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2007 Math Abstracts

2007 Research Fair - Math Abstracts

Adaptation of the Ray Tracing Computer Graphics Algorithm
to Visualize Hyperbolic Three Dimensional Space
by Christopher Davis
(Faculty Sponsor:  Richard Wellman)

Linear functions on a Cartesian coordinate system are generally used when parameterizing the paths which light follows. This makes intuitive sense and essentially states that light follows Euclidian lines. But if one selects another parameterization for the path of light other, less familiar conceptualizations can be seen.

One example of an alternative parameterization can be found in the Poincare disk, a common construction of hyperbolic geometry. In this construction, the universe is defined as the interior of a sphere. Lines are arcs of circles which intersect this sphere at right angles or diameters of it. If one can define light to follow just such a line, a new set of images can be visualized in which many of the familiar results of Euclidian geometry fail to hold.

Following this construction for hyperbolic light, one must introduce objects into the space; otherwise there is nothing to be seen. Initially one could take this universe and place in it Euclidian objects, described in the usual manner, like planes, triangles, and spheres. With some added complexity of construction and algorithm design one can add analogous hyperbolic objects which belong natively in this space. In this project I have done both. The Ray tracer can now visualize these shapes in this new space.

The images that result from these objects have a warped, bent appearance. From this abstract feeling of distortion one can gain a degree of intuition for what it means to exist in hyperbolic space, and how much the assumption of the Euclidian parallel postulate impacts the field of graphics.