Shorter or Rounder, or More Evenly Bent?

A Visual Tour of Curve Shortening

Theron J Hitchman

University of Northern Iowa

Motivation

A fancy geometry problem.

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Motivation

Other Interesing Geometry Problems which are related:
  • Finding "geodesics" in general spaces
  • Modelling "heat flow"
  • Finding harmonic maps
  • Mean Curvature Flow of surfaces
  • Hamilton's "Ricci Flow"
     $\rightarrow$ Perelman's work on the Poincaré conjecture

"Easier" Problem

A Hippopede

Parametrized Curves

Parametrized Curves

Parametrized Curves

Parametrized Curves: Problems

This gives us Calculus. Wooo-Hoo!

What problems do we inherit?

 

The motion may not reflect the curve, so much as some strange wandering along it.
How far is it to Dallas from Here?
How far to Houston?

ArcLength Parametrization

Idea of Curvature

We want to make a curve "less bendy."

How can we (use Calculus to) measure how much a particular curve bends?

Calculus?

The Tangent Vector

To find the tangent vector to $$\gamma(t) = (\,x(t), \,y(t)\, ),$$ we simply compute $$\gamma'(t) = (\,x'(t),\, y'(t)\,)$$ and then scale it to make it length $1$. Only the direction matters!

The Normal Vector

Now just turn 90 degrees counter-clockwise from the direction you are facing.

We have to have ONE proof

Lemma: The derivative of the tangent vector is parallel to the normal vector.
Proof: The tangent vectors all have unit length (arclength!), so $T \cdot T = 1$ is constant. Hence $$ 0 = T' \cdot T + T \cdot T' = 2 T' \cdot T .$$ This means $T'$ is orthogonal to $T$, and thus must be parallel to $N$.

Curvature

We write $$T' = \kappa N,$$ and call $\kappa$ the curvature.

Interpretations of Curvature

Natural geometric descriptions:
  1. $\kappa$ is the rate of change of the angle $T$ makes with the $x$ axis.
  2. $\kappa$ is the reciprocal of the radius of the circle which best approximates the curve at that point.

The Frenet-Serret Equations

$$ \begin{cases} T' & = \kappa N \\ N' & = - \kappa T \\ \end{cases}$$

Qualitative Meaning of Curvature

Curve Shortening Flow

$$ \dfrac{\partial}{\partial s} \gamma_s(t) = \kappa_s(t) N_s(t). $$
Here $t$ moves along each curve, and $s$ is the "flow direction" we wish to generate, changing $s$ moves us from one curve to another.

Educated Guess Time: Part One

What does curve shortening flow do to a line?

Educated Guess Time: Part Two

What does curve shortening flow do to a circle?

Gage & Hamilton

Suppose that your curve is convex, then
  • CSF makes sense: the differential equation has a solution.
  • In a finite time, the curve shrinks to a single point.
  • As the curve shrinks to a point, it gets rounder.
  • If we rescale so area enclosed is constant, the limit is a circle.

Convex

Not Convex

Isoperimetric Ratio

A way to test how circular your (simple) curve is:
If $A$ is the area enclosed, and $L$ is the length, then $$ \dfrac{L^2}{A} \geq 4\pi .$$
And equality holds exactly when the curve is a circle.

Some other stuff we know.

Grayson's work

As long as your curve is simple, that is, it does not cross itself, then CSF will keep the curve simple, and eventually the CSF makes the curve convex.

Aubresch and Langer Examples

Hard questions

  • Suppose your curve crosses itself. What does the CSF do to it? Can singularities form, past which CSF doesn't make sense?
  • What happens if we perturb one of the Aubresch-Langer examples?

Easier Questions

Let's ask easier questions we still can't solve.
Can we do something for polygons?

Chow-Glickenstein

$$ \dfrac{dB}{dt} = (A-B) + (C-B)$$

Jecko-Léger: Menger Curvature

$$ \dfrac{dB}{dt} = M(A,B,C) $$

What don't we know about these?

  • Do convex polygons stay convex?
  • Do simple polygons stay simple?
  • Are there special homothetic solutions?
  • How stable are the special solutions?
  • Can singularities form?

My Students

This is where I come in. Well, where my students come in.

Jesse Moeller and Ben Castle at CURM

Abbie Parker and Jonny Krein

Our Mission:

Study basic cases and look for structure: triangles, quadrilaterals, cases with symmetry.

Animations: Triangle

Animations: Quadrilateral

Animations: Quadrilateral 2

Animations: Some Polygon

Animations: A Parallelogram

Animations: That Parallelogram, rescaled

Some Expected Results

  • Triangles are easy.
    1. Shrink to a point.
    2. Maintain shape
    3. Computable extinction time
  • Cyclic polygons are not much harder.

The Unexpected

  • Parallelograms maintain their shape, and Spin!
  • kites are harder to deal with

The Ugly Truth

We might not have the right flow. There are figures which maintain their shapes, but are not regular polygons.

Time to search for another way.

Thank you