Applications of Morse Theory and Catastrophe Theory to Computer Graphics

莫尔斯理论和突变理论提供了实函数临界点邻域的有力模型。在适当的条件下，这些模型实际上能够识别函数域的拓扑类型。该项目扩展了发现和跟踪关键点以保证隐式曲面多边形的拓扑类型的软件，并将其应用于计算机图形学的其他领域。这些技术可以解决当前计算机图形学的各种开放问题。最直接的应用是拓扑正确的隐式曲线参数化。通过提供在高度函数的Morse临界值上和之间在表面渲染和二维切片之间切换的能力，该技术可以帮助实现复杂流形浸入的可视化。该技术可以在转换过程中保持形状的拓扑类型，例如，将甜甜圈转换为咖啡杯的中间形状都由具有一个孔的单个组件组成。对一个临界点的邻域建模也应该有助于在一个属的变化中保持一致的纹理映射。在数据压缩算法中结合临界点产生的数据压缩可以避免改变重构数据拓扑类型的错误。类似的技术可以确保复杂几何的简化以更快的显示，而不会牺牲其拓扑类型的定性性质。最后，将突变理论应用于全局照明将产生一个更快的算法来确定焦散反射区域。

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.