and revisit the theory of relation algebras and non-associate algebras underlying qualitative spatial reasoning. Furthermore, we investigate quotient of calculi on a general level as well as for the dipole calculi. Section 3 provides a condensed semantics for the dipole calculus. A condensed semantics, as we name it, provides spatial domain knowledge to the calculus in the form of an abstract symbolic model of a specific fragment of the spatial domain. In this model, possible configurations of very few of the basic spatial entities of a calculus are enumerated. In our case, we use orbits in the affine group GA(R
2). This provides a useful abstraction for reasoning about qualitatively different configurations in Euclidean space. We use affine geometry at a rather elementary level and appeal to pictures instead of complete analytic arguments, whenever it is easy to fill in the details - however, at key points in the argument, careful analytic treatments are provided. Further, we calculate the composition tables for the dipole calculi using the condensed semantics and we investigate properties of the composition.

In Section 4 we answer the question whether the standard constraint resoning method algebraic closure decides consistency for the dipole calculi. After the presentation of the technical details of dipole calculi and some of their properties, a sample application of dipole calculi using a spatial reasoning toolbox is presented in Section 5. The example uses the reasoning capabilities of a dipole calculus based on constraint reasoning. Our paper ends with a summary and conclusion and pointers to future work.

## 2 Representation Of Dipole Relations And Relation Algebras

In this section, we first present a set of spatial relations between dipoles, then variants of this set of spatial relations. The final subsection shows mathematical structures for constraint reasoning about dipole relations.

## 2.1 Basic Representation Of Dipole Relations

The basic entities we use are dipoles, i.e. oriented line segments formed by a pair of two points, a start point and an end point. Dipoles are denoted by *A, B, C, . . .*,
start points by sA and end points by eA, respectively (see Fig. 1). These dipoles are used for representing spatial objects with an intrinsic orientation. Given a set of dipoles, it is possible to specify many different relations of different arity, e.g. depending on the length of dipoles, on the angle between different dipoles, or on the dimension and nature of the underlying space. When examining different relations, the goal is to obtain a set of jointly exhaustive and pairwise disjoint atomic or *base* relations, such that exactly one relation holds between any two dipoles. The elements of the powerset of the *base* relations are called *general* relations. These are used to express uncertainty about the relative position of dipoles. If these relations form an algebra which fulfills certain requirements, it is possible to apply standard constraint-based reasoning mechanisms that