A ternary Relation Algebra of directed lines

We define a ternary Relation Algebra (RA) of relative position relations on two-dimensional directed lines (d-lines for short). A d-line has two degrees of freedom (DFs): a rotational DF (RDF), and a translational DF (TDF). The representation of the RDF of a d-line will be handled by an RA of 2D orientations, CYC_t, known in the literature. A second algebra, TA_t, which will handle the TDF of a d-line, will be defined. The two algebras, CYC_t and TA_t, will constitute, respectively, the translational and the rotational components of the RA, PA_t, of relative position relations on d-lines: the PA_t atoms will consist of those pairs <t,r> of a TA_t atom and a CYC_t atom that are compatible. We present in detail the RA PA_t, with its converse table, its rotation table and its composition tables. We show that a (polynomial) constraint propagation algorithm, known in the literature, is complete for a subset of PA_t relations including almost all of the atomic relations. We will discuss the application scope of the RA, which includes incidence geometry, GIS (Geographic Information Systems), shape representation, localisation in (multi-)robot navigation, and the representation of motion prepositions in NLP (Natural Language Processing). We then compare the RA to existing ones, such as an algebra for reasoning about rectangles parallel to the axes of an (orthogonal) coordinate system, a ``spatial Odyssey'' of Allen's interval algebra, and an algebra for reasoning about 2D segments.