50 Motion

Motion is ubiquitous in the physical world, yet its study is much less developed than that of another common physical modality, namely shape. While we have several standardized mathematical shape descriptions, and even entire disciplines devoted to that area–such as Computer-Aided Geometric Design (CAGD)—the state of formal motion descriptions is still in flux. This in part because motion descriptions span many levels of detail; they also tend to be intimately coupled to an underlying physical process generating the motion (dynamics). Thus, until recently, proper abstractions were lacking and there was only limited work on algorithmic descriptions of motion and their associated complexity measures. This chapter aims to show how an algorithmic study of motion is intimately tied to discrete and computational geometry. After a quick survey of earlier work (Sections 50.2 and 50.3), we devote the bulk of this chapter to discussing the framework of Kinetic Data Structures (Section 50.4) [Gui98, BGH99]. We also briefly discuss methods for querying moving objects (Section 50.5).