“…In this higher dimensional setting, the calculus is based on the transformations of the so-called Frenet-Serret frames [28]. Other applications of the double-cross formalism and its generalisation OPRA n include the modelling of traffic situations [29,30].…”
Section: Introduction and Summary Of Resultsmentioning
Abstract:We study the double-cross matrix descriptions of polylines in the two-dimensional plane. The double-cross matrix is a qualitative description of polylines in which exact, quantitative information is given up in favour of directional information. First, we give an algebraic characterization of the double-cross matrix of a polyline and derive some properties of double-cross matrices from this characterisation. Next, we give a geometric characterization of double-cross similarity of two polylines, using the technique of local carrier orders of polylines. We also identify the transformations of the plane that leave the double-cross matrix of all polylines in the two-dimensional plane invariant.
“…In this higher dimensional setting, the calculus is based on the transformations of the so-called Frenet-Serret frames [28]. Other applications of the double-cross formalism and its generalisation OPRA n include the modelling of traffic situations [29,30].…”
Section: Introduction and Summary Of Resultsmentioning
Abstract:We study the double-cross matrix descriptions of polylines in the two-dimensional plane. The double-cross matrix is a qualitative description of polylines in which exact, quantitative information is given up in favour of directional information. First, we give an algebraic characterization of the double-cross matrix of a polyline and derive some properties of double-cross matrices from this characterisation. Next, we give a geometric characterization of double-cross similarity of two polylines, using the technique of local carrier orders of polylines. We also identify the transformations of the plane that leave the double-cross matrix of all polylines in the two-dimensional plane invariant.
“…However, to obtain certainty that these calculi indeed violate R 4 , one has to find concrete counterexamples and verify them using the original definition of the respective calculus. For DRA f and INDU, this has been done in the literature [23,3]. Interestingly, the violation of associativity has been attributed to the absence of strong converse and strong composition, respectively.…”
Section: Algebraic Properties Of Existing Calculimentioning
confidence: 97%
“…In addition, weak composition implies that each cell contains exactly those t. If composition is strong, then Rel and ϕ even have to ensure that whenever ϕ(t) intersects with ϕ(r) • ϕ(s), it is contained in ϕ(r) • ϕ(s) -i.e., the composition of the interpretation of any two base relations has to be the union of interpretations of certain base relations. [35] directions from a point 9 P a Variant DRAc is not based on a weak partition scheme -JEPD is violated [23].…”
Qualitative spatial and temporal reasoning is based on socalled qualitative calculi. Algebraic properties of these calculi have several implications on reasoning algorithms. But what exactly is a qualitative calculus? And to which extent do the qualitative calculi proposed meet these demands? The literature provides various answers to the first question but only few facts about the second. In this paper we identify the minimal requirements to binary spatio-temporal calculi and we discuss the relevance of the according axioms for representation and reasoning. We also analyze existing qualitative calculi and provide a classification involving different notions of relation algebra.
“…Examples for such RST calculi are the double-cross (Freksa, 1992), the flip-flop (Ligozat, 1993) (also called in Scivos and Nebel (2004), as shown in Figure 9) and the Ternary Point Configuration Calculus (TPCC; Moratz & Ragni, 2008). However, it has been proved (Wolter & Lee, 2010) that a qualitative calculus expressive enough to distinguish from left to right of including flip-flop (Ligozat, 1993), double-cross (Freksa, 1992), dipole (Moratz et al ., 2000), (Moratz, 2006) and TPCC (Moratz & Ragni, 2008), existing relation algebraic approach is too weak for deciding consistency problems and all reasonable sub-algebras remain NP-hard, that is, directional relation calculi are inherently intractable.Figure 9Pictorial illustration of seven base relations in the calculus when . l, f, r, b and c stand for left, further, right, back and closer (Scivos & Nebel, 2004: 285)…”
Section: Aspects Of Qualitative Spatial Relationsmentioning
Representation and reasoning with qualitative spatial relations is an important problem in artificial intelligence and has wide applications in the fields of geographic information system, computer vision, autonomous robot navigation, natural language understanding, and spatial databases etc. The reasons for this interest in using qualitative spatial relations include cognitive comprehensibility, efficiency and computational facility. This paper summarizes progress in qualitative spatial representation by describing key calculi representing different types of spatial relationships. The paper concludes with a discussion of current research and glimpse of future work.
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