Creating a Tool to Demonstrate Hyperbolic Geometry And Its Uses for Data Structures

In this paper, we will show how hyperbolic geometry can be presented to computer science students by using technology and a graphical approach. We have developed an applet by interfacing Visual Basic and Mathematica that can be used as a graphics calculator in understanding concepts studied in hyperbolic geometry. In particular, the applet draws hyperbolic lines, hyperbolic triangles, animates transformations in hyperbolic geometry, and computes hyperbolic distances and measurements of hyperbolic triangles. We will point out that the Poincare disk, a model for hyperbolic geometry, is a natural medium on which one can lay out a data structure such as a tree or directed graph. As we know, the leaves and nodes in a tree may represent links to databases, URL addresses, or other kinds of data. Since a tree may have several nodes or leaves, and consequently contain huge amounts of information, we will discuss how transformations in hyperbolic geometry allow a user to traverse the branches, nodes, leaves of a tree, and even provide a user different visual perspectives of the data structure.