In this animation a "snail-shaped" surface, circular in cross-section, unwinds and deforms into a planar disk, then returns to its original shape and orientation. There is a lot going on geometrically. Topologically, not so much.
Geometrically speaking, the surface starts out as a tapering, winding cone that is everywhere smooth except at one point, the tip of its "tail." After the surface unwinds, it flattens out into a circular disk. At just that one instant of time when it is a circular disk, the surface is everywhere smooth. (When I say "smooth," I am referring to the ideal surface, not the "mesh surface" approximation that is what we actually see in the plot.)
Topologically speaking, the surface is homeomorphic with a circular disk in two-dimensional space, as can be seen when the animation reaches its midpoint. The controlling variable is FRAME, as with any Mathcad animation.
The multiplier f is a function of FRAME. As FRAME goes from zero to 100, f goes from 1 to zero and back to 1. This is how I made the surface deform into a planar disk (for which f = 0), and then return to its original appearance (for which f = 1).
As with my Lorenz Attractor animation, the idea for this Mathcad 3D plot and animation came from a figure in the Mathcad PLUS 6 for Macintosh brochure. Below is attached the Mathcad 13 worksheet that I used to create the 3D plot and animation.
Also attached is a scan of the Mathcad PLUS 6.0 for Macintosh sales brochure that has been such an inspiration to me after all these years. I did not attach the old brochure to be nostalgic, but rather, to recall what a powerful application Mathcad was even back at Mathcad PLUS 6.
According to PTC, Mathcad 15 is to be released this summer (2010), and will have the Design of Experiments (DOE) functionality that I helped to test last year. My own special interest here is in the area of multi-dimensional function fitting, specifically, the new DOE function "multidfit."
After more than fifteen years as a Mathcad single user, I signed up for a node-locked license, with annual maintenance, just last month (May 2010). I'm looking forward to the release of Mathcad 15 so that I can do some more work with multidfit.