differential geometer, I will really have to learn some more about geometric analysis fundamentals. CSF seems like the most accessible version of a large class of geometric evolution equations which are very important. (For example, Perelman's work on the Poincaré conjecture is essentially a control over the singularities of the Ricci flow introduced by Hamilton.)
2. There is this big project I never finished that would likely go easier if I knew some more about these kinds of things. Maybe this would be a good way to get back into it.
3. Discretized versions of the problem seem to be a place for getting students involved in mathematics. I have tried lots of different angles for getting students involved in research, and I have learned the hard way that I need a class of problems that satisfy these requirements:
1. easy to get started (low floor)
2. possibility of exciting new developments (high ceiling)
3. I have to know something about it.
4. I am teaching differential geometry this term, and I like to take the opportunity when teaching a course to deepen my knowledge of the material in some significant way.
5. I am giving a talk at an undergraduate conference pretty soon, and I want to talk about these ideas. (I gave them an abstract that would push me to learn more.)
6. The stuff is just cool.

So, I have started reading the primary literature already. I will write about what I am learning in small pieces, so this is an opportunity for me to organize the ideas into a sequence of blog posts. 

The list below is limited a bit by my current state of knowledge. I know more details about the beginning of the list (items 1-7), and less about the end which will likely expand quite a bit. 

Here is what I think will happen:

1. Fundamentals on smooth plane curves: adaptations of the Frenet-Serret apparatus to 2d; some basic formulae for computation
2. convex plane curves, in particular the idea of a support function; new expressions for length and area
3. some isoperimetric inequalities (following a paper by Harley Flanders). Possibly several posts.
4. the idea of curve shortening flow
5. Gage's isoperimetric inequality
6. The evolution of the isoperimetric profile under CSF.
7. The Hausdorff metric on closed sets
8. CSF shrinking convex curves to "round points." Likely several posts. 
9. About parabolic PDEs and their solutions.
10. Grayson's whisker lemma and the case of embedded non-convex curves.
11. Curves on surfaces.

This project is mainly for my own benefit. I want to get this stuff into my head, and writing it really helps me. If you find it interesting, too, come along.

Links for Sage Animations

I figure it is about time I learn to make some animations with Sage. I spent about five minutes poking around, and I found these web links which look useful. I want to come back to this later.

1. http://www.sagemath.org/doc/reference/sage/plot/animate.html
2. http://ask.sagemath.org/question/1437/animated-sphere
3. http://ask.sagemath.org/question/1422/3d-animation-with-tachyon
4. http://ask.sagemath.org/question/99/animate-3d-plots
5. http://www.liafa.univ-paris-diderot.fr/~labbe/blogue/2012/01/faire-une-animation-en-3d-avec-sage/

Evolutes

An Evolute of a Circular Helix

In a previous post, we studied the involute of a curve. Now we will reverse the direction of the problem: Given a regular curve $\alpha$, another curve $\beta$ is called an evolute of $\alpha$ when $\alpha$ is an involute of $\beta$.

This means that

1. $\beta (s)-\alpha (s)$ lies along the tangent to $\beta$ at $\beta (s)$.
2. ${\beta }^{\prime }(s)$ is orthogonal to ${\alpha }^{\prime }(s)$.

Our goal is to understand what we can say about evolutes.

Involutes

Let $\alpha ,\beta :(a,b)\to {\mathbb{R}}^3$ be regular curves. We say that $\beta$ is an involute of $\alpha$ when

1. $\beta (t)$ lies on the tangent line to $\alpha$ at $\alpha (t)$, and 
2. the tangents to $\alpha$ and $\beta$ at $\alpha (t)$ and $\beta (t)$ are orthogonal.

In this post, we will explore the basics of what one can say about an involute of a given curve.

Generalized Helices: Curves of Constant Slope

A regular curve in space is said to have constant slope when its tangents make a constant angle with a given fixed line. Such a curve is also called a generalized helix, or even just a helix amongst the cognoscenti. The line is called the axis of the helix.


An analytic way of stating the condition is as follows: Let $A$ be a unit vector parallel to the axis $\ell$. Denote the parametrized curve by $X$ and its arclength parameter by $s$. We use standard notation for the Frenet-Serret apparatus. Then the definition means that
$$\begin{array}{}\text{(1)}& ⟨T,A⟩=\cos \alpha .\end{array}$$
Here $\alpha$ is the angle mentioned because $T$ and $A$ are both unit vectors.

Our goal is to describe the consequences of equation (1) for the shape of the curve.

Non-Unit Speed Curves: Curvature and Torsion

We have seen that any regular curve can be reparametrized by arclength. Of course, the process involves computing an integral and then describing the inverse of some increasing function. A little bit of Calculus tells us that it is theoretically possible, but it doesn't make it computationally feasible.

The first place we feel the pinch of this trouble is in computing curvature and torsion. We have convenient formulae for finding these functions for a unit speed curve, but if some guy from a dark alley steps out and opens his trenchcoat to reveal... lots of beautiful geometric space curves, and says, "Hey, buddy. Wanna buy a curve?", the chance that you get a curve which is already unit speed is pretty small. I mean, its probably not a real swiss timepiece, even if the guy claims its a Bernoulli Brachistochrone.

So, how can we deal with such a thing?

THE CHAIN RULE!

Now, we can get by for any single curve with the chain rule, but then we have to compute the whole Frenet-Serret apparatus, too. It would be much nicer to have compact and easy to compute formulae for the general case. That is our goal. (Note: I don't recall learning this when I first encountered the subject. I take the below from Struik [1].)