We now turn to a discussion of selected visualization tasks and the approaches that have been used to carry them out. We first consider the depiction of surfaces or 2-manifolds, beginning with familiar surfaces such as the Klein bottle; we are led immediately to consider surfaces in 4D space as well as in 3D because the self-intersections of many classical surfaces in 3D disappear when they are represented in 4D coordinates. We then proceed to a discussion of volumes or 3-manifolds, methods for exploiting 4D lighting models, and the visualization of non-Euclidean geometry. Finally, we turn to surfaces that obey geometric constraints such as minimizing an energy functional, and then to the question of surfaces that change in time subject to the constraint that the curvature remains finite at every point; deformations obeying this constraint, the so-called regular homotopies, play a special role in the development of descriptive topology.

Thu Sep 21 19:17:33 CDT 1995