**For a better version of the sf vignettes see** https://r-spatial.github.io/sf/articles/

Simple
features or *simple
feature access* refers to a formal standard (ISO 19125-1:2004)
that describes how objects in the real world can be represented in
computers, with emphasis on the *spatial* geometry of these
objects. It also describes how such objects can be stored in and
retrieved from databases, and which geometrical operations should be
defined for them.

The standard is widely implemented in spatial databases (such as PostGIS), commercial GIS (e.g., ESRI ArcGIS) and forms the vector data basis for libraries such as GDAL. A subset of simple features forms the GeoJSON standard.

R has well-supported classes for storing spatial data (sp) and interfacing to the above mentioned environments (rgdal, rgeos), but has so far lacked a complete implementation of simple features, making conversions at times convoluted, inefficient or incomplete. The package sf tries to fill this gap, and aims at succeeding sp in the long term.

This vignette:

- explains what is meant by features, and by simple features
- shows how they are implemented in R
- provides examples of how you can work with them
- shows how they can be read from and written to external files or resources
- discusses how they can be converted to and from sp objects
- shows how they can be used for meaningful spatial analysis

A feature is thought of as a thing, or an object in the real world, such as a building or a tree. As is the case with objects, they often consist of other objects. This is the case with features too: a set of features can form a single feature. A forest stand can be a feature, a forest can be a feature, a city can be a feature. A satellite image pixel can be a feature, a complete image can be a feature too.

Features have a *geometry* describing *where* on Earth
the feature is located, and they have attributes, which describe other
properties. The geometry of a tree can be the delineation of its crown,
of its stem, or the point indicating its centre. Other properties may
include its height, color, diameter at breast height at a particular
date, and so on.

The standard says: “*A simple feature is defined by the OpenGIS
Abstract specification to have both spatial and non-spatial attributes.
Spatial attributes are geometry valued, and simple features are based on
2D geometry with linear interpolation between vertices.*” We will
see soon that the same standard will extend its coverage beyond 2D and
beyond linear interpolation. Here, we take simple features as the data
structures and operations described in the standard.

All geometries are composed of points. Points are coordinates in a 2-, 3- or 4-dimensional space. All points in a geometry have the same dimensionality. In addition to X and Y coordinates, there are two optional additional dimensions:

- a Z coordinate, denoting altitude
- an M coordinate (rarely used), denoting some
*measure*that is associated with the point, rather than with the feature as a whole (in which case it would be a feature attribute); examples could be time of measurement, or measurement error of the coordinates

The four possible cases then are:

- two-dimensional points refer to x and y, easting and northing, or longitude and latitude, we refer to them as XY
- three-dimensional points as XYZ
- three-dimensional points as XYM
- four-dimensional points as XYZM (the third axis is Z, fourth M)

The following seven simple feature types are the most common, and are for instance the only ones used for GeoJSON:

type | description |
---|---|

`POINT` |
zero-dimensional geometry containing a single point |

`LINESTRING` |
sequence of points connected by straight, non-self intersecting line pieces; one-dimensional geometry |

`POLYGON` |
geometry with a positive area (two-dimensional); sequence of points form a closed, non-self intersecting ring; the first ring denotes the exterior ring, zero or more subsequent rings denote holes in this exterior ring |

`MULTIPOINT` |
set of points; a MULTIPOINT is simple if no two Points in the MULTIPOINT are equal |

`MULTILINESTRING` |
set of linestrings |

`MULTIPOLYGON` |
set of polygons |

`GEOMETRYCOLLECTION` |
set of geometries of any type except GEOMETRYCOLLECTION |

Each of the geometry types can also be a (typed) empty set,
containing zero coordinates (for `POINT`

the standard is not
clear how to represent the empty geometry). Empty geometries can be
thought of being the analogue to missing (`NA`

) attributes,
NULL values or empty lists.

The remaining geometries 10 are rarer, but increasingly find implementations:

type | description |
---|---|

`CIRCULARSTRING` |
The CIRCULARSTRING is the basic curve type, similar to a LINESTRING in the linear world. A single segment requires three points, the start and end points (first and third) and any other point on the arc. The exception to this is for a closed circle, where the start and end points are the same. In this case the second point MUST be the center of the arc, i.e., the opposite side of the circle. To chain arcs together, the last point of the previous arc becomes the first point of the next arc, just like in LINESTRING. This means that a valid circular string must have an odd number of points greater than 1. |

`COMPOUNDCURVE` |
A compound curve is a single, continuous curve that has both curved (circular) segments and linear segments. That means that in addition to having well-formed components, the end point of every component (except the last) must be coincident with the start point of the following component. |

`CURVEPOLYGON` |
Example compound curve in a curve polygon: CURVEPOLYGON(COMPOUNDCURVE(CIRCULARSTRING(0 0,2 0, 2 1, 2 3, 4 3),(4 3, 4 5, 1 4, 0 0)), CIRCULARSTRING(1.7 1, 1.4 0.4, 1.6 0.4, 1.6 0.5, 1.7 1) ) |

`MULTICURVE` |
A MultiCurve is a 1-dimensional GeometryCollection whose elements are Curves, it can include linear strings, circular strings or compound strings. |

`MULTISURFACE` |
A MultiSurface is a 2-dimensional GeometryCollection whose elements are Surfaces, all using coordinates from the same coordinate reference system. |

`CURVE` |
A Curve is a 1-dimensional geometric object usually stored as a sequence of Points, with the subtype of Curve specifying the form of the interpolation between Points |

`SURFACE` |
A Surface is a 2-dimensional geometric object |

`POLYHEDRALSURFACE` |
A PolyhedralSurface is a contiguous collection of polygons, which share common boundary segments |

`TIN` |
A TIN (triangulated irregular network) is a PolyhedralSurface consisting only of Triangle patches. |

`TRIANGLE` |
A Triangle is a polygon with 3 distinct, non-collinear vertices and no interior boundary |

Note that `CIRCULASTRING`

, `COMPOUNDCURVE`

and
`CURVEPOLYGON`

are not described in the SFA standard, but in
the SQL-MM part 3
standard. The descriptions above were copied from the PostGIS
manual.

