This vignette describes what spherical geometry implies, and how package sf uses the s2geometry library (https://s2geometry.io) for geometrical measures, predicates and transformations.

Spatial coordinates either refer to projected (or Cartesian) coordinates, meaning that they are associated to points on a flat space, or to unprojected or geographic coordinates, when they refer to angles (latitude, longitude) pointing to locations on a sphere (or ellipsoid). The flat space is also referred to as R2, the sphere as S2.

Package sf implements simple features, a standard for point, line, and polygon geometries where geometries are built from points (nodes) connected by straight lines (edges). The simple feature standard does not say much about its suitability for dealing with geographic coordinates, but the topological relational system it builds upon (DE9-IM) refer to R2, the two-dimensional flat space.

Yet, more and more data are routinely served or exchanged using geographic coordinates. Using software that assumes an R2, flat space may work for some problems, and although sf up to version 0.9-x had some functions in place for spherical/ellipsoidal computations (from package lwgeom, for computing area, length, distance, and for segmentizing), it has also happily warned the user that it is doing R2, flat computations with such coordinates with messages like

although coordinates are longitude/latitude, st_intersects assumes that they are planar

hinting to the responsibility of the user to take care of potential problems. Doing this however leaves ambiguities, e.g. whether LINESTRING(-179 0,179 0)

  • passes through POINT(0 0), or
  • passes through POINT(180 0)

and whether it is * a straight line, cutting through the Earth's surface, or * a curved line following the Earth's surface

Starting with sf version 1.0, sf uses the new package s2 (Dunnington, Pebesma, Rubak 2020) for spherical geometry, which has functions for computing pretty much all measures, predicates and transformations on the sphere. This means:

  • no more hodge-podge of some functions working on R2, with annoying messages, some on the ellipsoid
  • a considerable speed increase for some functions
  • no computations on the ellipsoid (which are considered more accurate, but are also slower)

The s2 package is really a wrapper around the C++ s2geometry library which was written by Google, and which is used in many of its products (e.g. Google Maps, Google Earth Engine, Bigquery GIS) and has been translated in several programming other languages.

Fundamental differences

Compared to geometry on R2, and DE9-IM, the s2 package brings a few fundamentally new concepts, which are discussed first.

Polygons on S2 divide the sphere in two parts

On the sphere (S2), any polygon defines two areas; when following the exterior ring, we need to define what is inside, and the definition is the left side of the enclosing edges. This also means that we can flip a polygon (by inverting the edge order) to obtain the other part of the globe, and that in addition to an empty polygon (the empty set) we can have the full polygon (the entire globe).

Simple feature geometries should obey a ring direction too: exterior rings should be counter clockwise, interior (hole) rings should be clockwise, but in some sense this is obsolete as the difference between exterior ring and interior rings is defined by their position (exterior, followed by zero or more interior). sf::read_sf has an argument check_ring_dir that checks, and corrects, ring directions and many (legacy) datasets have wrong ring directions. With wrong ring directions, many things still work.

For S2, ring direction is essential. For that reason, st_as_s2 has an argument oriented = FALSE, which will check and correct ring directions, assuming that all exterior rings occupy an area smaller than half the globe:

library(sf)
nc = read_sf(system.file("gpkg/nc.gpkg", package="sf")) # wrong ring directions
library(s2)
s2_area(st_as_s2(nc, oriented = FALSE)[1:3]) # corrects ring direction, correct area:
## [1] 1137107793  610916077 1423145355
s2_area(st_as_s2(nc, oriented = TRUE)[1:3]) # wrong direction: Earth's surface minus area
## [1] 5.100649e+14 5.100655e+14 5.100646e+14
nc = read_sf(system.file("gpkg/nc.gpkg", package="sf"), check_ring_dir = TRUE)
s2_area(st_as_s2(nc, oriented = TRUE)[1:3]) # no second correction needed here:
## [1] 1137107793  610916077 1423145355

Here is an example where the oceans are computed as the difference from the full polygon,

as_s2_geography(TRUE)
## <s2_geography[1]>
## [1] <POLYGON ((0 -90, 0 -90))>

and the countries, and shown in an orthographic projection:

co = s2_data_countries()
oc = s2_difference(as_s2_geography(TRUE), s2_union_agg(co)) # oceans
b = s2_buffer_cells(as_s2_geography("POINT(-30 52)"), 9800000) # visible half
i = s2_intersection(b, oc) # visible ocean
plot(st_transform(st_as_sfc(i), "+proj=ortho +lat_0=52 +lon_0=-30"), col = 'blue')

Half-closed polygon boundaries

Polygons in s2geometry can be * CLOSED: they contain their boundaries, and a point on the boundary intersects with the polygon * OPEN: they do not contain their boundaries, points on the boundary do not intersect with the polygon * SEMI-OPEN: they contain part of their boundaries, but no boundary of non-overlapping polygons is contained by more than one polygon.

