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## MA595 Fall 2015 Mini-Course in Geometry:

COMPUTER GRAPHICS & GEOMETRICAL VISUALIZATION

Lawson's Snail. Crossed eye stereo left two, straight stereo right two images.
Professor
George Francis

1MWF 102 Altgeld Hall
First 8 weeks, 2 credit hours.

Text: Francis, A Topological Picturebook, Springer Paper Back, 2006.

We present the geometry of computer graphics, emphasizing
real-time interactive computer animation
(RTICA)
for
mathematical visualization, in particular for an immersive
virtual environment, the Cube, CAVE, and CANVAS
of the Illinois Simulator Lab (ISL)
of the Beckman Institute.
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. The course also includes
a survey of classical topics including binocular
optics and color theory, Haussdorff dimension and fractals, chaos and
strange attractors, Wolfram's cellular automata, Barnsley's iterated
function systems, Julia and Mandelbrot sets, discrete and continuous
logistic equations, and the Lienard-VanderPol dynamical system.

Prospective students should have a good spatial intuition,
some artistic abilities or ambitions, and a solid grounding in
linear algebra and vector calculus. Students may participate in
a concurrent tutorial, on useful line and surface graphics tools
for the classroom and the dissertation,
that does not require programming. Students with experience programming in
any computer language, such as BASIC, Pascal, C/C++, Java, Python,
or Mathematica, may gain 2 additional credits of independent study
for a graphics programming project
appropriate to the course and tailored to the proficiency
of the student. In particular, projects suitable for a virtual
environment at
ISL are not only supported
but strongly encouraged.