Coordinates can only be placed on the Earth’s surface when their coordinate reference system (CRS) is known; this may be a spheroid CRS such as WGS84, a projected, two-dimensional (Cartesian) CRS such as a UTM zone or Web Mercator, or a CRS in three-dimensions, or including time. Similarly, M-coordinates need an attribute reference system, e.g. a measurement unit.

Package `sf`

represents simple features as native R
objects. Similar to PostGIS, all
functions and methods in `sf`

that operate on spatial data
are prefixed by `st_`

, which refers to *spatial type*;
this makes them easily findable by command-line completion. Simple
features are implemented as R native data, using simple data structures
(S3 classes, lists, matrix, vector). Typical use involves reading,
manipulating and writing of sets of features, with attributes and
geometries.

As attributes are typically stored in `data.frame`

objects
(or the very similar `tbl_df`

), we will also store feature
geometries in a `data.frame`

column. Since geometries are not
single-valued, they are put in a list-column, a list of length equal to
the number of records in the `data.frame`

, with each list
element holding the simple feature geometry of that feature. The three
classes used to represent simple features are:

`sf`

, the table (`data.frame`

) with feature attributes and feature geometries, which contains`sfc`

, the list-column with the geometries for each feature (record), which is composed of`sfg`

, the feature geometry of an individual simple feature.

We will now discuss each of these three classes.

As we usually do not work with geometries of single simple features,
but with datasets consisting of sets of features with attributes, the
two are put together in `sf`

(simple feature) objects. The
following command reads the `nc`

dataset from a file that is
contained in the `sf`

package:

```
library(sf)
## Linking to GEOS 3.10.2, GDAL 3.4.3, PROJ 8.2.1; sf_use_s2() is TRUE
<- st_read(system.file("shape/nc.shp", package="sf"))
nc ## Reading layer `nc' from data source
## `/tmp/RtmpBEPzlm/Rinst65efa124af231/sf/shape/nc.shp' using driver `ESRI Shapefile'
## Simple feature collection with 100 features and 14 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
## Geodetic CRS: NAD27
```

(Note that users will not use `system.file`

but give a
`filename`

directly, and that shapefiles consist of more than
one file, all with identical basename, which reside in the same
directory.) The short report printed gives the file name, the driver
(ESRI Shapefile), mentions that there are 100 features (records,
represented as rows) and 14 fields (attributes, represented as columns).
This object is of class

```
class(nc)
## [1] "sf" "data.frame"
```

meaning it extends (and “is” a) `data.frame`

, but with a
single list-column with geometries, which is held in the column with
name

```
attr(nc, "sf_column")
## [1] "geometry"
```

If we print the first three features, we see their attribute values and an abridged version of the geometry

`print(nc[9:15], n = 3)`

which would give the following output:

In the output we see:

- in green a simple feature: a single record, or
`data.frame`

row, consisting of attributes and geometry - in blue a single simple feature geometry (an object of class
`sfg`

) - in red a simple feature list-column (an object of class
`sfc`

, which is a column in the`data.frame`

) - that although geometries are native R objects, they are printed as well-known text

Methods for `sf`

objects are

```
methods(class = "sf")
## [1] $<- [
## [3] [[<- aggregate
## [5] as.data.frame cbind
## [7] coerce dbDataType
## [9] dbWriteTable filter
## [11] identify initialize
## [13] merge plot
## [15] print rbind
## [17] show slotsFromS3
## [19] st_agr st_agr<-
## [21] st_area st_as_s2
## [23] st_as_sf st_as_sfc
## [25] st_bbox st_boundary
## [27] st_break_antimeridian st_buffer
## [29] st_cast st_centroid
## [31] st_collection_extract st_convex_hull
## [33] st_coordinates st_crop
## [35] st_crs st_crs<-
## [37] st_difference st_drop_geometry
## [39] st_filter st_geometry
## [41] st_geometry<- st_inscribed_circle
## [43] st_interpolate_aw st_intersection
## [45] st_intersects st_is
## [47] st_is_valid st_join
## [49] st_line_merge st_m_range
## [51] st_make_valid st_minimum_rotated_rectangle
## [53] st_nearest_points st_node
## [55] st_normalize st_point_on_surface
## [57] st_polygonize st_precision
## [59] st_reverse st_sample
## [61] st_segmentize st_set_precision
## [63] st_shift_longitude st_simplify
## [65] st_snap st_sym_difference
## [67] st_transform st_triangulate
## [69] st_union st_voronoi
## [71] st_wrap_dateline st_write
## [73] st_z_range st_zm
## [75] transform
## see '?methods' for accessing help and source code
```

It is also possible to create `data.frame`

objects with
geometry list-columns that are not of class `sf`

, e.g. by

```
<- as.data.frame(nc)
nc.no_sf class(nc.no_sf)
## [1] "data.frame"
```

However, such objects:

- no longer register which column is the geometry list-column
- no longer have a plot method, and
- lack all of the other dedicated methods listed above for class
`sf`

The column in the `sf`

data.frame that contains the
geometries is a list, of class `sfc`

. We can retrieve the
geometry list-column in this case by `nc$geom`

or
`nc[[15]]`

, but the more general way uses
`st_geometry`

:

```
<- st_geometry(nc))
(nc_geom ## Geometry set for 100 features
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
## Geodetic CRS: NAD27
## First 5 geometries:
## MULTIPOLYGON (((-81.47276 36.23436, -81.54084 3...
## MULTIPOLYGON (((-81.23989 36.36536, -81.24069 3...
## MULTIPOLYGON (((-80.45634 36.24256, -80.47639 3...
## MULTIPOLYGON (((-76.00897 36.3196, -76.01735 36...
## MULTIPOLYGON (((-77.21767 36.24098, -77.23461 3...
```

