  We have developed a Java-based teaching tool, "Appreciating Hubble at
Hyper-speed" ($\textit{AHaH}$), intended for use by students and instructors in
beginning astronomy and cosmology courses, which we have made available online.
This tool lets the user hypothetically traverse the Hubble Ultra Deep Field
(HUDF) in three dimensions at over $\sim500\!\times\!10^{12}$ times the speed
of light, from redshifts $z\!=\!0$ today to $z\!=\!6$, about 1 Gyr after the
Big Bang. Users may also view the Universe in various cosmology configurations
and two different geometry modes - standard geometry that includes expansion of
the Universe, and a static pseudo-Euclidean geometry for comparison. In this
paper we detail the mathematical formulae underlying the functions of this Java
application, and provide justification for the use of these particular
formulae. These include the manner in which the angular sizes of objects are
calculated in various cosmologies, as well as how the application's coordinate
system is defined in relativistically expanding cosmologies. We also briefly
discuss the methods used to select and prepare the images in the application,
the data used to measure the redshifts of the galaxies, and the qualitative
implications of the visualization - that is, what exactly users see when they
"move" the virtual telescope through the simulation. Finally, we conduct a
study of the effectiveness in this teaching tool in the classroom, the results
of which show the efficacy of the tool, with over $\sim$90% approval by
students, and provide justification for its further use in a classroom setting.
