## Minimal Surfaces Gallery

A *minimal surface* is a surface the locally minimizes areas.
That is, consider a small, simple, closed curve that lies on the
surface. Then, among all the surfaces that have that curve as boundary,
the minimal surface is the one that has the smallest area. For a
minimal surface, the mean curvature is zero at each point on the
surface. (In 3D-XplorMath-J, you can see this by setting the
"Surface Coloration" in the "Action" menu to "Hue = Mean Curvature";
for a minimal surface, the surface will be a uniform red color, indicating
a constant mean curvature.)

Several of the minimal surface exhibits (those above the separator in
the menu) are defined as simple parametric surfaces; that is,
as a triple of real-valued functions *(x(u,v),y(u,v),z(u,v))*, for *(u,v)*
lying in some rectangle in the plane. See the documentation for
the Surfaces Gallery for more information on
parametric surfaces.

The remaining exhibits in this gallery are so-called
"Weierstrass Minimal Surfaces." These surfaces are defined
using Weierstass data, which consists of a triple of
**complex-valued** functions. The functions are defined
as solutions of a set of complex partial differential equations.
In 3D-XplorMath-J, the parital differential equations are solved
numerically to give a grid of three-dimensional complex vectors
*(w*_{ij},u_{ij},v_{ij}). The
surface is defined by the real part of these complex vectors.
The imaginary part of the same vectors form another minimal
surface, called the *conjugate surface*. You can view
the conjugate surface by selecting the "Show Conjugate Surface"
option in the "Actions" menu. In fact, an entire one-parameter
family of minimal surfaces is obtained by using appropriate
combinations of the real and imaginary part. The "Associate
Family Morph" command in the "Animate" menu shows this one-parameter
family, with the main surface at the beginning of the animation
morphing to the conjugate surface at the end.

For some of the Weierstrass minimal surfaces, the Weierstrass
data defines only a fundamental piece of the surface. More
pieces of the surfaces can be obtained by transforming the
fundamental piece, that is by translating, rotating, and reflecting
it. The surface as a whole is "tiled" by copies of the fundamental
piece. An example is the "Riemann's Minimal Family" exhibit.
Exhibits of this type might have a "Show More Copies" or
"Number of Copies" command in the "Action" menu to control the number
of copies of the fundamental piece that are displayed. The conjugate
surface might also be tiled in a similar way, but in general, the
other surfaces in the one-parameter associated family are not.