Geometries are printed in abbreviated form, but we can view a complete geometry by selecting it, e.g. the first one by

```
1]]
nc_geom[[## MULTIPOLYGON (((-81.47276 36.23436, -81.54084 36.27251, -81.56198 36.27359, -81.63306 36.34069, -81.74107 36.39178, -81.69828 36.47178, -81.7028 36.51934, -81.67 36.58965, -81.3453 36.57286, -81.34754 36.53791, -81.32478 36.51368, -81.31332 36.4807, -81.26624 36.43721, -81.26284 36.40504, -81.24069 36.37942, -81.23989 36.36536, -81.26424 36.35241, -81.32899 36.3635, -81.36137 36.35316, -81.36569 36.33905, -81.35413 36.29972, -81.36745 36.2787, -81.40639 36.28505, -81.41233 36.26729, -81.43104 36.26072, -81.45289 36.23959, -81.47276 36.23436)))
```

The way this is printed is called *well-known text*, and is
part of the standards. The word `MULTIPOLYGON`

is followed by
three parentheses, because it can consist of multiple polygons, in the
form of `MULTIPOLYGON(POL1,POL2)`

, where `POL1`

might consist of an exterior ring and zero or more interior rings, as of
`(EXT1,HOLE1,HOLE2)`

. Sets of coordinates are held together
with parentheses, so we get
`((crds_ext)(crds_hole1)(crds_hole2))`

where
`crds_`

is a comma-separated set of coordinates of a ring.
This leads to the case above, where
`MULTIPOLYGON(((crds_ext)))`

refers to the exterior ring (1),
without holes (2), of the first polygon (3) - hence three
parentheses.

We can see there is a single polygon with no rings:

```
par(mar = c(0,0,1,0))
plot(nc[1], reset = FALSE) # reset = FALSE: we want to add to a plot with a legend
plot(nc[1,1], col = 'grey', add = TRUE)
```

but some of the polygons in this dataset have multiple exterior rings; they can be identified by

```
par(mar = c(0,0,1,0))
<- which(sapply(nc_geom, length) > 1))
(w ## [1] 4 56 57 87 91 95
plot(nc[w,1], col = 2:7)
```

Following the `MULTIPOLYGON`

datastructure, in R we have a
list of lists of lists of matrices. For instance, we get the first 3
coordinate pairs of the second exterior ring (first ring is always
exterior) for the geometry of feature 4 by

```
4]][[2]][[1]][1:3,]
nc_geom[[## [,1] [,2]
## [1,] -76.02717 36.55672
## [2,] -75.99866 36.55665
## [3,] -75.91192 36.54253
```

Geometry columns have their own class,

```
class(nc_geom)
## [1] "sfc_MULTIPOLYGON" "sfc"
```

Methods for geometry list-columns include

```
methods(class = 'sfc')
## [1] Ops [
## [3] [<- as.data.frame
## [5] c coerce
## [7] format identify
## [9] initialize print
## [11] rep show
## [13] slotsFromS3 st_area
## [15] st_as_binary st_as_grob
## [17] st_as_s2 st_as_sf
## [19] st_as_text st_bbox
## [21] st_boundary st_break_antimeridian
## [23] st_buffer st_cast
## [25] st_centroid st_collection_extract
## [27] st_convex_hull st_coordinates
## [29] st_crop st_crs
## [31] st_crs<- st_difference
## [33] st_geometry st_inscribed_circle
## [35] st_intersection st_intersects
## [37] st_is st_is_valid
## [39] st_line_merge st_m_range
## [41] st_make_valid st_minimum_rotated_rectangle
## [43] st_nearest_points st_node
## [45] st_normalize st_point_on_surface
## [47] st_polygonize st_precision
## [49] st_reverse st_sample
## [51] st_segmentize st_set_precision
## [53] st_shift_longitude st_simplify
## [55] st_snap st_sym_difference
## [57] st_transform st_triangulate
## [59] st_union st_voronoi
## [61] st_wrap_dateline st_write
## [63] st_z_range st_zm
## [65] str summary
## [67] vec_cast.sfc vec_ptype2.sfc
## see '?methods' for accessing help and source code
```

Coordinate reference systems (`st_crs`

and
`st_transform`

) are discussed in the section on coordinate reference systems. `st_as_wkb`

and
`st_as_text`

convert geometry list-columns into
well-known-binary or well-known-text, explained below. `st_bbox`

retrieves the coordinate
bounding box.

Attributes include

```
attributes(nc_geom)
## $n_empty
## [1] 0
##
## $crs
## Coordinate Reference System:
## User input: NAD27
## wkt:
## GEOGCRS["NAD27",
## DATUM["North American Datum 1927",
## ELLIPSOID["Clarke 1866",6378206.4,294.978698213898,
## LENGTHUNIT["metre",1]]],
## PRIMEM["Greenwich",0,
## ANGLEUNIT["degree",0.0174532925199433]],
## CS[ellipsoidal,2],
## AXIS["latitude",north,
## ORDER[1],
## ANGLEUNIT["degree",0.0174532925199433]],
## AXIS["longitude",east,
## ORDER[2],
## ANGLEUNIT["degree",0.0174532925199433]],
## ID["EPSG",4267]]
##
## $class
## [1] "sfc_MULTIPOLYGON" "sfc"
##
## $precision
## [1] 0
##
## $bbox
## xmin ymin xmax ymax
## -84.32385 33.88199 -75.45698 36.58965
```

The class of `nc_geom`

is
`c("sfc_MULTIPOLYGON", "sfc")`

: `sfc`

is shared
with all geometry types, and `sfc_TYPE`

with
`TYPE`

indicating the type of the particular geometry at
hand.

