Measures each cell on the sphere: its area, the length of its boundary and how close its shape is to a circle.
Value
Data frame with one row per cell_id: cell_id,
area_km2 (as cell_area gives it),
normalized_area, perimeter_km, compactness and
ipq.
Details
The perimeter follows each wall as the grid draws it. An ISEA wall is straight on its face of the solid and curved on the sphere, so it is followed on the sphere until its length is exact to about 1e-8. An H3 wall is a great-circle arc between corners, with an extra corner where it crosses an edge of the icosahedron.
Compactness (White et al. 1998) is the perimeter of a spherical cap with the cell's area over the cell's perimeter, \(\sqrt{4\pi a - a^2/r^2} / p\) for area \(a\), perimeter \(p\) and radius \(r\). A circle scores 1; a small regular hexagon 0.952, pentagon 0.930, square 0.886 and triangle 0.778.
The normalized area is the cell's area over the mean area of the grid's
cells, the body's surface over the cell count; an equal-area grid of
\(N\) cells gives \(N / (N - 2)\) for its hexagons, 5/6 of that for
the pentagons of the icosahedron and 4/6 for the squares of the octahedron. The isoperimetric quotient
ipq is \(4\pi a / p^2\), for a circle 1, a small regular hexagon
0.907, square 0.785 and triangle 0.605. Kmoch et al. (2022) compare
grids by these two measures, reading area and perimeter in an equal-area
plane centred on each cell; here both are read on the sphere, which adds
\((a/(rp))^2\) to compactness^2, a few parts in a million for
cells of a few thousand square kilometres.
References
White, D., Kimerling, A. J., Sahr, K. and Song, L. (1998). Comparing area and shape distortion on polyhedral-based recursive partitions of the sphere. International Journal of Geographical Information Science 12(8), 805-827. doi:10.1080/136588198241518
Kmoch, A., Vasilyev, I., Virro, H. and Uuemaa, E. (2022). Area and shape distortions in open-source discrete global grid systems. Big Earth Data 6(3), 256-275. doi:10.1080/20964471.2022.2094926
See also
wall_metrics for the spacing of neighbouring cells,
cell_area