Second areal moment (i.e., second moment of inertia)

Computes the second moment of area (also known as the second moment of inertia) for polygon geometries. This is a measure of how the area of a shape is distributed relative to its centroid.

Usage

second_areal_moment(x, project = TRUE)

Arguments

x: An sf object, sfc geometry column, or sfg geometry.
project: Logical. If the geometries have geodetic coordinates, then they will be projected using an Albers Equal Area Conic projection centered on the data.

Value

Numeric vector of second areal moments

Details

The second moment of area is the sum of the inertia across the x and y axes:

The inertia for the x axis is: $$I_x = \frac{1}{12}\sum_{i=1}^{N} (x_i y_{i+1} - x_{i+1}y_i) (x_i^2 + x_ix_{i+1} + x_{i+1}^2)$$

While the y axis is in a similar form: $$I_y = \frac{1}{12}\sum_{i=1}^{N} (x_i y_{i+1} - x_{i+1}y_i) (y_i^2 + y_iy_{i+1} + y_{i+1}^2)$$

where $x_i, y_i$ is the current point and $x_{i+1}, y_{i+1}$ is the next point, and where $x_{n+1} = x_1, y_{n+1} = y_1$.

For multipart polygons with holes, all parts are treated as separate contributions to the overall centroid, which provides the same result as if all parts with holes are separately computed, and then merged together using the parallel axis theorem.

The code and documentation are adapted from the PySAL Python package (Ray and Anselin, 2007). See Hally (1987) and Li et al. (2013) for additional details.

References

Hally, D. 1987. "The calculations of the moments of polygons." Canadian National Defense Research and Development Technical Memorandum 87/209. https://apps.dtic.mil/sti/tr/pdf/ADA183444.pdf

Li, W., Goodchild, M.F., and Church, R.L. 2013. "An Efficient Measure of Compactness for Two-Dimensional Shapes and Its Application in Regionalization Problems." International Journal of Geographical Information Science 27 (6): 1227–50. doi:10.1080/13658816.2012.752093.

Rey, Sergio J., and Luc Anselin. 2007. "PySAL: A Python Library of Spatial Analytical Methods." Review of Regional Studies 37 (1): 5–27. https://doi.org/10.52324/001c.8285.

Examples

library(sf)
#> Linking to GEOS 3.13.0, GDAL 3.8.5, PROJ 9.5.1; sf_use_s2() is TRUE
poly <- st_polygon(list(matrix(c(0,0, 1,0, 1,1, 0,1, 0,0), ncol=2, byrow=TRUE)))
second_areal_moment(poly)
#> [1] 0.1666667