The exhibits in the Surfaces gallery are two-dimensional surfaces embedded in three-dimensional Euclidean space. The gallery is quite large and is divided into several sub-galleries, such as Parametric Surfaces, Minimal Surfaces, and Implicit Surfaces. (The Implicit Surface exhibits are quite a bit different from the other types of surface exhibit, and the rest of this document does not apply to them.)
In most cases, a surface exhibit is represented ultimately as a grid of points that lie on the surface. (This is not true for implicit surfaces, and some of the minimal surfaces are a little more complicated than this.) The visual representation of the surface is then obtained either by drawing lines between certain points in the grid -- giving a wireframe display representation of the surface -- or by drawing colored quadrilaterals whose corners are points in the grid -- giving a patch display. The commands for choosing between these two visual representations can be found in the "View" menu.
Many surfaces (and not just those in the parametric surfaces sub-gallery) are given in parametric form, that is, as a triple of functions (x(u,v),y(u,v),z(u,v)). The grid of points that represents the surface is then given as (x(uij,vij),y(uij,vij),z(uij,vij)) for 0 <= i <= n and 0 <= j < m, where the (uij,vij) form a rectangular grid of points in the plane. A surface that is represented in this way has six parameters that determine the set of points (uij,vij): Minimum value of u, Maximum value of u, Minimum value of v, Maximum value of v, U resolution, and V resolution. The minimum and maximum values determine the extent of the grid, and the u and v resolutions determine the number of points in each direction. The resolutions are used in a somewhat peculiar way: The number of points in the u direction is given by 1 plus 6*(U resolution), and the number of points in the v direction is given by 1 plus 6*(V resolution). (In the default view of a surface, grid lines are drawn only at every sixth point in the grid, so the "resolution" actually refers to the number of grid lines that are drawn by default; but note that the number can be changed using the "Grid Spacing" sub-menu of the "Action" menu.) Recall that all exhibit parameters can be set using the "Set Parameters" command in the "Settings" menu.
When a surface patch is drawn, its color is determined by the color of the surface itself together with the properties of the lights that shine on the surface. Furthermore, the front and the back of the patch can have a different appearance. By default, the color of the surface is white, the surface is illuminated by several differently-colored lights, and the back face of a patch is a little less reflective than the front face. The surface color can be set using the "Surface Coloration" sub-menu of the "Action" menu. Properties of the lights that illuminate the surface can be set using the "Light Settings" command in the "Settings" menu, and lighting can be turned on and off using the "Lighting Enabled" option in the "Settings" menu. (When lighting is disabled, the color that is used when drawing a patch is simply the surface color, unmodified by any lighting effects.) The meaning of "front" and "back" are determined by the orientation options in the "View" menu: With "Normal Orientation," "front" is in the direction of the standard normal to the surface; with "Reverse Orientation," the meaning of "front" and "back" are reversed; and with "No Orientation," both sides of a surface patch are treated as "front."