Spatial reasoning about qualitative shape compositions Composing Qualitative Lengths and Angles

Shape composition is a challenge in spatial reasoning. Qualitative Shape Descriptors (QSD) have proven to be rotation and location invariant, which make them useful in spatial reasoning tests. QSD uses qualitative representations for angles and lengths, but their composition operations have not been defined before. In this paper, the Qualitative Model for Angles (QMAngles) and the Qualitative Model for Lengths (QMLengths) are presented in detail by describing their arity, reference systems and operators. Their operators are defined taking the well-known temporal model by Allen (Commun. ACM 26(11), 832-843 (1983). 10.1145/182.358434) as a reference. Moreover, composition tables are built, and the composition relations of qualitative angles and lengths are proved using their geometric counterparts. The correctness of these composition tables is also proved computationally using a logic program implemented using Swi-Prolog.