Annals of Mathematics and Artificial Intelligence 36: 437–457, 2002.  2002 Kluwer Academic Publishers. Printed in the Netherlands. Efficient querying and animation of periodic spatio-temporal databases ∗ Peter Revesz a and Mengchu Cai b a Department of Computer Science and Engineering, University of Nebraska–Lincoln, Lincoln, NE 68588, USA E-mail: revesz@cse.unl.edu b IBM Silicon Valley Lab., San Jose, CA 95141, USA E-mail: mcai@us.ibm.com We propose a representation of spatio-temporal objects with continuous and cyclic or acyclic periodic movements. We also describe an extended relational algebra query language for databases with such objects. We show that the new spatio-temporal databases are closed under the extended relational algebra queries, and each fixed relational algebra query can be evaluated in PTIME in the size of the input database. Keywords: spatio-temporal databases, animation, data model, relational algebra AMS subject classification: H.2.1, H.2.3 1. Introduction Many spatio-temporal objects such as clouds, cars, deserts, lakes, planets, ships and tornados change position or shape continuously and also sometimes periodically. Al- though in the last decade substantial research was done independently in spatial [26,36] and temporal [31] data modeling, continuously and periodically changing objects re- quire new data models that can capture the interdependency of the spatial and temporal extents of these objects. We introduce a new model for continuously and possibly periodically moving, growing or shrinking objects. The intuition for our model is that at any time instance the shape of a moving, growing or shrinking planar object can be approximated by the union of a set of rectangles whose sides are parallel to the axes. Further, as the object changes the rectangles also change. Such changing rectangles can be described as rectangles whose x and y dimensions are functions of time. Therefore, we call such rectangles parametric rectangles. We can also generalize from 2 to higher dimensional parametric rectangles (or rectangular bodies). While the shape of objects can be approximated by other ways, for example, by general polygons, the simplicity of the rectangular representation has several clear ad- ∗ This work was supported in part by NSF Grants IRI-9625055 and IRI-9632871 and by a Gallup Research Professorship.