This section provides a description of our work on the project.
History of the Project
To even animate the simplest of transformations, we first needed to have a solid visual grasp of the spacial manipulations we would later be using the computer to generate. For us, this meant that for each construction and transformation, we had to generate careful sketches and drawings, and sometimes even make clay or paper models of them. Most of the first three weeks of the project were consumed in work of this type.
Having established a solid base for the project, we then began to explore the 3D software available to us. We discovered early that the best model for our needs was the "Bezier surface" model. With a Bezier surface, you can place a number of "handles" on the surface to be manipulated. Handles can then be moved and the tangent plane at each handle can be set as needed. The end result is that "pulling" the surface at a point stretches the surface in exactly the way you would expect it to. With a little research, (after examining 3D Studio Max, Bryce 3D and a few smaller programs) we found that only Strata 3D provided this type of modelling in an easily accessible form.
With the software in hand, we then set about actually animating the sequences we had previously modelled. This, of course, is what the bulk of the time spent on this project went into - creating animations, rendering them (that is, turning them into Media Player and Quicktime files), and then retouching them near the end as we discovered better ways to present them.
The final step, the website, was actually an ongoing process throughout the project. A simple interface was built even before we had the 3D software, and as images and animations became available, we uploaded them along with their descriptive text and images. The site itself underwent three significant overhauls (reorganizing and retouching text and images) before it reached the current (and final) format.
Future of the Project
Unfortunately, we were unable to complete all the transformations we had originally hoped to for a variety of reasons. The most notable reason was the limitations of the Strata 3D software. Though clearly a step ahead of all the other software we had examined, the demo version which we used lacked the ability to cut holes in our surfaces (an integral part of the connected sum operation). This made creating surfaces like a torus connected-summed with a crosscap essentially impossible.
The other major flaw of Strata (and perhaps all 3D software) was the sheer time consumed by rendering the animations. Depending on the computer, a five second animation could take up to a full hour to convert to a movie file. After rendering thirty animations (many of them multiple times when something had to be corrected) it was really no surprise that most of the time allotted for the projected had been consumed.
A few unaccomplished things we had hoped to do:
- Provide additional examples of the connected sum operation.
- Animate the transformation from torus-plus-crosscap to klein-bottle-plus-crosscap. (See the discussion in Transformations.)
- Transform the figure eight klein bottle into one (and therefore all) of the other klein bottles.
- Include the Boy's and Roman surfaces (and animations transforming between them and the crosscap).