There are two “special” types: `GEOMETRYCOLLECTION`

, and
`GEOMETRY`

. `GEOMETRYCOLLECTION`

indicates that
each of the geometries may contain a mix of geometry types, as in

```
<- st_sfc(st_geometrycollection(list(st_point(1:2))),
(mix st_geometrycollection(list(st_linestring(matrix(1:4,2))))))
## Geometry set for 2 features
## Geometry type: GEOMETRYCOLLECTION
## Dimension: XY
## Bounding box: xmin: 1 ymin: 2 xmax: 2 ymax: 4
## CRS: NA
## GEOMETRYCOLLECTION (POINT (1 2))
## GEOMETRYCOLLECTION (LINESTRING (1 3, 2 4))
class(mix)
## [1] "sfc_GEOMETRYCOLLECTION" "sfc"
```

Still, the geometries are here of a single type.

The second `GEOMETRY`

, indicates that the geometries in
the geometry list-column are of varying type:

```
<- st_sfc(st_point(1:2), st_linestring(matrix(1:4,2))))
(mix ## Geometry set for 2 features
## Geometry type: GEOMETRY
## Dimension: XY
## Bounding box: xmin: 1 ymin: 2 xmax: 2 ymax: 4
## CRS: NA
## POINT (1 2)
## LINESTRING (1 3, 2 4)
class(mix)
## [1] "sfc_GEOMETRY" "sfc"
```

These two are fundamentally different: `GEOMETRY`

is a
superclass without instances, `GEOMETRYCOLLECTION`

is a
geometry instance. `GEOMETRY`

list-columns occur when we read
in a data source with a mix of geometry types.
`GEOMETRYCOLLECTION`

*is* a single feature’s geometry:
the intersection of two feature polygons may consist of points, lines
and polygons, see the example below.

Simple feature geometry (`sfg`

) objects carry the geometry
for a single feature, e.g. a point, linestring or polygon.

Simple feature geometries are implemented as R native data, using the following rules

- a single POINT is a numeric vector
- a set of points, e.g. in a LINESTRING or ring of a POLYGON is a
`matrix`

, each row containing a point - any other set is a
`list`

Creator functions are rarely used in practice, since we typically bulk read and write spatial data. They are useful for illustration:

```
<- st_point(c(1,2)))
(x ## POINT (1 2)
str(x)
## 'XY' num [1:2] 1 2
<- st_point(c(1,2,3)))
(x ## POINT Z (1 2 3)
str(x)
## 'XYZ' num [1:3] 1 2 3
<- st_point(c(1,2,3), "XYM"))
(x ## POINT M (1 2 3)
str(x)
## 'XYM' num [1:3] 1 2 3
<- st_point(c(1,2,3,4)))
(x ## POINT ZM (1 2 3 4)
str(x)
## 'XYZM' num [1:4] 1 2 3 4
st_zm(x, drop = TRUE, what = "ZM")
## POINT (1 2)
```

This means that we can represent 2-, 3- or 4-dimensional coordinates.
All geometry objects inherit from `sfg`

(simple feature
geometry), but also have a type (e.g. `POINT`

), and a
dimension (e.g. `XYM`

) class name. A figure illustrates six
of the seven most common types.

With the exception of the `POINT`

which has a single point
as geometry, the remaining six common single simple feature geometry
types that correspond to single features (single records, or rows in a
`data.frame`

) are created like this

```
<- rbind(c(3.2,4), c(3,4.6), c(3.8,4.4), c(3.5,3.8), c(3.4,3.6), c(3.9,4.5))
p <- st_multipoint(p))
(mp ## MULTIPOINT ((3.2 4), (3 4.6), (3.8 4.4), (3.5 3.8), (3.4 3.6), (3.9 4.5))
<- rbind(c(0,3),c(0,4),c(1,5),c(2,5))
s1 <- st_linestring(s1))
(ls ## LINESTRING (0 3, 0 4, 1 5, 2 5)
<- rbind(c(0.2,3), c(0.2,4), c(1,4.8), c(2,4.8))
s2 <- rbind(c(0,4.4), c(0.6,5))
s3 <- st_multilinestring(list(s1,s2,s3)))
(mls ## MULTILINESTRING ((0 3, 0 4, 1 5, 2 5), (0.2 3, 0.2 4, 1 4.8, 2 4.8), (0 4.4, 0.6 5))
<- rbind(c(0,0), c(1,0), c(3,2), c(2,4), c(1,4), c(0,0))
p1 <- rbind(c(1,1), c(1,2), c(2,2), c(1,1))
p2 <-st_polygon(list(p1,p2))
pol <- rbind(c(3,0), c(4,0), c(4,1), c(3,1), c(3,0))
p3 <- rbind(c(3.3,0.3), c(3.8,0.3), c(3.8,0.8), c(3.3,0.8), c(3.3,0.3))[5:1,]
p4 <- rbind(c(3,3), c(4,2), c(4,3), c(3,3))
p5 <- st_multipolygon(list(list(p1,p2), list(p3,p4), list(p5))))
(mpol ## MULTIPOLYGON (((0 0, 1 0, 3 2, 2 4, 1 4, 0 0), (1 1, 1 2, 2 2, 1 1)), ((3 0, 4 0, 4 1, 3 1, 3 0), (3.3 0.3, 3.3 0.8, 3.8 0.8, 3.8 0.3, 3.3 0.3)), ((3 3, 4 2, 4 3, 3 3)))
<- st_geometrycollection(list(mp, mpol, ls)))
(gc ## GEOMETRYCOLLECTION (MULTIPOINT ((3.2 4), (3 4.6), (3.8 4.4), (3.5 3.8), (3.4 3.6), (3.9 4.5)), MULTIPOLYGON (((0 0, 1 0, 3 2, 2 4, 1 4, 0 0), (1 1, 1 2, 2 2, 1 1)), ((3 0, 4 0, 4 1, 3 1, 3 0), (3.3 0.3, 3.3 0.8, 3.8 0.8, 3.8 0.3, 3.3 0.3)), ((3 3, 4 2, 4 3, 3 3))), LINESTRING (0 3, 0 4, 1 5, 2 5))
```

