and Surfaces > Surfaces > Types
Maps from R2 into R3.
Some surfaces, like the torus, are defined in terms of two parameters (often called u and v) that, when varied over a region of the uv plane, create a set of points from equations of the form
Other surfaces, however, are defined implicitly. That is, they are simply equations in three variables whose solutions form a surface in R3. Although it can also be defined parametrically, the sphere is an example of such a surface.
Although many surfaces are specified in xyz (rectangular) coordinates, it is occasionally more convenient to define a surface in cylindrical, spherical, or other coordinate systems. Plücker's conoid is one example of a surface that lends itself well to cylindrical coordinates. In cylindrical coordinates, one identifies a point by its distance from the origin in the xy plane (denoted r), the angle a line between that point and the origin makes with the positive x axis (denoted by the Greek letter theta), and the point's height above the xy plane (denoted z).
A surface with nonzero surface normals everywhere is said to be smooth. Because the surface normal is the cross product of the u and v tangent vectors, this means that the tangent vectors to the surface must also be nonzero. A paraboloid is an example of a smooth surface.
Roughly speaking, a regular surface is one that has well-defined tangent planes at all points. It should not have sharp corners, cusps, or self-intersections. An example of such a surface is the catenoid.
An abstract surface is one which is not embedded in Rn, but can still be analyzed with the tools of differential geometry. Any abstract surface comes with a metric that makes this possible. All of the surfaces currently in the library are embedded in R3.
Surfaces often have global properties that depend strongly on their behavior as a whole, rather than their behavior in a specific region on the surface. One example of such a property is orientability; can we define a positive direction for the surface normal that works for the whole surface? When the surface is, say, a sphere, we can. We can make all the normals point either outward or inward. But what if it is a surface like the non-orientable Möbius strip? It is impossible to assign a consistent direction to the surface normals.