Parametric Surfaces

When the three coordinates depend on two parameters, a surface in three dimensions is swept out as the graph is drawn. To graph these the three coordinates are entered as a vector (in many programs, including Grapher, the one I used).
We begin by typing the parametric equations of a sphere of radius 1.

x = cos(t) cos(u)
y = cos(t) sin(u)
z = sin(t)

where the domain is u = 0…2π and t = 0…2π. Here the u variable is the equivalent of longitude and the t variable is the equivalent of latitude.

Color is used in 3D graphing because the eye is used to seeing images in color. In this case I have added extra information by making the z coordinate determine the color. This is usually an option for a 3D graphing program.

By using parametric equations to produce surfaces, we are freed from the function method. Since functions have to pass a vertical line test, closed shapes, such as the sphere, are not easily available to us. Parametric equations let us have more access (actually a lot more access) to possible shapes. In this example the use of the circular functions, sine and cosine, give a closed spherical surface. As shown below, simple changes to these three equations produce very interesting shapes.

Variations on the Parametric Equations of the Sphere: The Best Way to go to 4D!

The picture at the top of the page shows a sphere with its parametric equations. The other four pictures were made by my students from simple variations in these equations. These examples illustrate an important method for exploring graphs. Begin with something that seems familiar, then make small changes. See what happens.

In this first equation we have made a small change to the y coordinate functions.

This last one used slightly more complicated equations and a more complicated color scheme.