Qualitative spatial reasoning for orientation relations in a 3-D context

Our previous work focuses on how the nine tiles in the 2-D projection-based model for cardinal directions can be partitioned into sets based on horizontal and vertical constraints (called Horizontal and Vertical Constraints Model). In this paper, the 2-D Horizontal and Vertical Constraints model is adapted and extended into a 3-D Horizontal and Vertical Constraints Block model so that it facilitates easy reasoning with 3-D volumetric regions (i.e. without holes and single-pieced) in the real physical world (e.g. intelligent robotics, building construction, etc…). This model partitions a 3-D Euclidean space of a 3-D reference region into 9 blocks, namely, left, middlex, right, above, middley, below, left, middlez, right. The additional central block (or the Minimum Bounding Box of the 3-D reference region) is an intersection of the three blocks, namely, middlex, middley, and middlez. The added value of the 3-D Horizontal and Vertical Constraints Block model is the use of intuitive (i.e. commonsense) knowledge representation for 3-D orientation relations. However, the underlying formal representation of the model is facilitated through the use of the 3-D Cartesian Coordinate system, first order logic, and boolean algebraic expressions. The novel contribution of this research work is fostering reasoning with partial orientation relation related knowledge (note: these are called weak relations) and also integrating mereology into the 3-D model in order to render the representation of the model more expressive. Finally, composition of relations is the technique employed in this research to general new knowledge. Mereology is integrated into the model in order to render the model more expressively. Finally, several examples will demonstrate how the model could be used to make inferences about 3-D orientation relations.