Curves
and Surfaces > Surfaces > Types
of Surfaces Maps from R^{2} into R^{3}. |
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Parametric vs. ImplicitSome 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 R^{3}. Although it can also be defined parametrically, the sphere is an example of such a surface. Different CoordinatesAlthough 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). Smooth vs. Non-smoothA 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. Regular vs. Non-regularRoughly 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. Abstract SurfacesAn abstract surface is one which is not embedded in R^{n}, 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 R^{3}. Locally vs. Globally DefinedSurfaces 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. |