Abstract. We address a fundamental question concerning spatio-temporal database systems: "What are exactly spatio-temporal queries?" We define spatio-temporal queries to be computable mappings that are also generic, meaning that the result of a query may only depend to a limited extent on the actual internal representation of the spatio-temporal data. Genericity is defined as invariance under transformations that preserve certain characteristics of spatio-temporal data (e.g., collinearity, distance, velocity, acceleration, ...) that are relevant to a database user. These transformations also respect the monotone nature of time. We investigate different genericity classes relative to the constraint database model for spatio-temporal databases and we identify sound and complete languages for the first-order, respectively the computable, queries in these genericity classes.

In the geometric data model [6], spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatiotemporal triangulation of geometric data objects. This algorithm generates new geometric objects that form a partition both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the coordinate system chosen when modelling the spatio-temporal data. We can show that our algorithm works correctly and has a polynomial time complexity (in the number of input triangles and the maximal degree of the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation.

We address a fundamental question concerning spatio-temporal database systems: "What are exactly spatio-temporal queries?" We define spatio-temporal queries to be computable mappings that are also generic, meaning that the result of a query may only depend to a limited extent on the actual internal representation of the spatio-temporal data. Genericity is defined as invariance under groups of geometric transformations that preserve certain characteristics of spatio-temporal data (e.g., collinearity, distance, velocity, acceleration, ...). These groups depend on the notions that are relevant in particular spatio-temporal database applications. These transformations also have the distinctive property that they respect the monotone and unidirectional nature of time.We investigate different genericity classes with respect to the constraint database model for spatio-temporal databases and we identify sound and complete languages for the first-order and the computable queries in these genericity classes. We distinguish between genericity determined by time-invariant transformations, genericity notions concerning physical quantities and genericity determined by time-dependent transformations. This paper is organized as follows. In Section 2, we define spatio-temporal databases, spatio-temporal queries, and the constraint query languages FO and FO + While. In Section 3, we define a number of genericity notions. In Section 4, we present sound and complete first-order query languages for the different notions of genericity. In Section 5, we present sound and complete languages for the computable queries satisfying the different notions of genericity. We end with a discussion in Section 6. DEFINITIONS AND PRELIMINARIESWe denote the set of the real numbers by R and the n-dimensional real space by R n .Throughout this paper, we use the following notational convention. Variables that range over real numbers are

Abstract:In the geometric data model for spatio-temporal data, introduced by Chomicki and Revesz , spatio-temporal data are modelled as a finite collection of triangles that are transformed by time-dependent affinities of the plane. To facilitate querying and animation of spatio-temporal data, we present a normal form for data in the geometric data model. We propose an algorithm for constructing this normal form via a spatio-temporal triangulation of geometric data objects. This triangulation algorithm generates new geometric data objects that partition the given objects both in space and in time. A particular property of the proposed partition is that it is invariant under time-dependent affine transformations, and hence independent of the particular choice of coordinate system used to describe the spatio-temporal data in. We can show that our algorithm works correctly and has a polynomial time complexity (of reasonably low degree in the number of input triangles and the maximal degree of the polynomial functions that describe the transformation functions). We also discuss several possible applications of this spatio-temporal triangulation. The application of our affine-invariant spatial triangulation method to image indexing and retrieval is discussed and an experimental evaluation is given in the context of bird images.