Topological relations between directed lines and simple geometries

Abstract

Directed lines are fundamental geometric elements to represent directed linear entities. The representations of their topological relations are so different from those of simple lines that they cannot be solved exactly with normal methods. In this paper, a new model based on point-set topology is defined to represent the topological relations between directed lines and simple geometries. Through the intersections between the start-points, end-points, and interiors of the directed lines and the interiors, boundaries, and exteriors of the simple geometries, this model identifies 5 cases of topological relations between directed lines and points, 39 cases of simple lines, and 26 cases of simple polygons. Another 4 cases of simple lines and one case of simple polygons are distinguished if considering the exteriors of the directed lines. All possible cases are furthermore grouped into an exclusive and complete set containing 11 named predicts. And the conceptual neighborhood graph is set up to illustrate their relationship and similarity. This model can provide a basis for natural language description and spatial query language to present the dynamic semantics of directed lines relative to the background features.