A Parameterized Surface Applet

A Parameterized Surface Applet

This applet is designed to build intuition about parameterized surfaces.

The applet has a control window, a Parameterized Surface window, and a Domain window.
The control window lets you define x, y, and z as functions of s and t. One can also scroll the values of s0 and t0. Readouts let you see the values of the point P0 = (x(s0,t0), y(s0,t0), z(s0,t0)).
The Parameterized Surface window shows the surface along with the point P0 and curves (x(s0,t), y(s0,t), z(s0,t)) and (x(s,t0), y(s,t0), z(s,t0)) in red and blue respectively. The gridlines checkbox adds lines in pink and teal for other values of s and t.
The Domain window follows the same color scheme, but has the lines drawn in the s-t plane.

Things to notice:

We can build a parameterized surface as a wireframe of the graphs of slices in the s-t plane, much as we originally did surfaces by constructing a wireframe model in x and y.
The easiest surfces to parameterize are grafph of z=f(x,y). They parameterize as (s, t, f(s, t)).
The next set of parameterizations to consider are graphs in another standard coordinate system, like rho=g(phi, theta) in spherical coordinates. It is straightforward to parameterize in spherical, and then to write x, y, and z as functions of rho, phi, and theta.

This applet was designed as by modifying an applet from a demo by Tom Banchoff at Brown University. It is used with permission. Go to the Banchoff Applet Help page.

Return to the Banchoff Applet page.
Return to the SLU Calculus Applet page.

Return to the Saint Louis University Department of Mathematics and Computer Science home page

Last updated By Mike May, S.J. , October 6, 2006.