The Flash Forum article The Beauty of Parametric Curves offers two applets. The one for designers encourages experimentation with patterns, and the one for mathematicians allows a user to enter any definition of a parametric curve with parameters corresponding to sliders values.  Commented source code and required custom classes are included in the downloadable zip file.

Click the picture on the left to go to the to this Flash Forum article.

The NSF-funded project that funded the development of much of the material on the MathDL Flash Forum website uses the website www.flashandmath.com for ongoing dissemination.

Click the picture on the right to go to the flashandmath.com website

This video tutorial with supporting files highlights the exciting capabilities of the open source project PaperVision3D. The particular example uses Adobe Flex 3, but since the underlying programming language is ActionScript 3, the broad idea is applicable to Flash as well.

Click the picture on the left to go to this archived Flash Forum tutorial.

This article Motion in 3D provides examples and Flash CS3 source code for plotting a parametrically defined curve in space and tracing velocity and acceleration components as arrows on the graph. Any functions can be entered by the user and the applet also contains a series of interesting examples for teaching.

Click the picture on the right to go to this archived Development article.