Morse theory and catastrophe theory provide powerful models of the neighborhood of a critical point of a real function. Under suitable conditions these models are in fact capable of discerning the topological type of the domain of the function. This project extends software that finds and tracks critical points to guarantee the topological type of an implicit surface polygonization, and applies it to other areas of computer graphics. These techniques can solve a variety of currently open problems of computer graphics. The most immediate application is a topologically-correct implicit curve parameterization. The techniques can aid in the visualization of complicated manifold immersions by providing the ability to switch between surface renderings and 2-D slices on and between Morse critical values of a height function. The techniques can maintain the topological type of a shape during transformation, such that for example the intermediate shapes in the transformation of a donut into a coffee cup all consist of a single component with one hole. Modeling the neighborhood of a critical point should also aid in maintaining a consistent texture mapping across a change of genus. Incorporating critical points in a data compression algorithm yields a data compression that avoids errors that alter the topological type of the reconstructed data. Similar techniques can insure that the simplification of complex geometry for faster display does not sacrifice the qualitative nature of its topological type. Finally, the application of catastrophe theory to global illumination will yield a faster algorithm for determining regions of caustic reflection.