The objects created are shown here:

Geometries can also be empty, as in

```
<- st_geometrycollection())
(x ## GEOMETRYCOLLECTION EMPTY
length(x)
## [1] 0
```

Well-known text (WKT) and well-known binary (WKB) are two encodings for simple feature geometries. Well-known text, e.g. seen in

```
<- st_linestring(matrix(10:1,5))
x st_as_text(x)
## [1] "LINESTRING (10 5, 9 4, 8 3, 7 2, 6 1)"
```

(but without the leading `## [1]`

and quotes), is
human-readable. Coordinates are usually floating point numbers, and
moving large amounts of information as text is slow and imprecise. For
that reason, we use well-known binary (WKB) encoding

```
st_as_binary(x)
## [1] 01 02 00 00 00 05 00 00 00 00 00 00 00 00 00 24 40 00 00 00 00 00 00 14 40
## [26] 00 00 00 00 00 00 22 40 00 00 00 00 00 00 10 40 00 00 00 00 00 00 20 40 00
## [51] 00 00 00 00 00 08 40 00 00 00 00 00 00 1c 40 00 00 00 00 00 00 00 40 00 00
## [76] 00 00 00 00 18 40 00 00 00 00 00 00 f0 3f
```

WKT and WKB can both be transformed back into R native objects by

```
st_as_sfc("LINESTRING(10 5, 9 4, 8 3, 7 2, 6 1)")[[1]]
## LINESTRING (10 5, 9 4, 8 3, 7 2, 6 1)
st_as_sfc(structure(list(st_as_binary(x)), class = "WKB"))[[1]]
## LINESTRING (10 5, 9 4, 8 3, 7 2, 6 1)
```

GDAL, GEOS, spatial databases and GIS read and write WKB which is
fast and precise. Conversion between R native objects and WKB is done by
package `sf`

in compiled (C++/Rcpp) code, making this a
reusable and fast route for I/O of simple feature geometries in R.

One of the attributes of a geometry list-column (`sfc`

) is
the `precision`

: a double number that, when non-zero, causes
some rounding during conversion to WKB, which might help certain
geometrical operations succeed that would otherwise fail due to floating
point representation. The model is that of GEOS, which copies from the
Java Topology Suite (JTS), and works like
this:

- if precision is zero (default, unspecified), nothing is modified
- negative values convert to float (4-byte real) precision
- positive values convert to
`round(x*precision)/precision`

.

For the precision model, see also here,
where it is written that: “… to specify 3 decimal places of precision,
use a scale factor of 1000. To specify -3 decimal places of precision
(i.e. rounding to the nearest 1000), use a scale factor of 0.001.” Note
that all coordinates, so also `Z`

or `M`

values
(if present) are affected. Choosing values for `precision`

may require some experimenting.

As we’ve seen above, reading spatial data from an external file can be done by

```
<- system.file("shape/nc.shp", package="sf")
filename <- st_read(filename)
nc ## Reading layer `nc' from data source
## `/tmp/RtmpBEPzlm/Rinst65efa124af231/sf/shape/nc.shp' using driver `ESRI Shapefile'
## Simple feature collection with 100 features and 14 fields
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
## Geodetic CRS: NAD27
```

we can suppress the output by adding argument `quiet=TRUE`

or by using the otherwise nearly identical but more quiet

`<- read_sf(filename) nc `

Writing takes place in the same fashion, using
`st_write`

:

```
st_write(nc, "nc.shp")
## Writing layer `nc' to data source `nc.shp' using driver `ESRI Shapefile'
## Writing 100 features with 14 fields and geometry type Multi Polygon.
```

If we repeat this, we get an error message that the file already exists, and we can overwrite by

```
st_write(nc, "nc.shp", delete_layer = TRUE)
## Deleting layer `nc' using driver `ESRI Shapefile'
## Writing layer `nc' to data source `nc.shp' using driver `ESRI Shapefile'
## Writing 100 features with 14 fields and geometry type Multi Polygon.
```

or its quiet alternative that does this by default,

`write_sf(nc, "nc.shp") # silently overwrites`

The `dsn`

and `layer`

arguments to
`st_read`

and `st_write`

denote a data source name
and optionally a layer name. Their exact interpretation as well as the
options they support vary per driver, the GDAL driver
documentation is best consulted for this. For instance, a PostGIS
table in database `postgis`

might be read by

`<- st_read("PG:dbname=postgis", "meuse") meuse `

where the `PG:`

string indicates this concerns the PostGIS
driver, followed by database name, and possibly port and user
credentials. When the `layer`

and `driver`

arguments are not specified, `st_read`

tries to guess them
from the datasource, or else simply reads the first layer, giving a
warning in case there are more.

`st_read`

typically reads the coordinate reference system
as `proj4string`

, but not the EPSG (SRID). GDAL cannot
retrieve SRID (EPSG code) from `proj4string`

strings, and,
when needed, it has to be set by the user. See also the section on coordinate reference systems.

