k: the signed curvature of the curve (positive where C is convex)
T’ = k N |X'|
Allow C to evolve according to Xt = kN. Each point will move along its unit normal at a rate proportional to the curvature.
Grayson (1987) proved that any smooth, simple, closed curve will eventually become convex while staying embedded.
Gage and Hamilton (1984) proved that any smooth, convex , closed curve evolves asymptotically to a circle, which shrinks to a point.
Over time, the arclength of the curve monotonically decreases (thus "curve shortening"):
Lt = -∫ k2 ds.
Arclength of the evolving flower shown in the second video below
The area of the lamina of the curve decreases linearly: At = -2*Π.
Note that the curvature satisfies the PDE kt = (1 / |X’|) ( k’ / |X’| )’ + k3.
In particular, if a curve B initially contains another curve C, both evolving according to Xt = kN, then the two curves never cross. If they were to cross, the two would first have to be tangent at a point. At that time and point, C would be inside B with a larger k, so C would evolve away from the point faster than B; C could not cross B.
Our question: The curve shrinks to a point. Where is that point relative to the original curve?
We hoped to use the fact that an evolving curve never crosses an evolving curve that contains it to define two evolving circles. Both circles would initally contain the curve and become tangent when and where the curve disappears.
Because of the rate of disappearance (At = -2p), we showed this to be impossible. Instead, two such circles define a "region of disappearance": the area inside both circles at the time of the disappearance of the curve. The point of disappearance is somewhere within this region.
We developed a simulation for the evolving curve. Instead of approximating the PDE, we decided to approximate the length of the curve to define the gradient descent equation.
Let x1, …, xn be points lying on the simulated curve C.
The length of the curve is approximately equal to the sum of the line segments connecting consecutive points on the curve. The closer the points are to each other, the better the approximation.
≈ f ( x1, …, xn ) = ∑ | xi - xi-1 |.
From here, we can define the gradient descent equation:
d/dt[ xi ] = Fi = - ∇ f ( x1, …, xn ) .
A few videos of evolving curves from our simulation:
An evolving circle
An evolving flower
An evolving spiral
An evolving flower within an evolving circle
If you are experiencing difficulty viewing the videos, try switching to Mozilla Firefox.
What We Do
Currently, we are studying the behavior of curves under a "curve flattening" evolution.