About My Master's Thesis

(I've added a link to my thesis. There aren't images with the text though, so have a look at the ones on this page and the ones linked to from here.)

For my master's thesis , I developed several methods of modeling continuous- and discrete-time, chaotic mathematical systems as surfaces. Using these methods, I developed a virtual reality research tool, the Fractal Explorer , that was shown at SIGGRAPH, '92 , at the unveiling of the CAVE virtual environment, also developed at EVL.

One of the methods I use in the Fractal Explorer is mathematical and is used to create this surface model of the Lorenz attractor. This model is derived mathematically from the set of ordinary differential equations that define the attractor system by a process of linearizing the system and then finding the eigenvectors of that linearized version. One of the eigenvectors represents "spreading", and is considered to lie tangent to the "surface". Also tangent to the "surface" is the vector along the trajectory. Their cross product yields a surface normal that is used for lighting.

Another surface-modeling method used in the Fractal Explorer is data-based and was used to create this ribbon that models part of the Lorenz attractor. The ribbon has been released from the CAVE's wand, and is being drawn into the attractor. The ribbon's surface is built from two trajectories of the attractor as they are generated, and is parallel to the direction in which the attractor tends to spread out -- the attractor's "surface".

The Fractal Explorer also allows mathematicians to explore fractals. This is a polyhedral model of a Sierpinski tetrahedron, the attractor of a three-dimensional iterated function system (IFS) (special thanks to EVL graduate student Alan Verlo and EVL alumnus Dr. John Hart). The Explorer allows the user to interactively transform the contraction mappings that generate the attractor in order to change the attractor's shape. Here, virtual spheres, designed to allow users to manipulate structures in the Explorer, have been used to twist the fractal on every level of its structure.

Other Images from Master's Thesis

Another Lorenz attractor image

Yet another Lorenz attractor image

Chua's double scroll attractor

Roessler attractor