`st_drivers()`

returns a `data.frame`

listing
available drivers, and their metadata: names, whether a driver can
write, and whether it is a raster and/or vector driver. All drivers can
read. Reading of some common data formats is illustrated below:

`st_layers(dsn)`

lists the layers present in data source
`dsn`

, and gives the number of fields, features and geometry
type for each layer:

```
st_layers(system.file("osm/overpass.osm", package="sf"))
## Driver: OSM
## Available layers:
## layer_name geometry_type features fields crs_name
## 1 points Point NA 10 WGS 84
## 2 lines Line String NA 9 WGS 84
## 3 multilinestrings Multi Line String NA 4 WGS 84
## 4 multipolygons Multi Polygon NA 25 WGS 84
## 5 other_relations Geometry Collection NA 4 WGS 84
```

we see that in this case, the number of features is `NA`

because for this xml file the whole file needs to be read, which may be
costly for large files. We can force counting by

```
Sys.setenv(OSM_USE_CUSTOM_INDEXING="NO")
st_layers(system.file("osm/overpass.osm", package="sf"), do_count = TRUE)
## Driver: OSM
## Available layers:
## layer_name geometry_type features fields crs_name
## 1 points Point 1 10 WGS 84
## 2 lines Line String 0 9 WGS 84
## 3 multilinestrings Multi Line String 0 4 WGS 84
## 4 multipolygons Multi Polygon 13 25 WGS 84
## 5 other_relations Geometry Collection 0 4 WGS 84
```

Another example of reading kml and kmz files is:

```
# Download .shp data
<- "http://coagisweb.cabq.gov/datadownload/biketrails.zip"
u_shp download.file(u_shp, "biketrails.zip")
unzip("biketrails.zip")
<- "http://coagisweb.cabq.gov/datadownload/BikePaths.kmz"
u_kmz download.file(u_kmz, "BikePaths.kmz")
# Read file formats
<- st_read("biketrails.shp")
biketrails_shp if(Sys.info()[1] == "Linux") # may not work if not Linux
<- st_read("BikePaths.kmz")
biketrails_kmz = "http://www.northeastraces.com/oxonraces.com/nearme/safe/6.kml"
u_kml download.file(u_kml, "bikeraces.kml")
<- st_read("bikeraces.kml") bikraces
```

GDAL provides the crud
(create, read, update, delete) functions to persistent storage.
`st_read`

(or `read_sf`

) are used for reading.
`st_write`

(or `write_sf`

) creates, and has the
following arguments to control update and delete:

`update=TRUE`

causes an existing data source to be updated, if it exists; this option is by default`TRUE`

for all database drivers, where the database is updated by adding a table.`delete_layer=TRUE`

causes`st_write`

try to open the the data source and delete the layer; no errors are given if the data source is not present, or the layer does not exist in the data source.`delete_dsn=TRUE`

causes`st_write`

to delete the data source when present, before writing the layer in a newly created data source. No error is given when the data source does not exist. This option should be handled with care, as it may wipe complete directories or databases.

Read and write functions, `st_read()`

and
`st_write()`

, can handle connections to spatial databases to
read WKB or WKT directly without using GDAL. Although intended to use
the DBI interface, current use and testing of these functions are
limited to PostGIS.

Coordinate reference systems (CRS) are like measurement units for
coordinates: they specify which location on Earth a particular
coordinate pair refers to. We saw above that `sfc`

objects
(geometry list-columns) have two attributes to store a CRS:
`epsg`

and `proj4string`

. This implies that all
geometries in a geometry list-column must have the same CRS. Both may be
`NA`

, e.g. in case the CRS is unknown, or when we work with
local coordinate systems (e.g. inside a building, a body, or an abstract
space).

`proj4string`

is a generic, string-based description of a
CRS, understood by the PROJ library. It
defines projection types and (often) defines parameter values for
particular projections, and hence can cover an infinite amount of
different projections. This library (also used by GDAL) provides
functions to convert or transform between different CRS.
`epsg`

is the integer ID for a particular, known CRS that can
be resolved into a `proj4string`

. Some
`proj4string`

values can resolved back into their
corresponding `epsg`

ID, but this does not always work.

The importance of having `epsg`

values stored with data
besides `proj4string`

values is that the `epsg`

refers to particular, well-known CRS, whose parameters may change
(improve) over time; fixing only the `proj4string`

may remove
the possibility to benefit from such improvements, and limit some of the
provenance of datasets, but may help reproducibility.

Coordinate reference system transformations can be carried out using
`st_transform`

, e.g. converting longitudes/latitudes in NAD27
to web mercator (EPSG:3857) can be done by

```
<- st_transform(nc, 3857)
nc.web_mercator st_geometry(nc.web_mercator)[[4]][[2]][[1]][1:3,]
## [,1] [,2]
## [1,] -8463267 4377519
## [2,] -8460094 4377510
## [3,] -8450437 4375553
```

`sf`

objects and objects deriving from
`Spatial`

(package `sp`

) can be coerced both
ways:

```
showMethods("coerce", classes = "sf")
## Function: coerce (package methods)
## from="Spatial", to="sf"
## from="sf", to="Spatial"
methods(st_as_sf)
## [1] st_as_sf.SpatVector* st_as_sf.Spatial* st_as_sf.data.frame*
## [4] st_as_sf.lpp* st_as_sf.map* st_as_sf.owin*
## [7] st_as_sf.ppp* st_as_sf.ppplist* st_as_sf.psp*
## [10] st_as_sf.s2_geography* st_as_sf.sf* st_as_sf.sfc*
## see '?methods' for accessing help and source code
methods(st_as_sfc)
## [1] st_as_sfc.SpatialLines* st_as_sfc.SpatialMultiPoints*
## [3] st_as_sfc.SpatialPixels* st_as_sfc.SpatialPoints*
## [5] st_as_sfc.SpatialPolygons* st_as_sfc.WKB*
## [7] st_as_sfc.bbox* st_as_sfc.blob*
## [9] st_as_sfc.character* st_as_sfc.dimensions*
## [11] st_as_sfc.factor* st_as_sfc.list*
## [13] st_as_sfc.map* st_as_sfc.owin*
## [15] st_as_sfc.pq_geometry* st_as_sfc.psp*
## [17] st_as_sfc.raw* st_as_sfc.s2_geography*
## [19] st_as_sfc.sf* st_as_sfc.tess*
## see '?methods' for accessing help and source code
# anticipate that sp::CRS will expand proj4strings:
<- "+proj=longlat +datum=NAD27 +no_defs +ellps=clrk66 +nadgrids=@conus,@alaska,@ntv2_0.gsb,@ntv1_can.dat"
p4s st_crs(nc) <- p4s
# anticipate geometry column name changes:
names(nc)[15] = "geometry"
attr(nc, "sf_column") = "geometry"
<- as(nc, "Spatial")
nc.sp class(nc.sp)
## [1] "SpatialPolygonsDataFrame"
## attr(,"package")
## [1] "sp"
<- st_as_sf(nc.sp)
nc2 all.equal(nc, nc2)
## [1] "Attributes: < Component \"class\": Lengths (4, 2) differ (string compare on first 2) >"
## [2] "Attributes: < Component \"class\": 1 string mismatch >"
## [3] "Component \"geometry\": Attributes: < Component \"crs\": Component \"input\": 1 string mismatch >"
## [4] "Component \"geometry\": Attributes: < Component \"crs\": Component \"wkt\": 1 string mismatch >"
```

