Computer Animation and the Geometry of Surfaces in 3- and 4-Space
Thomas Banchoff · 2010
Geometers have always used any available media to help them illustrate their work with diagrams, pictures, and models. Modern computer graphics provides a new medium with great potential both for teaching and research. Older methods of representing curves and surfaces by drawings on a blackboard or models in wire or plaster are frequently found to be inadequate in many important geometric problems, specifically those which involve objects undergoing transformations or objects which exist properly in the fourth dimension or higher. A high-speed graphics computer makes it possible to approach and solve such problems by methods which were unavailable only a few years ago. Producing 30 different pictures per second, such a computer can display on a television tube a sequence of images which the viewer readily interprets as the projections of an object rotating in 3-dimensional space. By turning dials, a mathematician can investigate a curve or surface by having it rotate about different axes and stopping it at especially interesting positions. He or she can fly inside the object to focus on some local behavior or proceed to examine some specific singularity by deforming the object through a one-parameter family of curves or surfaces.