This talk is, naturally, on the web at http://www.cs.princeton.edu/~burchard/cs/viz/drei96/.
Dynamical systems has been an area of mathematical research that has drawn enormous inspiration from interactive computation. Nonetheless, numerical computations have seldom led to proofs about the properties of dynamical systems because estimating the errors in calculations of trajectories is exceedingly difficult. We shall describe a novel approach to proving many properties of low dimensional dynamical systems with methods that rely upon transversality arguments and local coordinate transformations.
What does one mean by the phrase "the shape of space"? What are some possible shapes for our own 3-dimensional universe, and how might we determine which is correct? The talk will include a showing of the prize-winning Geometry Center video The Shape of Space, which takes the viewer on trips through universes which are finite, yet have no boundaries.
Rated G: No specific mathematical background is required, and students at all levels are encouraged to attend.
Newtonian mechanics completely describes a mechanical system's short term behavior. The system's long term behavior may be visualized in the so-called configuration and phase spaces. The talk will provide a gentle introduction to these spaces, using software developed for Bolt, Beranek and Newman's high school curriculum project "Teaching Mathematical Thinking Through Computer-Aided Space Explorations".
Rated G: No prior knowledge of mechanics is assumed.
Over the last few years there has been considerable interest in virtual environments and immersive technologies for conceptual design and evaluation. The immersive hardware offers us great potential to view conceptual design in full size and in an appropriate context. Short of building full-scale mechanical models, virtual environments appear to be the only methods which can provide this essential design evaluating capability. This outlines a number of challenges in terms of mathematical algorithms and software technology.
In this talk we briefly overview our recent work on interactive walkthrough and evaluation of large CAD models. We highlight algorithms and systems for model-building, display and interaction and utilize visualization techniques to demonstrate their performance. This include Boolean combination of NURBS models, interactive display on current serial and parallel graphics systems, visibility culling and use of textures. The resulting algorithms utilize a number of techniques from computational geometry, geometric and solid modeling, symbolic and numeric computing, physics and computer graphics. We demonstrate their application to walkthrough of architecture and submarine models.
I'll show highlites of this year's Video Review of Computational Geometry, including the future classic Four-Polytopes and a Funeral (for my conjecture), by Tamara Munzner and myself. I'll talk about making the video and about the future of the Video Review. I'll also show my new Cartoon Guide to Linear Programming Duality.
Surface equilibrium and motion problems are of interest both as mathematical problems and for their applicability to materials science. Motion problems include motion by weighted mean curvature, as in slow melting or annealing of materials; motion by surface diffusion, as for pores in a sintering ceramic, and dendritic crystal growth, as when metal is cast. A general program for motion of polygonal curves modelling all these motions has been developed (primarily by Andy Roosen). For surfaces, a program has been developed only for motion by weighted mean curvature and some of its variants. Visualization of results from these programs have contributed significantly to our understanding of these problems and our ablility to describe the work to others. Related work includes the use of Brakke's Surface Evolver to provide evidence for conjectures on the shapes of bodies of given volume minimizing various edge energies (Jenny Kelley) and to compute the torque exerted by a drop of water between two cylinders to make them line up (Karen Almgren). Videos on all these problems will be shown.
Brakke's Surface Evolver takes a polyhedral surface and moves it to minimize some geometric energy functional. By default this is surface area, but we have also modified it to study Willmore surfaces, knot energies, and other problems. The Evolver is well integrated with geomview for interactive graphics, and can also be used with RenderMan to create high-quality images of soap films. Various features allow us to work with just a single fundamental unit of a symmetric surface.
We are primarily interested in studying 4-dimensional space by looking at the surfaces in 4-space. We use visualization tools and mathematical tools. The focus of mathematical research is on topological knotting of surfaces.
We discuss our use of visualization tools using 3D graphics for calculus of several variables. This is not a simple matter of ``making illustrations with the computer''. In doing this it is important to understand the significance of interactive graphics and also the general nature of visualization.
Work of the last decade in the interaction between algebraic geometry and theoretical physics has increased interest in problems involving families of complex algebraic curves with additional structure (such as curves with special line bundles or vector bundles on these curves). The behavior of these bundles, as singularities develop in the underlying curves, has physical as well as mathematical significance. Even relatively simple questions are difficult to answer (for example, whether or not one of these curve-line bundle pairs can be deformed into any other such pair -- i.e., whether the space parameterizing all these pairs is connected). We present a technique to construct a triangle mesh on an implicit surface defined by a function of two complex variables. The technique, Contour Meshing, generalizes well to larger dimensions. We explore a family of surfaces with an interactive 4D viewer to view the neighborhood of a singularity.
