Computational topology and swept volumes

Some fundamental principles of computational topology are formulated and discussed. It is demonstrated that swept volumes comprise a class of geometric objects that is particularly well suited to the application and development of the advanced mathematics/computer science based foundations of computational topology. Two new results in this vein are presented: (1) every smooth swept volume is proved to have a Whitney regular stratification; and (2) it is shown how homology can be used to characterize surface intersections. Applications of the dynamical systems approach to swept volumes in virtual sculpting and modeling of heterogeneous material such as bones are also described.