Decomposing the airspace into segments and passages

We call a portion of airspace segment if it allows at most n vehicles having containment radius r to simultaneously pass through without intruding unavailable space.

The concept of a segment is adopted from the topological concept called pocket, which is the term that defines imperfect holes in computational geometry [9].

The configuration of a segment is determined by a width parameter α, where α=n∙r is the radius of a disc that can fit in the segment. Each segment is connected to the outside only via relatively narrow passages, where a passage is a part of airspace that n vehicles cannot enter or exit at once. Surrounded by passages, each segment can be viewed as a region having limited accessibility from neighboring regions.

Graphical abstract of segment configuration

The figure on the right shows the result of increasing α in segment configurations. The polytope in green represents available airspace where n vehicle with containment radius r can enter its space without touching any obstacles.

Increasing α=0 to α=3∙r results in splitting airspace into three disjoint segments. By tracking how the segment connectivity evolves with parameter α, one can construct a segment graph that encodes the adjacency between segments and passages.

• The figure on the right shows segment configurations and corresponding segment graphs at altitude of 60m and 80m.

• There are a greater number of segments (=40) at ℎ=60. One segment is connected to only about two segments on average, indicating low accessibility within this airspace layer.

• Segments at lower altitudes are generally smaller in size and are scattered throughout the airspace, as the lower airspace is more heavily constrained by geospatial elements.

• In contrast to the lower airspace at ℎ=60, the upper airspace at ℎ=80 has fewer segments (=18) and a higher average degree (=3.1). As the altitude increases, new segments are formed, lower altitude segments merge into larger ones, and lower altitude passages also develop into segments.

• The figure on the right shows the Reeb graph of airspace segments in the case study area.

• The Reeb graph structure clearly shows at which altitude individual segments are formed and connected to other segments. The formation of new segments is most frequent at 50≤h≤60, while the merging of lower-altitude segments to higher-altitude segments is at the highest level at 70≤h≤90.

• It also shows that a high proportion of narrow air passages exists below 70 m and that larger open segments with improved accessibility exist above 110 m.

• Airspace above 130m, which is mostly unobstructed, is rarely affected by the alpha shape filtration, indicating the capability of the proposed method to distinguish topologically complex and simple environment.