Andrew J. Hanson's Home Page

Computer Science Department School of Informatics OnCourse CL IU -- OneStart System Indiana University, Bloomington See also my page in the Indiana University Cognitive Science Program. Here is my one-page biographical sketch (postscript), (pdf), as well as a more detailed two-page biographical sketch (postscript) (pdf). Address: Andrew J. Hanson Professor and Chair Lindley Hall 215 Indiana University Bloomington, IN 47405 U.S.A. Chair's Office (812) 855-4510, FAX (812) 855-4829 hanson (at) cs.indiana.edu,   hansona (at) indiana.edu

Courses:

B581, Graduate Computer Graphics       Fall semester, 2008:

This is an OpenGL-based course on the mathematical foundations and practical interactive methods of modern interactive graphics. The course emphasizes creating interactive interfaces to visualize the graphics objects and techniques being studied. Lighting and simple material modeling are covered as an introduction to the creation of realistic images.

Overview of B581.

Publications

Visualizing Quaternions, is now published (Morgan-Kaufmann/Elsevier, 2006, ISBN 978-0-12-088400-1). This is a comprehensive approach to the significance and applications of quaternions, and focuses on the exploitation of Quaternion Fields, a tool developed primarily by the author. The official website for the book is maintained by the publisher, and provides background material, downloadable material from tables, and demonstration software. I maintain a local companion website here, which may have more recent updates pending upload to the official site. Updates will be accumulated on the update page. An example is the closed form double-reflection quaternion form   q = ( A · B, A × B ) inadvertantly omitted from the Clifford Algebra treatment in Chapter 31. Visualizing Relativity using complexified quaternions was part of the material covered by Andrew Hanson and Daniel Weiskopf in their Siggraph 2001 Course 15 Notes. The Solar Journey DVD contains an educational computer animated film on the astronomy of the local neighborhood of the Earth and the Sun developed as part of our NASA-sponsored research work. The DVD version containing the Solar Journey animation and supplementary science materials is distributed by Finley-Holiday Films at the web site linked above.

Recent Bibliography. (Way out of date...working on it.)

Newly available in machine-readable form: Constrained Hamiltonian Systems, by Hanson, Regge, and Teitelboim, originally published in 1976 by the Accademia Nazionale dei Lincei (Contributi del Centro Linceo Interdisc. di Scienze Matem. e loro Applic., No.22, Accademia Nazionale dei Lincei, Rome, 135 pages (1976)).

Media, including download information for the MeshView Four-Dimensional Viewer. Supported fully under X-windows/Motif only. Precompiled for Linux, Macintosh, SUN SOLARIS, and SGI IRIX. Recently available: reduced functionality Windows XP version. The shortcuts work in the Windows version, but you need to look at the Linux version to see what they are.

Research

My most recent research focuses on several areas: Mathematical Visualization, Virtual Reality, and Astronomy.

• Mathematical Visualization:
  

  A Tessellation for Algebraic Surfaces in CP3. This work presents an explicit algorithm for tessellating the algebraic surfaces (real 4-manifolds) F(n) embedded in CP3 defined by the equation   z0n + z1n + z2n + z3n = 0   in the standard homogeneous coordinates [z0, z1, z2, z3], where n is any positive integer. Note that F(4) in particular is a K3 surface. Thus this method provides what is essentially an elegant 24-point tessellation of K3. The arXiv posting is here: 0804.3218, and a local copy can be found here.

  Dual Five-Point Function Geometry. This work investigates the geometry of two-complex-variable contour integration using the classic 1960's Dual 5-Point Function of the early string model as the critical example. The investigation actually began with some questions introduced in a very early paper, Dual N-Point Functions in PGL(N-2,C)-Invariant Formalism (A. J. Hanson, Physical Review, 1972). A number of new insights are given in our paper "A Contour Integral Representation for the Dual Five-Point Function and a Symmetry of the Genus Four Surface in R6" by Andrew J. Hanson and Ji-Ping Sha, which is published in J. Phys. A: Mathematics and General., vol. 39, pages 2509-2537 (2006). There is also a version on the arXiv, math-ph/0510064. A local copy can be found here.

