Qualitative Description and Reasoning of Topological Relation in Three-Dimensional GIS

Topological relation is invariant under continuous transformations of space, which incarnates the global characteristics of entity. The topological relation is an important spatial relationship in GIS. The three-dimensional space can be described with the simplex model, and a complex entity can be divided into simplexes whose dimensions are equal to or less than the dimension of entity in GIS. The topological relations among 0-simplix (point), 1-simplex (line), 2-simplex (triangle), and 3-simplex (tetrahedron) in the three-dimensional GIS are described with the 9-intersition matrix model. Topological relations that the 9-intersection matrix can classify are 11 topological relations between lines, 13 topological relations between line and triangle, 11 topological relations between line and tetrahedron, 18 topological relations between triangles, 9 topological relations between triangle and tetrahedron, and 8 topological relations between tetrahedron. Base on the set theory, the qualitative reasoning of topological relation is studied, whose results are listed in the composition table.