Mathematical Visualization SS07 Technical University Berlin

Assignment 1

Theme: What makes a successful "interactive mathematical visualization"?

This assignment consists of three parts, all more or less preparatory in nature. For parts 2 and 3, you can write the results on paper and hand them in, or e-mail them to me at gunn [at] math.tu-berlin.de.

Part 1. Evaluation

This course is about interactive mathematical visualization. Before we begin doing our own visualization, it's useful to try out some existing examples. Based on this experience, I then want you to articulate what elements contribute to a successful (or unsuccessful) experience.

The images at the bottom of this page are linked to Java webstart applications that I have written in the last few years while here at the Technical University. They are by no means finished applications, but I have chosen them since they all have some rudimentary on-line help document (or manage to get along without it) , and each one treats a different theme. Choose one from the category Differential Geometry and one from the category Fractals. Run the applications (either at home or on a computer in the lab in MA 316). Finally, print out and fill in evaluation forms for each application, and bring them to our next meeting. r Fill out the evaluation forms I passed out in class (or pick up a couple at my office 319) and bring them to our next meeting.

Note: You are expected to read the on-line documentation to discover features of the application which the author intends you to know about.

Part 2. Virtual Reality: First Impressions

Thursday morning we visited the PORTAL and experienced a variety of demos, some mathematical and some not. I would like us to discuss this experience at our next meeting. In order to prepare for this discussion, here are some questions to consider:

  • How does the experience in the PORTAL compare to that provided by a desktop computer display? (for short, we refer this as the workstation environment) How does the viewer interact with the content? What sensory experiences does he/she have? etc.
  • You probably have all heard of virtual reality. Was this your first experience of it, and if so, how did it differ from your expectations?
  • What do you think might be some advantages/disadvantages of the PORTAL compared with the workstation, in the realm of mathematical visualization?
  • Did you have any spontaneous ideas about mathematical themes which would be particularly interesting to experience in the PORTAL?

Part 3. Project preparation

In preparation for your semester project, spend some time to review your relationship to mathematics in order to choose a theme for your project. (Of course this decision will also be influenced by the mathematical themes introduced in the lectures, but it's never to early to start to think about it). What sorts of mathematics appeal to you? Have you previously done some mathematical visualization that you would like to develop further? Did you have a particular topic in mind when you chose to attend this lecture? Write a page in the language of your choice to answer these questions, or similar ones of your choice, that help you begin to plan your semester project.

Differential Geometry

Space Curve Explorer

This application extends the previous one to allow specification of an arbitrary curve in Euclidean 3-space. The curvature and torsion graphs corresponding to the curve can be optionally viewed. Other options include display of the Frenet frame and osculating circle at a given point of the curve (controlled by a slider), and display of the evolute curve (locus of the centers of the osculating circle).

Surface Explorer

This application supports the exploration of arbitrary parametrized surface patches in Euclidean 3-space. The Gaussian and mean curvature graphs can be optionally viewed. The domain of the patch can be manipulated by dragging its corners or edges. One can optionally display the tangent plane (and/or second order Taylor approximation) to the surface at an arbitrary point of the domain. Surface definition is organized in three types: explicit, surface of revolution, and tangent developable. Each type has a set of built-in surfaces, or you can type in your own examples. As above, these expressions can be parametrized. There are a host of rendering options also available, including a variety of texture maps and color options.

Fractals

Elephant Trunk

In September 2005 I visited a freshman seminar at UNC-CH taught by Sue Goodman, and talked about iterating 3D transformations. I prepared this and the following application as webstart applications so that the students could play with these ideas. ElephantTrunk features a single iterated conformal map of Euclidean 3-space (i.e. a Euclidean isometry combined with a homogeneous scaling transformation). The iterates of this transformation are applied to a simple geometry (in this case, a bone-shaped surface of revolution). The user can interactively control this transformation.

Self-similar Tree

If instead of a single transformation you choose a pair of such transformations and construct all possible combinations of these two transformations (up to a certain length), and apply the resulting list of transformations to a simple cylinder, you get an object which deserves the name "self-similar tree". The coloring of a specific copy reflects the sequence of transformations responsible for the copy.

Punctured Torus

The following application is under development. It's a re-implementation of a program I wrote at the Geometry Center in 1993 to explore limit sets of Kleinian groups. This version was inspired by the wonderful book Indra's Pearls by Mumford, Series, and Wright. This application allows you to independently set the two complex values which determine a punctured torus, and then draws the limit set of the covering group, conceived of as acting on hyperbolic 3-space + plane at infinity. There are options for controlling how many group elements are calculated; and for controlling the rendering, such as whether it should be drawn as the Riemann sphere or flat; in the former case, one can display part of the pleated surface within hyperbolic space corresponding to the group. You can also display fixed points and words associated to the almost parabolic elements in the group, which play an important role in characterizing the limit set. Finally, once you set one of the parameters, it attempts to detemine and display the boundary of the Teichmuller space for the other parameter, but this feature does not work reliably.