Splitting Ratios: Metric Details of Topological Line-Line Relations

Konstantinos A. Nedas and Max J. Egenhofer

Within the geographic domain, an important class of problems relies on geometric abstractions in the form of lines where, for instance, transportation networks and trajectories of movements are typically perceived or modeled at such a generalized geometric level. To support querying and computational comparisons, oftentimes multi-resolution models are needed to guide users from coarser to finer details. Within such a setting topological properties are coarse spatial information, whereas metric refinements offer finer details. The 9-intersection distinguishes 33 topological relations between two lines. This paper develops a model that captures metric details for line-line relations through splitting ratios, which are normalized values of lengths and areas of intersections. These ratios apply to the 9-intersection’s nonempty values, thereby providing refinements of topological properties. Three such splitting ratios comprehensively refine 30 of the 33 topological relations: one for the lengths of common paths, one for the partitioning of lines through intersections, and another one for the areas enclosed by two lines with two or more common components. For the remaining three relations—disjoint, meet, and equal—no further metric refinements based on common parts are possible. The splitting ratios are integrated into a compact representation of detailed topological relations, thereby addressing topological and metric properties of arbitrarily complex line-line relations.

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