In principle the DE9-IM model deals with interior, boundary and exterior, and intersection predicates are sensitive to this (the difference between contains and covers is all about boundaries). DE9-IM however cannot uniquely assign points to polygons when polygons form a polygon coverage (no overlaps, but mostly common boundaries). This means that if we would count points by polygon, and some points fall on shared polygon boundaries, we either miss them (contains) or we count them double (covers); this leads to bias or need for post-processing. Using SEMI-OPEN non-overlapping polygons guarantees that every point is assigned to maximally one polygon in an intersection. This corresponds to e.g. how this would be handled in a grid (raster) coverage, where every grid cell (typically) only contains its upper-left corner and its upper and left sides.

a = as_s2_geography("POINT(0 0)")
b = as_s2_geography("POLYGON((0 0,1 0,1 1,0 1,0 0))")
s2_intersects(a, b, s2_options(model = "open")) 
## [1] FALSE
s2_intersects(a, b, s2_options(model = "closed"))
## [1] TRUE
s2_intersects(a, b, s2_options(model = "semi-open")) # a toss
## [1] FALSE
s2_intersects(a, b) # default: semi-open
## [1] FALSE

Cap, enclosing rectangle

Computing the minimum and maximum values over coordinate ranges, as sf does with st_bbox(), is of limited value for spherical coordinates because

  • small regions covering the antimeridian end up with a huge longitude range
  • regions including a pole will end up with a latitude range not extending to +/- 90

S2 has two alternatives: the cap and the lat_lng_rect:

fiji = s2_data_countries("Fiji")
aa = s2_data_countries("Antarctica")
s2_bounds_cap(fiji)
##        lng       lat    angle
## 1 178.7459 -17.15444 1.801369
s2_bounds_rect(c(fiji,aa))
##     lng_lo    lat_lo    lng_hi    lat_hi
## 1  177.285 -18.28799 -179.7933 -16.02088
## 2 -180.000 -90.00000  180.0000 -63.27066

The cap reports a bounding cap (circle) as a mid point (lat, lng) and an angle around this point. The rect reports the _lo and _hi bounds of lat and lng, as well as its center (_cnt) values. Note that for Fiji, lng_lo being higher than lng_hi indicates that the region covers (crosses) the antimeridian; the lng_cnt values is not the mean of lng_lo and lng_hi.

Switching between S2 and GEOS

The two-dimensional R2 library that was formerly used by sf is GEOS, and sf can be instrumented to use GEOS or sf. First we will ask if s2 is being used by default:

sf_use_s2()
## [1] TRUE

then we can switch it of (and use GEOS) by

sf_use_s2(FALSE)

and switch it on (and use S2) by

sf_use_s2(TRUE)

Measures

Area

library(sf)
library(units)
## udunits system database from /usr/share/xml/udunits
nc = read_sf(system.file("gpkg/nc.gpkg", package="sf"))
sf_use_s2(TRUE)
a1 = st_area(nc)
sf_use_s2(FALSE)
a2 = st_area(nc)
plot(a1, a2)
abline(0, 1)

summary((a1 - a2)/a1)
##       Min.    1st Qu.     Median       Mean    3rd Qu.       Max. 
## -2.638e-04 -1.650e-04 -7.133e-05 -6.448e-05  1.598e-05  2.817e-04

Length

nc_ls = st_cast(nc, "MULTILINESTRING")
l1 = st_length(nc_ls)
l2 = st_length(nc_ls, use_lwgeom = TRUE)
plot(l1 , l2)
abline(0, 1)

summary((l1-l2)/l1)
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0       0       0       0       0       0

Distances

d1 = st_distance(nc, nc[1:10,])
d2 = st_distance(nc, nc[1:10,], use_lwgeom = TRUE)
dim(d1)
## [1] 100  10
dim(d2)
## [1] 100  10
plot(as.vector(d1), as.vector(d2))
abline(0, 1)

summary(as.vector(d1)-as.vector(d2))
##    Min. 1st Qu.  Median    Mean 3rd Qu.    Max. 
##       0       0       0       0       0       0

Predicates

Transformations

References