
 
Surfaces in three dimensional space can be described in many ways  for example,
The surface at the right exemplifies all three as
On the other hand, some surfaces cannot be represented in any of these ways. The surface at the right, whose technical name is "torus," is an example. In this module, we explore a new  but familiar  way of describing surfaces, as parametric mappings of planar regions in space. In particular, we will see that there is a natural way to describe the torus as a parametric surface.
Often the two parameters are labeled u and v. The parameterized surface is a vector valued function r(u,v) of two variables, whether written in ijk vector notation or as an ordered triple of functions of u and v. Since each of the variables u and v ranges over an interval, the domain for r(u,v) is a coordinate rectangle, say [a,b] x [c,d], in the uvplane. (Either or both intervals may be infinitely long.)
Just as we can use Cartesian coordinates as parameters, we can use other coordinate systems as well. The next two steps illustrate this.
Now we consider a parameterization of the torus pictured above before step 1. We can visualize this surface by first imagining a circle of radius a in the xyplane that runs through the center of the "tube". From each point on this circle, we can reach a circle of points on the surface by making it the center of a circle of radius b, where b < a. This second circle is drawn in a vertical plane that includes the zaxis. In the figure at the right, we show the horizontal circle of radius a in blue and a typical circle of radius b in red. As the red circle travels around the blue one, it sweeps out the entire torus. We take as parameters u and v, respectively, the central angles for the blue and red circles. Then we can construct the parameterization for each point r(u,v) on the surface by adding a vector from the origin to a point on the blue circle and a vector from that point to a point on the corresponding red circle.

 
modules at math.duke.edu
Copyright
CCP and the author(s), 2002