Consider a closed, nonintersecting curve in the plane moving
with speed 
normal to itself. More precisely,
let 
be a smooth, closed initial curve
in 
, and let 
be the one-parameter family of curves
generated by moving 
along its normal vector field with
speed 
.
Here, 
is a given scalar function of the curvature 
.
Thus, 
, where x is the position vector of
the curve, t is time, and n is the unit normal to the curve.
Consider a speed function of the form 
, where 
is a constant.
An evolution equation for the curvature 
, see [23],
is given by
where we have taken the second derivative of the curvature 
with
respect to arclength 
.
This is a reaction-diffusion equation;
the drive toward singularities due to the reaction
term (
) is balanced by the smoothing effect of
the diffusion term (
).
Indeed, with 
, we have a pure reaction
equation 
.
In this case, the solution is 
, which
is singular in finite t if the initial curvature is anywhere negative.
Thus, corners can form in the moving curve when 
.
For 
, the front develops a sharp corner
in finite time as discussed above. In general, it is not clear how to
construct the normal at the corner and continue
the evolution, since the derivative is not defined there.
One possibility is the ``swallowtail'' solution formed by letting the
front pass through itself.
However, from a geometrical argument it seems clear that the
front at time t should consist of only the set of all points
located a distance t from the initial curve.
(This is known as the Huyghens principle construction, see [23]).
Roughly speaking, we want to remove the ``tail'' from the ``swallowtail''.
Another way to characterize this weak solution is through the
following ``entropy condition'' posed by Sethian (see [23]):
If the front is viewed as a burning flame,
then once a particle is burnt it stays burnt.
Careful adherence to this stipulation produces the Huyghens principle
construction.
Furthermore, this physically reasonable weak solution
is the formal limit of the smooth solutions 
as the curvature
term vanishes, (see [23]).
Extensive discussion of the role
of shocks and rarefactions in propagating fronts may be found
in [22].
Let us imagine now a very specific speed function, namely
. This case corresponds to a curve collapsing
under its curvature. It can be shown that for an arbitrary smooth
simple curve, (see Gage [8], Grayson [9]), such a curve
collapses to a single point.
In Figure 1a,b, we show a star-shaped region collapsing
under this flow.
Figure 1: Collapse of Star-Shaped Curve under Curvature
Here, we have evolved the curve using the Osher-Sethian level set
method, see [17]. Briefly, this technique works as follows.
Given a moving closed hypersurface 
, that is,
,
we wish to produce an Eulerian
formulation for the motion of the hypersurface propagating along its
normal direction with speed F, where F can be a function of
various arguments, including the curvature, normal direction, e.t.c.
The main idea is to embed this propagating interface as the
zero level set of a higher dimensional function 
.
Let 
, where 
is defined by
where d is the distance from x to 
, and
the plus (minus) sign is chosen if the point x is outside (inside)
the initial hypersurface 
.
Thus, we have an initial function 
with the property that
It can easily be shown that the equation of motion given by
is such that the evolution of the zero level set of 
always corresponds
to the motion of the initial hypersurface under the given speed
function F.
This evolution equation Eqn. 7 is solved by means
of difference operators on a fixed Eulerian grid. Care must be taken
in the case where the speed function F contains a hyperbolic
component. For details, see [17,24].
Since its introduction, this approach to front propagation has
been used to model a wide variety of problems, including
the generation of minimal surfaces [6],
fast interface techniques [1],
singularities and geodesics in moving curves and surfaces in [7],
flame propagation [26,27], shape reconstruction
[13,14,12], shape representation and
recognition [11], grid generation [25],
and semiconductor manufacturing [2].
