Representing 3D Topological Adjacencies between Volumes Using a 36-Intersection Model

Abstract

Topological properties of objects should be maintained and preserved to concisely represent objects. However, the implementation of 2D topological rules requires the decomposition of 3D objects into lower dimensions to determine topological relationships. This results in 2D topological relationships although the connected objects are in 3D. Hence, accurate representation of 3D connectivity in 3D models is limited. 3D topological rules can be implemented to include topological connectivity in 3D space. This paper implemented an extension of the 27-Intersection Model (27-IM) called the 36-Intersection Model (36-IM) to represent 3D topological adjacencies of two objects in 3D space. This resulted in a 12 × 3 intersection matrix or 36-IM that represented the intersections in terms of dimension and number of separations. Six cases were tested, consisting of “meets”, “disjoint” “intersects at a line”, “intersects at a point”, “contains”, and “overlaps”. The resulting 36-IM matrices provided an accurate representation of how the objects in 3D space were related and their dimension of intersections. The formalisms of the 36-IM matrices were also interoperable which allowed the interpretation of 36-IM using the 9IM and DE-9IM to determine general topological relationships. By examining the intersection of interiors, boundaries and exteriors of 3D objects without object decomposition, 3D topological relationships can be determined as well as the dimension and manner of intersection.