CHAPTER 4 A Representation of Relationships in Temporal Spaces CHRISTOPHE CLARAMUNT and BIN JIANG 4.1 INTRODUCTION Time is probably one of the most essential and paradoxical concepts that human beings face. Time is always present in our everyday life from the perception of events to the development of human thinking behaviours. However, time is still a difficult concept to describe and formalise as it has no obvious physical characteristics and properties. We can only establish a temporal statement from the observation or prevision of changes. The relationship between time and space is a consequence of the observation of changes as the perception of spatial alterations denotes the existence of time. The representation of time within Geographical Information Systems (GIS) is still an important and expected development to make these systems more suited to the temporal analysis of real-world phenomena. Over the past years, the representation of spatio- temporal data has been extensively discussed by different research communities such as the Artificial Intelligence domain that provides a mathematical foundation to the representation of changes in space (Vieu, 1997), temporal database approaches that develop database models and query languages for the description and manipulation of spatio-temporal objects (Wu et al., 1997) and studies oriented to the temporal extension of current spatial data models within GIS (Langran, 1992; Cheylan and Lardon, 1993; Peuquet, 1994; Frank, 1994; Worboys, 1994; Claramunt and Thériault, 1995 and 1996). This chapter proposes a new reasoning and computational approach that integrates space and time within an integrated temporal space referential. The principles underlying temporal spaces are derived from time geography concepts, spatial and temporal reasoning formalisms. A set of minimal relationships and configurations in a temporal space are identified from the possible combinations of relationships in time and geographical space. Such a model allows the representation and computational study of independent trajectories in space and time. The algebra is illustrated with a case study that outlines some potential benefits of the temporal space model. The remainder of this chapter is organised as follows. The next section briefly reviews current approaches in the combined representation of space and time. Then we introduce temporal and spatial relationships used for the development of our model. The following sections develop the concept of temporal space, propose a formal model for representing relationships in a temporal space, and illustrate the application of the