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 ). https://doi.org/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.

中文翻译：

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关于定性形状组合的空间推理

形状组合是空间推理中的一个挑战。定性形状描述符 (QSD) 已被证明是旋转和位置不变的，这使得它们在空间推理测试中非常有用。QSD 使用角度和长度的定性表示，但它们的合成操作之前没有定义。本文详细介绍了角定性模型（QMAngles）和长度定性模型（QMLengths）的数量、参考系统和算子。他们的算子是参考 Allen (Commun. ACM 26 (11), 832–843 (1983 ) 的著名时间模型定义的。https://doi.org/10.1145/182.358434) 作为参考。此外，建立了组合表，并使用它们的几何对应物证明了定性角度和长度的组合关系。