Applications of Morse Theory and Catastrophe Theory to Computer Graphics

莫尔斯理论和突变理论提供了真实的函数临界点邻域的强有力的模型。在适当的条件下，这些模型实际上是能够辨别的功能域的拓扑类型。这个项目扩展了寻找和跟踪临界点的软件，以保证隐式曲面的拓扑类型，并将其应用于计算机图形学的其他领域。这些技术可以解决计算机图形学的各种当前开放的问题。最直接的应用是拓扑正确的隐式曲线参数化。该技术可以通过提供在高度函数的莫尔斯临界值上和之间在表面渲染和2-D切片之间切换的能力来帮助复杂的流形浸入的可视化。该技术可以在变换期间保持形状的拓扑类型，使得例如在甜甜圈到咖啡杯的变换中的中间形状全部由具有一个孔的单个组件组成。对临界点的邻域进行建模也应该有助于在属的变化中保持一致的纹理映射。在数据压缩算法中消除临界点产生避免改变重构数据的拓扑类型的错误的数据压缩。类似的技术可以确保简化复杂的几何图形以获得更快的显示不会牺牲其拓扑类型的定性性质。最后，应用突变理论的全球照明将产生一个更快的算法来确定焦散反射的区域。

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.