Take a piece of wire, connect the two ends together, and dip it in
a bath of soapy water. The thin membrane that spans the wire
boundary is a minimal surface; of all possible
surfaces that span the boundary, it is the one with minimal energy.
One way to think of this "minimal energy; is that to imagine the surface
as an elastic rubber membrane: the minimal shape is the one that in which
the rubber membrane is the most relaxed.
For example, a minimal surface that spans two rings is given by the
catenoid surface below:
Notice that this is not the only possible minimal surface: another one
consists of the two disks; one spanning each ring. This means
minimal surfaces exist locally ;
each one only has
to be most relaxed membrane of all the ones close by.
Level Set Methods for Minimal Surfaces
Given a wire frame, how can one find a minimal surface that
spans the frame.
Motion under curvature provides
an appealing way to find the surface.
To illustrate the idea, what is the "minimal" curve connecting two points?
Although its obvious that the straight line
connecting the two points is the minimal curve, a different
way to construct the minimal surfaces begins with
any curve at all connecting the two points.
Any curve
Curvature flow
Minimal curve
If this initial curve moves according to
its curvature , and is forced to stay attached to the
endpoints, then the kinks will disappear and the curve will
straighten itself out.
A level set perspective
can
be used to find such a minimal surface.
Start with any membrane at all that spans the wire frame,
and then let that surface evolve under
curvature flow .
As long as it stays attached to the frame,
it will become a minimal surface when it is through moving.
Because the level set approach allows a surface to break and move as it
evolves, this approach can find minimal surfaces that are very different from
the initial shape. For example, if the two rings in the above
example are pulled slightly apart, the only minimal surface is the two disks:
Evolution of Catenoid Splitting into two disks
Movies
Initial Rings Close Together:
Initial Rings Farther Apart
(Movie: not yet constructed)
(Movie: not yet constructed)
Details
The calculations were made using a
level set method to
track the motion of evolving surface. An initial surface is attached to the
wire frame, and
then embedded as the zero level set of the signed distance function
in all of space. This level set function is evolved under mean curvature,
with boundary conditions enforced to always attach the front
to the boundary. The level set function is periodically re-initialized
to maintain uniformity in its spacing. The final state of the calculation
is the minimal surface.
References
Chopp, D.L., Computing Minimal Surfaces via Level Set Curvature Flow ,
Jour. of Comp. Phys., Vol. 106, pp. 77--91, 1993.