CEC Theses and Dissertations


The Dynamics of Complex Surfaces in n-Dimensions Using Computer Graphics

Date of Award


Document Type


Degree Name

Doctor of Philosophy (PhD)


Graduate School of Computer and Information Sciences


S. Rollins Guild

Committee Member

Lee J. Leitner

Committee Member

Gregory Simco


Visualizing the dynamics of n-dimensional graphics is made possible by high speed, high quality computer graphics, and special techniques. One can visualize the dynamics of a complex surface in n-dimensions by differential manifold segmentation theory and special techniques that utilize surface subdivision algorithms. Techniques like collapsibility, decomposability, separation, and object projection allow a complex surface of multivariate composition to be defined in n-dimensions using computer graphics. These techniques look at n-dimensional manifolds as locally Euclidean in that each of its points has some sufficiently small neighborhood that looks like n-dimensional Euclidean space. These techniques recursively subdivide the complex surface into smaller parts until the projection of a part covers at most one pixel on the screen. The intensity of this pixel is set to the average intensity of the corresponding subarea in the parameter range. The part of the surface corresponding to this subrange is then considered to be displayed. The process stops when the whole surface is displayed.

Surfaces in four space exhibit properties that are prohibited in three space. As an example, non-orientable surfaces may be free of self-intersection in four space. The goal of this dissertation is to marry the classical Gaussian models, Euclidean n-space, and Markovian decision process to interactive computer graphics, and to provide a formal geometric foundation for the dynamics of Complex Surfaces in n-dimensions. These structures are used to describe the dimensional shape properties of objects. Various methods such as collapsing, and decomposing are used to make sense of the shapes of objects in a larger dimensional space than the familiar 3 dimensional world. This dissertation describes solutions to several problems associated with manipulating n-dimensional surfaces, and presents visualization techniques for multivariate systems.

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