As the `Spatial*`

objects only support
`MULTILINESTRING`

and `MULTIPOLYGON`

,
`LINESTRING`

and `POLYGON`

geometries are
automatically coerced into their `MULTI`

form. When
converting `Spatial*`

into `sf`

, if all geometries
consist of a single `POLYGON`

(possibly with holes), a
`POLYGON`

and otherwise all geometries are returned as
`MULTIPOLYGON`

: a mix of `POLYGON`

and
`MULTIPOLYGON`

(such as common in shapefiles) is not created.
Argument `forceMulti=TRUE`

will override this, and create
`MULTIPOLYGON`

s in all cases. For `LINES`

the
situation is identical.

The standard for simple feature access defines a number of geometrical operations.

`st_is_valid`

and `st_is_simple`

return a
boolean indicating whether a geometry is valid or simple.

```
st_is_valid(nc[1:2,])
## [1] TRUE TRUE
```

`st_distance`

returns a dense numeric matrix with
distances between geometries. `st_relate`

returns a character
matrix with the DE9-IM
values for each pair of geometries:

```
= st_transform(nc, 32119)
x st_distance(x[c(1,4,22),], x[c(1, 33,55,56),])
## Units: [m]
## [,1] [,2] [,3] [,4]
## [1,] 0.00 312184.9 128341.85 475623.3
## [2,] 440561.15 114939.7 590434.80 0.0
## [3,] 18944.03 352719.1 78756.89 517527.8
st_relate(nc[1:5,], nc[1:4,])
## although coordinates are longitude/latitude, st_relate assumes that they are
## planar
## [,1] [,2] [,3] [,4]
## [1,] "2FFF1FFF2" "FF2F11212" "FF2FF1212" "FF2FF1212"
## [2,] "FF2F11212" "2FFF1FFF2" "FF2F11212" "FF2FF1212"
## [3,] "FF2FF1212" "FF2F11212" "2FFF1FFF2" "FF2FF1212"
## [4,] "FF2FF1212" "FF2FF1212" "FF2FF1212" "2FFF1FFF2"
## [5,] "FF2FF1212" "FF2FF1212" "FF2FF1212" "FF2FF1212"
```

The commands `st_intersects`

, `st_disjoint`

,
`st_touches`

, `st_crosses`

,
`st_within`

, `st_contains`

,
`st_overlaps`

, `st_equals`

,
`st_covers`

, `st_covered_by`

,
`st_equals_exact`

and `st_is_within_distance`

return a sparse matrix with matching (TRUE) indexes, or a full logical
matrix:

```
st_intersects(nc[1:5,], nc[1:4,])
## Sparse geometry binary predicate list of length 5, where the predicate
## was `intersects'
## 1: 1, 2
## 2: 1, 2, 3
## 3: 2, 3
## 4: 4
## 5: (empty)
st_intersects(nc[1:5,], nc[1:4,], sparse = FALSE)
## [,1] [,2] [,3] [,4]
## [1,] TRUE TRUE FALSE FALSE
## [2,] TRUE TRUE TRUE FALSE
## [3,] FALSE TRUE TRUE FALSE
## [4,] FALSE FALSE FALSE TRUE
## [5,] FALSE FALSE FALSE FALSE
```

The commands `st_buffer`

, `st_boundary`

,
`st_convexhull`

, `st_union_cascaded`

,
`st_simplify`

, `st_triangulate`

,
`st_polygonize`

, `st_centroid`

,
`st_segmentize`

, and `st_union`

return new
geometries, e.g.:

```
<- c(1,5,14)
sel = st_geometry(nc.web_mercator[sel,])
geom <- st_buffer(geom, dist = 30000)
buf plot(buf, border = 'red')
plot(geom, add = TRUE)
plot(st_buffer(geom, -5000), add = TRUE, border = 'blue')
```

Commands `st_intersection`

, `st_union`

,
`st_difference`

, `st_sym_difference`

return new
geometries that are a function of pairs of geometries:

```
par(mar = rep(0,4))
<- st_union(nc)
u plot(u)
```

The following code shows how computing an intersection between two
polygons may yield a `GEOMETRYCOLLECTION`

with a point, line
and polygon:

```
<- par(mfrow = c(1, 2))
opar <- st_polygon(list(cbind(c(0,0,7.5,7.5,0),c(0,-1,-1,0,0))))
a <- st_polygon(list(cbind(c(0,1,2,3,4,5,6,7,7,0),c(1,0,.5,0,0,0.5,-0.5,-0.5,1,1))))
b plot(a, ylim = c(-1,1))
title("intersecting two polygons:")
plot(b, add = TRUE, border = 'red')
<- st_intersection(a,b))
(i ## GEOMETRYCOLLECTION (POLYGON ((7 0, 7 -0.5, 6 -0.5, 5.5 0, 7 0)), LINESTRING (4 0, 3 0), POINT (1 0))
plot(a, ylim = c(-1,1))
title("GEOMETRYCOLLECTION")
plot(b, add = TRUE, border = 'red')
plot(i, add = TRUE, col = 'green', lwd = 2)
```

