Distortion of a grid's face projection

Tissot's indicatrix of the projection that maps the sphere onto the faces of the grid's solid (Snyder's equal-area projection, or Fuller's): a small circle on the sphere maps to an ellipse on the face, with semi-axes a and b times the circle's radius.

Usage

projection_distortion(x, lon, lat)

Arguments

x

A HexGridInfo object from hex_grid with an ISEA grid. H3 is built on a gnomonic projection of its own and is not covered.

lon, lat

Longitudes and latitudes in degrees.

Value

A data frame with one row per point: lon, lat, face (from 0), the scale factors a (largest) and b (smallest), the maximum angular deformation angular \(2 \arcsin((a - b) / (a + b))\) in degrees, the areal scale areal \(a b\), and angle, the direction of the longer axis on the face plane in degrees anticlockwise from the face's x axis.

Details

The scale factors are the singular values of the projection's derivative, computed exactly by forward-mode automatic differentiation of the projection code. Face-plane lengths are measured on the plane triangle that has the face's area, so the areal scale a * b is 1 everywhere under Snyder's projection, and its mean over a face is 1 under Fuller's.

Snyder's projection bends along the arcs from each face's centre to its corners, where its derivative jumps; there and on face edges the result is the derivative on one side. Its angular deformation is largest at the face centre, approached along those arcs: 17.27 degrees, with scale factors 1.163 and 0.860 (Snyder 1992, Table 1). At a face centre the projection has a different derivative along each direction, and the result is the limit along the azimuth the point's coordinates give.

References

Snyder, J. P. (1992). An equal-area map projection for polyhedral globes. Cartographica 29(1), 10-21. doi:10.3138/27H7-8K88-4882-1752

van Leeuwen, D. and Strebe, D. (2006). A "slice-and-dice" approach to area equivalence in polyhedral map projections. Cartography and Geographic Information Science 33(4), 269-286. doi:10.1559/152304006779500687

See also

plot with distortion and tissot to map it

Examples

g <- hex_grid(resolution = 3, aperture = 3)
projection_distortion(g, c(0, 16.37), c(0, 48.21))

gf <- hex_grid(resolution = 3, aperture = 3, projection = "fuller")
d <- projection_distortion(gf, runif(1000, -180, 180),
                           asin(runif(1000, -1, 1)) * 180 / pi)
range(d$areal)