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report 5aug01

Summer Research Experience: illiMath2001

11 June - 3 August
Program Director: George Francis

In this program students with strong interest and some experience in mathematical graphics participated in research programs of their associated mentor (AM). The program provided the facilities (and any necessary training in their use) for the student to assist their AM in creating visual materials illustrating research in a variety of media, including the production of custom soft-ware, web-pages, animations, print quality images and virtual environments (CAVE and CUBE). The program also provided the student with tutorials in the mathematical background and lectures by specialists in the subjects of their projects. A similar REU program, the Audible Sketchpad for the CAVE, was conducted for the past two years. Here are the titles for the projects this summer, their student principal investigator (SPI), associated mentor (AM) and corresponding mentor (CM), and brief synospis are here.

"Alice on the Eightfold-Way"; SPI: Matthew Woodruff and Ben Bernard; AM: Dr. Ben Schaeffer, ISL, Beckman, UIUC; CM: Dr. Jeff Weeks, Canton, NY and Prof. John Sullivan, UIUC

The Integrated Systems Lab (ISL) of the Beckman Institute is building the CUBE, which is the first rigid walled 6-sided CAVE in the US. This long-range project to implement Thurston's Geometries in the CUBE got underway this summer. The collaboration between im2001 students, ISL scientists, the Math Dept, and Dr. Jeff Weeks, achieved proof-of-concept implementations of five of Thurston's eight geometries, programmed in {\tt syzygy}, the experimental CUBE library for distributed graphics computing on Linux clusters. For this purpose a syzygy linux cluster was created in Altgeld Hall. See our preprint of preliminary results.

Kairomone"; SPI: Lorna Salaman and Matthew Woodruff; AM: Dr. Karen Shuman, VIGRE Post-doc, UIUC Math Dept; CM: Prof. Robert Acar, Math Dept, U. Puerto Rico at Maiaguez.

The corn root worm (diabrotica) is a serious agricultural pest. The late UIUC entomologist, Robert Metcalf, his coworkers and students, have developed a model for integrated pest management involving naturally occurring chemical food attractants, the kairomones, that influence the behavior of the diabrotica beetle in a corn field ringed with kairomone baited traps. A real-time interactive computer animation modelling diabrotica behavior begun last summer was completed, and extensive documentation, useage instructions and background material was assembled this summer. A summary was presented as a PME talk at MathFest, Madison, WI.

"Narnia"; SPI: Alison Ortony; AM: Elizabeth Denne, graduate student UIUC Math Dept; and Stuart Levy, senior research programmer, NCSA; (CM) Prof. John Sullivan, Math Dept, UIUC

Some years ago Stuart Levy initiated a project of adapting to the CAVE the software package polycut by Prof. Ken Brakke of Susquehanna College. Polycut simulates navigating through 3-manifolds which are branched coverings of 3-space, with knotted and linked branching curves. The mathematics of this subject generalizes the classical theory of Riemann surfaces by one dimension, involving topology, geometry and group theory. Though a complete solution to this visualization problem still eludes us, extensive geometrical documentation and evaluation of extant software was undertaken this summer and presented as a PME talk at MathFest, Madison, WI.

"CAVE Navigator" AM: Dr. Volodymyr Kindratenko, NCSA

This project will systematize the scores of different modes people have devised over the past decade for navigating the virtual worlds in the CAVE immersive virtual environments at the NCSA and elsewhere. The mathematics here involves Lie Groups, in particular the double covering of SO(2) by the unit quaternion group. But a thorough command of the 3D calculus of curves and surfaces is an adequate beginning. Additional information will be provided by the project staff.

"Bishop Coaster"; SPI: Ben Farmer; AM: Michael Pelsmajer, graduate student UIUC Math Dept; and Prof. Paul McCreary, Xavier University, New Orleans;

As we learned in Advanced Calculus, the Frenet frame along a differentiable space curve consists of a unit tangent vector, the unit normal vector pointing towards the center of curvature, and their cross-product, the binormal. At inflection points and along straight sections of the 3D track the Frenet frame is undefined. For these and other reasons, it yields to a more robust, less temperamental frame invented by Richard Bishop. This project investigates the relationship between these two framings and explores applications of the Bishop Frame to common problems in 3D-geometry.

"carniBeats, soniBlui"; SPI: Ben Shanbaum; AM: Michael Pelsmajer, graduate student UIUC Math Dept; and Prof. Guy Garnett, UIUC Music Department.

Two complementary solutions to a problem posed by Guy Garnett on how to explore sonic space in the CAVE. The first is an application of the Bishop Frame algorithms to a carneval ride with a series of swiveling space shuttles attached to a 3D-Lissajou based track. The second is a sonification of the gesture based 3D-painting RTICA, ``Alaska Blui'', by Chris Hartman.

"Quaternions"; SPI: Robert Shuttleworth; AM: George Francis.

This RTICA implements an efficient quaternion-to-matrix, and matrix-to-quaternion translation algorithm, and applies it to compare a quadratic Bezier spline to succesive SLERPs in a camera path.

"Silhouette"; SPI: Doug Nachand; AM: Prof. John Hart, UIUC CS Dept.

"Phillips and DeWitt Eversions"; SPI: Doug Nachand; AM: George Francis.

There remains several sphere eversions that have not yet been made computational. Two (technically related) of these are the Tony Phillips eversion (Scientific American, 1966) which is the first published one, and Bryce DeWitt (maybe non-) eversion, which was proposed (but never "proved") in 1967 at the same Batelle conference which spawned the Froissart-Morin eversion. Both eversions are based on horizontal slices (plane curves) of a surface undergoing a regular homotopy.

"Schprel: Special Relativity Project"; SPI: Mark Flider; AM: George Francis.

The purpose of this project is to show the apparent distortion of objects to a moving observer when the speed of light is slow enough to become non-negligible in relativistic physics. It was developed as an RTICA (Real-Time Interactive Computer Animation) for Prof. George Francis' Math 198 and is now being developed in the NCSA's CAVE and the Beckman Institute's CUBE.