This is joint work with Chris Weigle and Tyler Jarvis.
The coordinate frame concept of classical three-dimensional geometry admits a representation in terms of the group SU(2), which is in turn representable as the manifold S3 of unit quaternions. A number of classical calculations for curves and surfaces can be reformulated in this quaternion frame space. We describe some general properties of curve framings in terms of quaternions, and present a quaternion version of the Frenet frame as an example. We explore also a curve's induced frame map in quaternion space, which exposes many essential features. The extension of the treatment to surfaces produces a suggestive quaternion generalization of the classical Gauss map. We note several applications of quaternion frame fields to the visualization of curves and surfaces, to the representation of anisotropic shading in computer graphics, and to constrained navigation in virtual reality environments.
Oorange is a response to the "Babel" dilemma: too often others cannot share the fruits of a mathematical program due to incompatible software environments. Oorange is intended as a common frame on which to conduct mathematical experiments. Its hybrid design tries to combine the best features of several existing systems. Here is some of the "infrastructure" that it provides:
More information: http://www_sfb288.math.tu-berlin.de/oorange.html
This talk describes a Web-based algorithm animation system. Our system augments the expressive power of Web pages for publishing passive multimedia information with a full-fledged interactive algorithm animation system. It improves on previous Web-based algorithm animations by providing a framework that makes it easy to construct new animations, including those that involve multiple views. Because views of the same running algorithm may reside on different machines, our system is particularly well-suited for electronic classrooms. This strategy is an improvement over the electronic classroom systems we are aware of, which simply display the same X window on multiple machines. We believe our framework generalizes to electronic textbooks in arbitrary domains.
Software is invisible, disappearing into files on disks. The invisible nature of software can contribute to low programmer productivity by hiding system complexity, particularly for large team-oriented projects. Visualization can help software engineers cope with this complexity and thereby increase programmer productivity. I will describe several visual representations of software that scale to production-sized systems and illustrate their use in a number of software case studies involving: version history, program comparison, static properties of code, performance profiles, trace analysis, and dynamic program slices.
A triangulation of P, a set of n points in the plane, is a decomposition of the convex hull of P into triangles whose set of vertices is exactly P. A point set P may admit many triangulations; a minimum-weight triangulation (MWT) is one in which the sum of edge lengths is minimum. No general, polynomial-time algorithm for MWTs is known.
Keil and Dickerson have independently suggested an approach to identify edges that are possible in some MWT and certain in any MWT, and conjectured that the edges identified in this way are expected to form a connected graph. We discuss a fast implementation of their method, its installation in the drawing program "ipe", and some of the resulting counterexamples and conjectures.
This is joint work with Michael McAllister and Patrice Belleville.
Convex shapes are easiest to represent, manipulate, and render. Even though they form the building blocks of bottom-up solid modelers, it is more often the case that the convex structure of a geometric shape is lost in its representation. This talk addresses the problem of decomposing a complex polyhedral surface into a small number of ``convex'' patches (ie, boundary parts of convex polyhedra). The corresponding optimization problem is shown to be NP-complete and an experimental search for good heuristics is described as well as visualized.
Address structures are a natural generalization of both the self-similar structure of the real numbers, and group representations. These structures have many interesting connections and are perhaps of utility in a variety of concrete and abstract applications. Recently, such structures were used to provide matching rules for a very wide class of aperiodic hierarchical tilings.
In this talk I would cover animation of computational geometry algorithms (e.g., convex hull, Delaunay triangulation).
The World-Wide Web and hypertext provide a uniquely rich environment for the exposition of mathematics, both at a research and an educational level. In this talk we demonstrate some of their possibilities by viewing an interactive research paper that presents a recent solution to a 35-year-old problem in the geometry of surfaces. Tha author will explain the key points of the question and its answer while indicating how the hypertext format adds to the readers understanding of the paper. This talk should be accessable to the non-specialist.
Trellis is a display technique for multivariate data. It depends primarily on the notion of conditioning, that is, showing many simultaneous views of two variables while carefully varying the values of the other variables from one display to the next. It is surprisingly effective in revealing hidden structure, perhaps because it is so well tuned to both the sorts of questions that are of interest to scientists when analyzing data, and to the sorts of perceptual tasks that humans do well. In this talk I will describe Trellis, primarily through examples.