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by Rodney R. Gayle
Nestled in a small corner of southeast Minneapolis, just off Interstate 35W, lies the Geometry Center. Its location has kept one of the University's best assets hidden, but the technology emerging from the Center is bringing it into the limelight.
The Geometry Center, funded by the National Science Foundation (NSF), was established in 1991 for the purpose of promoting and developing visualization and communication techniques for mathematics and related sciences. In accord with the NSF's directives, the Center focuses on software development, video production, and the integration of technology into the classroom.
In addition to these objectives, the Geo-metry Center invites professors and researchers from around the world to participate in its research and software development programs. Researchers travel to the University of Minnesota to work individually, collaborate with others, or attend conferences and workshops hosted by the Geometry Center. They consult with the Center's technical staff and have access to state-of-the-art equipment. This unique environment allows workshops and conferences at the Geometry Center to have a strong focus on software use and development and facilitate interaction and computer exploration.
The staff at the Geometry Center includes the director, mathematics professor Richard McGehee, six technical and six administrative professionals, and a small number of post-doctoral fellows, graduate students, and student programmers.
The foundation of the Geometry Center's visualization efforts lies in a software package called Geomview, an interactive three-dimensional (3D) program that the staff wrote to enhance the capabilities of existing mathematics and engineering software. Commercial mathematics software, such as Mathematica and Maple, can perform complex calculations but lack an extensive interactive 3D graphics package. Conversely, engineering programs such as AutoCAD provide sophisticated 3D graphics but are not designed for mathematical problems. Geomview fills this gap by offering advanced 3D graphics that work with data-generating programs.
Geomview differs from standard computer-aided design (CAD) packages because it only handles the display of data. When used alone, Geomview allows the user to view a geometric object from various angles and change its location and dimensions, but it does not include functions to construct or otherwise operate on that object. Geomview links to both commercial and custom-made products, however, so that the data generated from those programs can be displayed.
One recent product based on Geomview is a spacecraft design program called Crafter, developed by the Center's staff for use by scientists at NASA's Jet Propulsion Laboratory (JPL) in Pasadena, California. The scientists at JPL needed a way to quickly create approximate 3D models of spacecraft for use in making animations during the planning stage of a mission. Using the traditional CAD packages that create the actual blueprints was too awkward. Crafter, while not able to construct detailed specifications, facilitates the creation of a simplistic model needed for demonstrating the initial draft of a mission.
Another recent project at the Center is a program for visualizing satellite orbits. Several communications companies are interested in launching "constellations" of low earth orbit satellites to allow telephones and computers to communicate with each other directly through satellites, rather than routing through ground wires or cellular stations. The goal of this system is to be able to connect any two points on the earth with a series of satellite connections. This network routing problem is complicated because in order to be close enough to the earth to allow rapid communication, the satellites must move with respect to the surface of the earth. The particular route chosen for a certain connection thus changes quickly with respect to time. Another type of satellite orbit, called geosynchronous, arranges for a satellite to remain over one spot on the ground continuously. However, because of the large distance from the surface of the earth, the geosynchronous satellites would not be practical for interactive telecommunications.
Patrick Worfolk and Robert Thurman, two Geometry Center postdoctoral fellows, have written a program called SaVi (Satellite Visualization) which allows planners to view constellations of satellite orbits in motion. This program also allows the planners to compute various statistics about the orbits, such as the amount of ground each satellite covers.
In addition to visualization, the other major component of the Geometry Center's objectives is to facilitate the communication of mathematics and science. The Center had one of the very first Web sites and has achieved international recognition for its innovative use of the Web to present interactive mathematical materials. WebEQ, a system which allows researchers to put mathematical equations in their Web pages, is a current project related to this effort. Standard HTML, the language in which Web pages are written, does not provide for mathematical equations with special symbols, superscripts, or subscripts. Using Java, a new platform-independent programming language designed for use over the Internet, Geometry Center staff have developed a working prototype of a system which enables mathematical equations to appear on Web pages. Although WebEQ is not intended to be the final answer to the problem of mathematical typesetting on the Web, it is a key experiment in the process of determining that solution.
For more information about Geomview, Crafter, WebEQ, SaVi, or other projects at the Geometry Center, visit the Center's Web site at http://www.geom.umn.edu.
