The large configuration space for the rrrr relation is visualized in Fig. 8. The

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other analogous relations which are extremely coarse are llrr, rrll and llll. In many applications, this unwanted coarseness of four relations can lead to problems9. Therefore, we introduce an additional qualitative feature by considering the angle spanned by the two dipoles. This gives us an important additional distinguishing feature with four distinctive values. These qualitative distinctions are parallelism (P) or anti-parallelism (A) and mathematically positive and negative angles between A and B, leading to three refining relations for each of the four above-mentioned relations (Fig. 9).

We call this algebra DRAfp as it is an extension of the fine-grained relation algebra DRAf with additional distinguishing features due to "parallelism". For the other relations, a '+' or '−', 'P' or 'A' respectively, is already determined by the original base relation and does not have to be mentioned explicitly. These base relations then have the same relation symbol as in DRAf .

The introduction of parallelism into dipole calculi not only has benefits in certain applications. The algebraic features also benefit from this extension 9An investigation by Dylla and Moratz into the first cognitive robotics applications of dipole relations integrated in situation calculus [37] showed that the coarseness of DRAf compared to DRAfp would indeed lead to rather meandering paths for a spatial agent.