Microworld: Implicit
Surfaces
Click the Hyperlink
above to visit the Microworld.
Author:James
E. White
A
function f of three variables may define a surface in R(3) as the solution
to the equation f(x,y,z) = 0. Then again, it may not. The condition (the implicit
function theorem) for it to define a surface is that it have nonvanishing
derivative (or gradient).
We
call the points where the derivative vanishes singular points, and
a great variety of interesting things can happen at such points. The red cone
pictured above has a single singular point at its vertex. The yellow surface
has no singular points. This tool can help you explore the singularities and
bifurcations of smooth functions of 3 variables. You can get very fine detail
if you draw in a small box, then fly into it.
Implicit
Surfaces is a utility that students may use to draw the solution of an
equation: f(x,y,z) = 0. They define the function f(x,y,z) by writing any expression
in x,y and z. It is automatically set to 0 and the surface is drawn.
Read
the information under the Instructions button for more detailed information
on how to use this tool. You will learn how to draw your surfaces, rotate
and translate them, draw them solid or wireframe, fly through them, and bring
them to life in other ways. If you have a picture you would like to keep,
take a snapshot (Workbook menu), and save it.
Return to the listing of MathwrightWeb Microworlds
| - James E. White, Ph.D. , Library Director, | ||
| author of this website, Mathwright 2000, MindScapes, | ||
| MathwrightWeb, and Mathwright32 |