  Other topics. Various long-term projects deal with techniques for modeling, depicting, and interacting with geometric structures of extreme complexity. Subject domains of interest range from mathematical objects in four dimensions to exploiting quaternions to represent orientation fields of geometric objects. Recent work concerns rephrasing some of the classical differential geometry of curves and surfaces directly in terms of quaternion fields; an application is the determination of optimal framings of curves and surfaces by minimizing appropriate energies of the quaternion frame fields ("quaternion Gauss maps") in the 3-sphere.



• User Interfaces for Virtual Reality Applications: Work with Eric Wernert on constrained navigation methods for desktop and immersive Virtual Reality systems appear in the Proceedings of IEEE Visualization '97 and '99. Browse our Vis97 paper and our Vis99 paper.



• Scientific Visualization in Astronomy: The Solar Journey project was undertaken with NASA support (NAG5-8163, NAG5-11999) in collaboration with the University of Chicago beginning in 1999. This has led to a number of projects in virtual astronomy. An example of our work on star rendering may be found in this QuickTime movie depicting simulated stars, which compares favorably to real images such as Akira Fujii's Orion.

  A Transparently Scalable Visualization Architecture for Exploring the Universe, TVCG, Jan/Feb 2007, is a full description of work done mainly by Chi-Wing Fu in my laboratory. This framework supports transparent interactive navigation across enormous scale ranges such as those naturally occurring in astronomy. Paper and web site.

  Scalable WIM: Effective Exploration in Large-scale Astrophysical Environments, TVCG, Sept/Oct 2006, 12, pp. 1005-1011; Proceedings of IEEE Visualization 2006. Describes a World-in-Miniature interface design for astrophysical exploration whose development was led by Yinggang Li in my laboratory. Paper and web site.

  Approach to the Black Hole at the Galactic Core. This animation represents an assembly of data collected by astronomers all over the world to examine the surroundings of the Black Hole suspected to be present at the center of our Milky Way galaxy. A variety of scientific instruments, using different methods and different wavelengths have been utilized to get many orders of magnitude of image precision. This permits us to make a continuous fly-in starting from the familiar constellations and going all the way in to the stars actually orbiting the Black Hole itself.

  The Sun's interaction with its environment. This image of the heliosphere, representing the interaction of the Solar wind with the surrounding interstellar material, is taken from our short film "Solar Journey;" an extended version of the film will be produced for public distribution on videotape and DVD during the coming year. The shapes depicted here utilize a theoretical model by Timur Linde from the University of Chicago. The image has appeared as the Astronomy Picture of the Day, APOD 2002 June 24, and was used as an illustration in a recent astronomy news article in Science Magazine, page 2005 of Vol. 300, 27 June 2003. (The image credit is very obscure, in tiny vertically-aligned print along the spine of page 2005.)

  Satellites in our Sky (GMT June 24th 2003 2:21pm). We have over a thousand satellites flying through the sky over our heads. This image is from a brief animation representing a user's interaction with our Earthday graphics program. The animation shows a large portion of these at a selected time, and then zooms in for a closeup of the International Space station (ISS). We can clearly see the ring structure of geo-stationary (deep-space) satellites rotating with the Earth, located 38,500km above the Earth's surface (about 6 times the radius of the Earth). The entire animation appeared as the Astronomy Picture of the Day on 14 July 2003. See APOD 2003 July 14.

  
      Our initial work on handling very large scales of spacetime in interactive virtual reality environments is described in our paper, Very large scale visualization methods for astrophysical data, which appears in Proceedings of Joint Eurographics-IEEE TVCG Symposium on Visualization, May 29-31, 2000, Amsterdam, the Netherlands. This paper is part of the published proceedings, © Springer-Verlag.

  Our most recent work, to appear in TVCG, January 2007, describes the maturation of the scaling framework developed in Philip Chi-Wing Fu's PhD thesis, and is entitled "A Transparently Scalable Visualization Architecture for Exploring the Universe." For details, see the Scaling Framework Web Site.

I have also created a variety of graphics images derived from the Fermat Equation (see below) that are relevant to the Calabi-Yau spaces that may lie at the smallest scales of the unseen dimensions in String Theory; these have appeared in Brian Greene's books, The Elegant Universe and The Fabric of the Cosmos,, and in the book by Callender and Huggins, Physics Meets Philosophy at the Planck Scale. The writhing purple shapes in the October/ November 2003 NOVA production Elegant Universe, as well as the cover of the November 2003 Scientific American, were derived from software models I supplied to the NOVA graphics providers.