`par(opar)`

Non-simple geometries are for instance self-intersecting lines (left); non-valid geometries are for instance polygons with slivers (middle) or self-intersections (right).

```
library(sf)
<- st_linestring(cbind(c(0,1,0,1),c(0,1,1,0)))
x1 <- st_polygon(list(cbind(c(0,1,1,1,0,0),c(0,0,1,0.6,1,0))))
x2 <- st_polygon(list(cbind(c(0,1,0,1,0),c(0,1,1,0,0))))
x3 st_is_simple(st_sfc(x1))
## [1] FALSE
st_is_valid(st_sfc(x2,x3))
## [1] FALSE FALSE
```

Where possible geometric operations such as
`st_distance()`

, `st_length()`

and
`st_area()`

report results with a units attribute appropriate
for the CRS:

```
<- st_area(nc[1,])
a attributes(a)
## $units
## $numerator
## [1] "m" "m"
##
## $denominator
## character(0)
##
## attr(,"class")
## [1] "symbolic_units"
##
## $class
## [1] "units"
```

The **units** package can be used to convert between
units:

```
::set_units(a, km^2) # result in square kilometers
units## 1137.108 [km^2]
::set_units(a, ha) # result in hectares
units## 113710.8 [ha]
```

The result can be stripped of their attributes if needs be:

```
as.numeric(a)
## [1] 1137107793
```

(This will eventually be the topic of a new vignette; now here to
explain the last attribute of `sf`

objects)

The standard documents about simple features are very detailed about
the geometric aspects of features, but say nearly nothing about
attributes, except that their values should be understood in another
reference system (their units of measurement, e.g. as implemented in the
package **units**).
But there is more to it. For variables like air temperature,
interpolation usually makes sense, for others like human body
temperature it doesn’t. The difference is that air temperature is a
field, which continues between sensors, where body temperature is an
object property that doesn’t extend beyond the body – in spatial
statistics bodies would be called a point pattern, their temperature the
point marks. For geometries that have a non-zero size (positive length
or area), attribute values may refer to the every sub-geometry (every
point), or may summarize the geometry. For example, a state’s population
density summarizes the whole state, and is not a meaningful estimate of
population density for a give point inside the state without the context
of the state. On the other hand, land use or geological maps give
polygons with constant land use or geology, every point inside the
polygon is of that class. Some properties are spatially extensive,
meaning that attributes would summed up when two geometries are merged:
population is an example. Other properties are spatially intensive, and
should be averaged, with population density the example.

Simple feature objects of class `sf`

have an *agr*
attribute that points to the *attribute-geometry-relationship*,
how attributes relate to their geometry. It can be defined at creation
time:

```
<- st_read(system.file("shape/nc.shp", package="sf"),
nc agr = c(AREA = "aggregate", PERIMETER = "aggregate", CNTY_ = "identity",
CNTY_ID = "identity", NAME = "identity", FIPS = "identity", FIPSNO = "identity",
CRESS_ID = "identity", BIR74 = "aggregate", SID74 = "aggregate", NWBIR74 = "aggregate",
BIR79 = "aggregate", SID79 = "aggregate", NWBIR79 = "aggregate"))
## Reading layer `nc' from data source
## `/tmp/RtmpBEPzlm/Rinst65efa124af231/sf/shape/nc.shp' using driver `ESRI Shapefile'
## Simple feature collection with 100 features and 14 fields
## Attribute-geometry relationship: 0 constant, 8 aggregate, 6 identity
## Geometry type: MULTIPOLYGON
## Dimension: XY
## Bounding box: xmin: -84.32385 ymin: 33.88199 xmax: -75.45698 ymax: 36.58965
## Geodetic CRS: NAD27
st_agr(nc)
## AREA PERIMETER CNTY_ CNTY_ID NAME FIPS FIPSNO CRESS_ID
## aggregate aggregate identity identity identity identity identity identity
## BIR74 SID74 NWBIR74 BIR79 SID79 NWBIR79
## aggregate aggregate aggregate aggregate aggregate aggregate
## Levels: constant aggregate identity
data(meuse, package = "sp")
<- st_as_sf(meuse, coords = c("x", "y"), crs = 28992, agr = "constant")
meuse_sf st_agr(meuse_sf)
## cadmium copper lead zinc elev dist om ffreq
## constant constant constant constant constant constant constant constant
## soil lime landuse dist.m
## constant constant constant constant
## Levels: constant aggregate identity
```

When not specified, this field is filled with `NA`

values,
but if non-`NA`

, it has one of three possibilities

value | meaning |
---|---|

constant | a variable that has a constant value at every location over a spatial extent; examples: soil type, climate zone, land use |

aggregate | values are summary values (aggregates) over the geometry, e.g. population density, dominant land use |

identity | values identify the geometry: they refer to (the whole of) this and only this geometry |

With this information (still to be done) we can for instance

- either return missing values or generate warnings when a
*aggregate*value at a point location inside a polygon is retrieved, or - list the implicit assumptions made when retrieving attribute values
at points inside a polygon when
`relation_to_geometry`

is missing. - decide what to do with attributes when a geometry is split: do
nothing in case the attribute is constant, give an error or warning in
case it is an aggregate, change the
`relation_to_geometry`

to*constant*in case it was*identity*.

Further reading:

- S. Scheider, B. Gräler, E. Pebesma, C. Stasch, 2016. Modelling spatio-temporal information generation. Int J of Geographic Information Science, 30 (10), 1980-2008. (open access)
- Stasch, C., S. Scheider, E. Pebesma, W. Kuhn, 2014. Meaningful Spatial Prediction and Aggregation. Environmental Modelling & Software, 51, (149–165, open access).