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2000 Course: Topics in Geometry


Lawson's Snails. Crossed stereo left two, straight stereo right two images.

Professor George Francis
3:00 - 5:00 pm MWF
UIMATH.grafiXlab -- 102 Altgeld Hall, and
NCSA Renaissance Experimental Lab (REL) -- 3414 Beckman Institute

We treat the geometrical basis of computer graphics, emphasizing real-time interactive computer animation (RTICA) and its use in computational science and engineering (CSE) applications, in particular, for mathematical visualization. We will use Unix (Irix and Linux) workstations in the Renaissance Experimental Lab (REL) of the NCSA to program short RTICAs in C/C++/Java, using the OpenGL graphics library. We will use the CAVE totally immersive virtual environment.

Topics include the structure of the OpenGL graphical pipeline, the polyhedral encoding of surfaces as triangular meshes, the geometry of linear and aerial perspective (ligh and shade), the representation of the 3-D affine group in 4-D homogeneous coordinates, the algebra of 3-D rotations in terms of unit quaternions, projective spaces and their Euclidean, spherical and Minkowski (hyperbolic) metrics. We will explore non-Euclidean splines and morphing techniques, real time interactive texture mapping, and other advanced graphics techniques for innovative mathematical application.

Prospective students should have a good spatial intuition, some artistic abilities or ambitions, a solid grounding in linear algebra and 3-D calculus. In addition, it is advisable that students either already have a good command of C/Unix and be prepared to learn some higher geometry, OR program fluently in some language but already have a strong geometrical background. Students will be expected to share their expertise with their colleagues. Credit is based on active participation in the tutorials, timely submission of documented programming assignments, and the completion of a substantial semester projects appropriate to the course and tailored to the proficiency of the student.

The .dvi and .ps files for the early chapters of the M428 classnotes are on the web. They are "old" but possibly useful. On some browsers you can just click on the .ps files and read them on line without downloading them first.