gueting@fernuni-hagen.de, schneide@fernuni-hagen.de

Praktische Informatik IV, FernUniversität Hagen, Postfach 940, D-5800 Hagen, Germany

**Abstract:** Spatial data types or algebras for database systems should (i) be fully general (which means, closed under set operations, hence e.g. a region value can be a set of polygons with holes), (ii) have formally defined semantics, (iii) be defined in terms of finite representations available in computers, (iv) offer facilities to enforce geometric consistency of related spatial objects, and (v) be independent of a particular DBMS data model, but cooperate with any. We offer such a definition. A central idea is to use *realms* as geometric domains underlying spatial data types. A realm as a general database concept is a finite, dynamic, user-defined structure underlying one or more system data types. A geometric realm defined here is a planar graph over a finite resolution grid. Problems of numerical robustness and topological correctness are solved below and within the realm layer so that spatial algebras defined above a realm enjoy very nice algebraic properties. Realms also interact with a DBMS to enforce geometric consistency on object creation or update. The ROSE algebra is defined on top of realms and offers general types to represent point, line, and region features together with a comprehensive set of operations. It is described within a polymorphic type system and interacts with a DBMS data model and query language through an abstract *object model interface*. An example integration of ROSE into the object-oriented data model O2 and its query language is presented.

**Keywords:** Spatial data types, algebra, realm, finite resolution, numerical robustness, topological correctness, geometric consistency, object model interface, ROSE.

**Published:** *VLDB Journal 4 (1995)*, 100-143.