These images show equivalent renderings of a 2D cross-section of the 6D manifold embedded in CP4 described in string theory calculations by the homogeneous equation in five complex variables:
z1⁵ + z2⁵ + z3⁵ + z4⁵ + z5⁵ = 0
The surface is computed by assuming that some pair of complex inhomogenous variables, say z3/z5 and z4/z5, are constant (thus defining a 2-manifold slice of the 6-manifold), normalizing the resulting inhomogeneous equations a second time, and plotting the solutions to
z1⁵ + z2⁵ = 1
The resulting surface is embedded in 4D and projected to 3D using Mathematica (left image) and our own interactive MeshView 4D viewer (right image). If you have CosmoPlayer, you can also interact with this VRML version of the quintic Calabi-Yau cross-section.

In the right-hand image, each point on the surface where five different-colored patches come together is a fixed point of a complex phase transformation; the colors are weighted by the amount of the phase displacement in z1 (red) and in z2 (green) from the fundamental domain, which is drawn in blue and is partially visible in the background. Thus the fact that there are five regions fanning out from each fixed point clearly emphasizes the quintic nature of this surface.

For further information, see: A.J. Hanson. A construction for computer visualization of certain complex curves. Notices of the Amer.Math.Soc., 41(9):1156-1163, November/December 1994.

Arbitrary Genus Surfaces:

This image shows my computer graphics construction of a four-hole torus described by an equation in complex two-space given by H. Blaine Lawson, "Complete Minimal Surfaces in S³," Ann. of Math. 92, pp.~335--374 (1970), with m = n = 2,

Im z1^{(m + 1)} + |z2|^(m-n) Im z2⁽ⁿ⁺¹⁾ = 0

and

|z1|² + |z2|² = 1

In general, the genus is m*n, and this surface is not actually minimal in S³ except for
m = n = 0 and m = n = 1.

Review article

Cover picture: IEEE Computer 27 (July 1994)

For more information about mathematical visualization in general, see the Web version of the review article Interactive Methods for Visualizable Geometry, by A.J. Hanson, T. Munzner, and G. Francis, published in IEEE Computer 27 , No. 7, pp. 73--83 (IEEE Computer Society Press, Los Alamitos, CA, July, 1994).

Mathematics and Physics Animations

We have produced a number of short video animations with mathematical and physical content. Some of my favorite projects are the following:

Cosmic Clock:
Observing the Universe using the finite speed of light to place measured objects in their correct temporal context.

The Cosmic Bloom excerpt from the movie as available here; it seems to play OK on PC's with QuickTime, but has troubles on some other platforms.

This 3:35 minute animation contains a visualization of the entire Universe from three different points of view: the time spectrum of observable photon radiation arriving at the earth, the constant-time shells of light sources represented in "comoving coordinates " (as though the Universe had always been the size it is today), and in "physical coordinates" (which incorporate the Hubble expansion since the "Great Flash", when the Universe was about 300,000 years old). This film was one of a select few chosen for showing at the Siggraph 2000 Electronic Theater in July 2000, and appears in Siggraph Video Review 134, Scene 5 (2000).

4Dice: A Glimpse into the 4th Dimension

This short (1:00 minute) animation of a back-face-culled 4D die or hypercube, which has eight sphere-containing cubes as the analogs of faces, was shown at the Siggraph '95 Computer Animation Festival and appears in Siggraph Video Review 114, Scene 14 (1995).

Visualizing Fermat's Last Theorem: (NOTE: the old links are invalid, pointing to a retired server, from which we are attempting (unsuccessfully) to restore this data at this time.)

Fermat's last theorem was proven at last in 1995 by Andrew Wiles, but the mystique lives on. This film was made in 1990, when it was still unclear that Fermat's conjecture would ever actually be proven --- in a way it was more romantic when we could believe that, wasn't it?

The film was shown in the Siggraph '90 Animation Screening Room, and appears in Siggraph Video Review 61, Scene 4 (1990).