The inversion of directions is an important operation with directions which plays an important role in qualitative spatial reasoning and spatial queries. In this work, we address on the inversion operation of the basic cardinal direction relations in the model of Goyal. The direction relation matrix model proposed by Goyal is a projection-based model for spatial direction relations between regions. This model is simple in calculation and easy to carry out formal reasoning, which is considered as currently one of the most excellent models for representation and qualitative reasoning with cardinal direction relations in two-dimensional space. This work aims to realize the automatic inference and calculation of the inverse of the basic cardinal direction relations in the model of Goyal and further to improve the ability of spatial reasoning and spatial analysis of spatial database. In order to avoid the complicated manual reasoning, an algorithm for automatically performing the inverse operation on this model is devised by means of the operations of direction relation matrix. Theorems are provided to prove formally that our algorithm is correct and complete, which is also verified by comparing the result of our algorithm with that of manual reasoning for each basic cardinal direction relation. This study realized the automatic inference and calculation of the inverse of the basic cardinal direction relations in the model of Goyal and further improved the ability of spatial reasoning and spatial analysis of spatial database.

Aims: A Comprehensive Model Incorporating Multiple Spatial Relations in 3D Space. Background: At present, the research on two-dimensional spatial relation expression and inference models is relatively mature, but these models cannot be used to deal with three-dimensional spatial orientation relations. With the application of spatial orientation relations, three-dimensional spatial orientation relations are involved in many fields such as urban architectural design, robotics, image processing, etc. Two-dimensional spatial orientation relations models cannot satisfy the needs of three-dimensional spatial applications, so there is an urgent need to research three-dimensional spatial object orientation relations expression and inference models. Objective: This work aims to provide a comprehensive model incorporating multiple spatial relations in 3D space. The joint representation of direction, distance, and topological relations describes more complex spatial position relations among objects. Methods: Based on this comprehensive model, the computational properties of interval algebra are used to combine the directional and topological relations. Result: The study lays a good foundation for the formal representation and reasoning of spatial relations between regions, enhances the analyzability of spatial relations between objects, and improves the accuracy of spatial analysis. Conclusion: The main novel contribution of this paper is that we propose a comprehensive orientation relation model, called 3D-TRD, which considers three spatial orientation relations simultaneously. The paper gives examples to represent the position relations of two spatial objects by comparing the RCC8 model, the 3DR46 model, and the comprehensive model to highlight the advantages of our proposed model. Based on the model, the bidirectional mapping association method is also used to represent the location of the spatial objects. The first advantage of the 3D-TRD model is that it represents spatial location relations more accurately than 3DR46, RCC8, and five qualitative distances. The second advantage of the 3D-TRD model is that it proposes a bidirectional mapping representation in three-dimensional space. The third advantage of the 3D-TRD model is that it provides a good basis for the formal representation and inference study of the spatial relations between regions.