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 (wij,uij,vij). 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.