As we have seen previously, z=f(x,y) describes a surface in xyz space. Alternatively, a surface can be described in parametric form:
where the points (u,v) lie in some region R of the uv plane.
Let us compare and contrast the parameterization of a surface with that of a space curve. A space curve is described by the vector function:
where a<=t<=b. A space curve is a one-dimensional object, similar to a piece of string. Each coordinate x,y and z depends only on one parameter, t. Each point on the curve corresponds to a different value of t.
A surface is a two dimensional object. One needs TWO pieces of information to uniquely define a point on a surface. These two pieces of information are the parameters u and v. These notions will hopefully become clearer as you go through the examples below.
Examples
A natural example is a sphere. It takes two pieces of information to describe a point on a sphere: the latitude and longitude. Let u, with 0<=u<=2*pi be the longitude. Let v, with 0<=v<=pi be the latitude. Here u and v correspond, respectively, to the the spherical coordinates theta and phi. Using the formulas for spherical coordinates we have
Here a is a constant, not a variable.
This is an example of surface that CANNOT be described by a single function z=f(x,y). A sphere of radius a centered at the origin can be defined by the relationship
The top half of the sphere is defined by the surface
and the bottom half is the defined by the surface
A second example is a cone, as shown in the figure. The height is 3, the base radius is 2, and the cone is centered at the origin.
Two parameters are required to define a point on the surface. We choose them to be u, the height from the base, and v, the angle with respect to the x-axis. v is the same as the polar angle theta. We can describe any point on the surface by:
0<=u<=3, and 0<=v<=2*pi. Here r is the radius, the same as the variable in polar coordinates. The variable r must be eliminated since it is a third parameter.
For a cone the radius and the height from the base are related by a linear function. In this example, the radius is 2 when the height is 0 and is 0 when the height is 3. If we let u denote the height, r=2-2u/3. Hence, we have
Copyright © 1996 Department of Mathematics, Oregon State University
If you have questions or comments, don't hestitate to contact us.