Figure 5.1 The 9-intersection model of topological relations

figures
code
R

This page attempts to show the various spatial predicates of the 9-intersection model presented in Figure 5.1 in code.

First we make some shapes, and map them.

Code 
library(sf)
library(dplyr)
library(tmap)
library(stringr)

p1 <- st_point(c(-2, 0)) %>% st_buffer(1, nQuadSegs = 90)
p2 <- p1
p3 <- p1 + c(2, 0)
p4 <- p3 * matrix(c(0.5, 0, 0, 0.5), 2, 2)
p5 <- p4 + c(0, -0.5)
p6 <- st_difference(p3, p4) + c(2.1, 0)
p7 <- p4 * matrix(c(1.5, 0, 0, 1.5), 2, 2) + c(2, -1)

polys <- st_sfc(list(p1, p2, p3, p4, p5, p6, p7)) %>% 
  st_sf() %>%
  mutate(ID = 1:7, 
         dx = c(0, 0, .5, 0, 0, 0, 0), 
         dy = c(-1, 1, 2, 0, 0, 0, 0))

Map and label these for reference.

Code 
lines <- polys %>% st_cast("MULTILINESTRING") # for placing text labels

tm_shape(polys) + 
  tm_fill(col = "ID", alpha = 0.5, style = "cat", 
          palette = c("red", "dodgerblue3", "darkgreen", 
                      "black", "yellow", "violet", "brown"),
          legend.is.portrait = FALSE) +
  tm_shape(lines) +
  tm_text(text = "ID", xmod = "dx", ymod = "dy") +
  tm_layout(
    frame = FALSE, legend.outside = TRUE,
    legend.outside.position = "bottom", 
    legend.position = c(0.5, 0.5), 
    legend.just = c(0.5, 0.5))

Now we can run various tests against the spatial predicates in turn. For example st_disjoint tells us the following.

Code 
polys %>% st_disjoint()
Sparse geometry binary predicate list of length 7, where the predicate
was `disjoint'
 1: 3, 4, 5, 6, 7
 2: 3, 4, 5, 6, 7
 3: 1, 2, 6, 7
 4: 1, 2, 6, 7
 5: 1, 2, 6, 7
 6: 1, 2, 3, 4, 5
 7: 1, 2, 3, 4, 5

We can tabulate the result of applying all the spatial predicates to these polygons into a table.

The details of how this is done don’t matter greatly, but if you are interested click into the code below. The core of it is based on the various st_* functions such as st_intersects or st_contains. These correspond, more or less to the relations identified in Figure 5.1. The spatial predicates available in the sf package in R correspond to those implemented in spatial databases such as PostGIS.

Code 
get_spatial_query_as_vector <- function(data, predicate) {
  data %>% 
    predicate() %>%
    lapply(str_c, collapse = " ") %>%
    unlist()
}

de9im <- data.frame(
  ID = polys$ID,
  disjoint          = get_spatial_query_as_vector(polys, st_disjoint),
  touches           = get_spatial_query_as_vector(polys, st_touches),
  equals            = get_spatial_query_as_vector(polys, st_equals),
  intersects        = get_spatial_query_as_vector(polys, st_intersects),
  contains_properly = get_spatial_query_as_vector(polys, st_contains_properly),
  contains          = get_spatial_query_as_vector(polys, st_contains),
  within            = get_spatial_query_as_vector(polys, st_within),
  covers            = get_spatial_query_as_vector(polys, st_covers),
  covered_by        = get_spatial_query_as_vector(polys, st_covered_by)
)

knitr::kable(de9im)
ID disjoint touches equals intersects contains_properly contains within covers covered_by
1 3 4 5 6 7 1 2 1 2 1 2 1 2 1 2 1 2
2 3 4 5 6 7 1 2 1 2 1 2 1 2 1 2 1 2
3 1 2 6 7 3 3 4 5 4 3 4 5 3 3 4 5 3
4 1 2 6 7 4 3 4 5 4 3 4 4 3 4
5 1 2 6 7 5 3 4 5 5 3 5 5 3 5
6 1 2 3 4 5 6 6 7 6 6 6 6
7 1 2 3 4 5 7 6 7 7 7 7 7
Code 
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© 2023 David O’Sullivan