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\s+2ABSTRACT\s0
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In the field of scientific visualization, the term
``four dimensional visualization'' usually refers to the process of
rendering a three dimensional field of scalar values.  While this
paradigm applies to many different data sets, there are also uses
for visualizing data that correspond to actual four-dimensional
structure.  Four dimensional structures have typically been
visualized via wireframe methods, but this process alone is usually
insufficient for an intuitive understanding; all but the most simple
datasets easily overwhelm the viewer.  This research covers the
visualization of four dimensional objects through wireframe methods
with extended visualization cues, and through raytracing methods.
Both the wireframe and raytracing methods employ true four-space
viewing parameters and geometry.
The raytracing approach easily solves the hidden surface and shadowing
problems of 4D objects, and yields an image in the form of a
three-dimensional field of RGB values, which can be rendered with a
variety of existing methods.  The 4D raytracer also supports true
four-dimensional lighting, reflections and refractions.
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\s+2ACKNOWLEDGEMENTS\s0
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In appreciation of the time, effort and patience extended to me in
the pursuit of this thesis, I would like to thank my committee chairman,
Dr. Thomas Foley.  In addition to his guidance in the topic of this
research, he has also been unfailing in helping me get the resources
and information I needed to complete both this research and this paper.
I would also like to acknowledge and thank Dr. Gregory Nielson and Dr.
Gerald Farin for being on my thesis committee and offering their
observations and suggestions.
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Thanks are also due to Kevin Price for letting me use his Amiga and
helping me produce animations of the raytrace data that aided my
understanding of the process and nature of four-dimensional raytracing.
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Finally, I would like to thank my family for their patience, confidence,
enthusiasm and financial support.
