Imagine a closed curve lying on this page, and don't let the curve
cross over itself. (This is known as a simple, closed curve).
For example, take a piece of rope, glue the two ends together, and
drop it on the ground, making sure that the rope doesn't cross over
itself.
Simple Closed Curve
Not Simple Curve
Even Less Simple
The curvature measures how fast a curve bends at any spot. For example, a
circle has a constant curvature
because it always is turning at the same rate; a smaller circle has a
higher constant curvature because it turns faster.
Now, suppose each piece of the curve moves perpendicular to the curve with
speed proportional to the curvature.
Since the curvature
can be either positive or negative (depending on whether the curve is
turning clockwise or counterclockwise), some parts of the curve move
outwards while others move inwards.
This is called "motion under curvature".
Blue arrows are where the curvature is negative
Green arrows are where the curvature is positive
A Famous Theorem
A famous theorem in differential geometry, proved less than ten years ago,
says that any simple closed curve moving under its curvature collapses nicely
to a circle and then disappears. That is, no matter how wildly twisting
a curve is, as long as it is simple, it will "relax" to a circle and then
disappear.
Movie of Weird Curve Collapsing under Its Curvature
(305K)
Why Care?
Although the theorem is interesting in its own right, it has wide applications.
The surface tension of an interface,
like a soap bubble, is
proportional to its curvature; for example, the dynamics of ink in an ink
jet plotter is affected by its surface tension.
Other applications include the evolution of
boundaries between fluids
and removing noise from an image.
The movie was made using a
level set method to track the motion of
an interface evolving according to its curvature. The curve is embedded
as the zero level set of the signed distance function. The level set
equation is updated using a first order in
time, central difference in space approximation. Periodic boundary conditions
are taken on a 200x200 grid.
References
Sethian, J.A., A Review of Recent Numerical Algorithms for Hypersurfaces
Moving with Curvature-Dependent Speed, Journal Differential Geometry,
31, pp. 131-161, 1989.