library(magrittr) library(knitr) library(rgl) library(ggplot2) knit_hooks$set(webgl = hook_rgl) view_matrix <- structure(c(0.586383819580078, 0.356217533349991, -0.727502763271332, 0, -0.810031354427338, 0.257360488176346, -0.526888787746429, 0, -0.000456457957625389, 0.898260772228241, 0.439460128545761, 0, 0, 0, 0, 1), .Dim = c(4L, 4L))
bleiglas is an R package that employs Voro++ for the calculation of three dimensional Voronoi diagrams from input point clouds. This is a special form of tessellation where each polygon is defined as the area closest to one particular seed point. Voronoi diagrams have useful applications in - among others - astronomy, material science or geography and bleiglas provides functions to make 3D tessellation more readily available as a mean for data visualisation and interpolation. It can be used for any 3D point cloud, but the output is optimized for spatiotemporal applications in archaeology.
vignette("bleiglas_case_study")
.If you have questions beyond this documentation feel free to open an issue here on Github. Please also see our contributing guide.
You can install bleiglas from github
if(!require('remotes')) install.packages('remotes') remotes::install_github("nevrome/bleiglas", build_vignettes = TRUE)
For the main function tessellate
you also have to install the Voro++ software. The package is already available in all major Linux software repositories (on Debian/Ubuntu you can simply run sudo apt-get install voro++
.). MacOS users should be able to install it via homebrew (brew install voro++
).
For this quickstart, we assume you have packages tidyverse
, sf
, rgeos
(which in turn requires the Unix package geos
) and c14bazAAR
installed.
I decided to use Dirk Seidenstickers Archives des datations radiocarbone d'Afrique centrale dataset for this purpose. It includes radiocarbon datings from Central Africa that combine spatial (x & y) and temporal (z) position with some meta information.
I selected dates from Cameroon between 1000 and 3000 uncalibrated BP and projected them into a worldwide cylindrical reference system (epsg 4088). As Cameroon is close to the equator this projection should represent distances, angles and areas sufficiently correct for this example exercise. As a minor pre-processing step, I here also remove samples with equal position in all three dimensions for the tessellation.Click here for the data preparation steps
# download raw data with the data access package c14bazAAR
# c14bazAAR can be installed with
# install.packages("c14bazAAR", repos = c(ropensci = "https://ropensci.r-universe.dev"))
c14_cmr <- c14bazAAR::get_c14data("adrac") %>%
# filter data
dplyr::filter(!is.na(lat) & !is.na(lon), c14age > 1000, c14age < 3000, country == "CMR")
# remove doubles
c14_cmr_unique <- c14_cmr %>%
dplyr::mutate(
rounded_coords_lat = round(lat, 3),
rounded_coords_lon = round(lon, 3)
) %>%
dplyr::group_by(rounded_coords_lat, rounded_coords_lon, c14age) %>%
dplyr::filter(dplyr::row_number() == 1) %>%
dplyr::ungroup()
# transform coordinates
coords <- data.frame(c14_cmr_unique$lon, c14_cmr_unique$lat) %>%
sf::st_as_sf(coords = c(1, 2), crs = 4326) %>%
sf::st_transform(crs = 4088) %>%
sf::st_coordinates()
# create active dataset
c14 <- c14_cmr_unique %>%
dplyr::transmute(
id = seq_len(nrow(.)),
x = coords[,1],
y = coords[,2],
z = c14age,
period = period
)
Data: c14
c14
Tessellation means filling space with polygons so that neither gaps nor overlaps occur. This is an exciting application for art (e.g. textile art or architecture) and an interesting challenge for mathematics. As a computational archaeologist I was already aware of one particular tessellation algorithm that has quite some relevance for geostatistical analysis like spatial interpolation: Voronoi tilings that are created with Delaunay triangulation. These are tessellations where each polygon covers the space closest to one of a set of sample points.
It turns out that Voronoi tessellation can be calculated not just for 2D surfaces, but also for higher dimensions. The Voro++ software library does exactly this for 3 dimensions. This makes it useful for spatio-temporal applications.
bleiglas::tessellate()
is a minimal wrapper function that calls the Voro++ command line interface (therefore you have to install Voro++ to use it) for datasets like the one introduced above. We can apply it like this:
raw_voro_output <- bleiglas::tessellate( c14[, c("id", "x", "y", "z")], x_min = min(c14$x) - 150000, x_max = max(c14$x) + 150000, y_min = min(c14$y) - 150000, y_max = max(c14$y) + 150000, unit_scaling = c(0.001, 0.001, 1) )
A critical step when using tessellation for spatio-temporal data is a suitable conversion scale between time- and spatial units. Since 3D tessellation crucially depends on the concept of a 3D-distance, we need to make a decision how to combine length- and time-units. Here, for the purpose of this example, we have 1 kilometre correspond to 1 year. Since after the coordinate conversion our spatial units are given in meters, we divide all spatial distances by a factor 1000 to achieve this correspondence: unit_scaling = c(0.001, 0.001, 1)
.
I decided to increase the size of the tessellation box by 150 kilometres to each (spatial) direction to cover the area of Cameroon. Mind that the scaling factors in unit_scaling
are also applied to the box size parameters x_min
, x_max
, ....
The output of Voro++ is highly customizable, and structurally complex. With the -v
flag, the voro++ CLI interface prints some config info, which is also the output of bleiglas::tesselate
:
Container geometry : [937.154:1936.57] [63.1609:1506.58] [1010:2990] Computational grid size : 3 by 5 by 6 (estimated from file) Filename : /tmp/RtmpVZjBW3/file3aeb5f400f38 Output string : %i*%P*%t Total imported particles : 392 (4.4 per grid block) Total V. cells computed : 392 Total container volume : 2.8563e+09 Total V. cell volume : 2.8563e+09
It then produces an output file (*.vol
) that contains all sorts of geometry information for the calculated 3D polygons. tesselate
returns the content of this file as a character vector with the additionally attached attribute unit_scaling
(attributes(raw_voro_output)$unit_scaling
), which is just the scaling vector we put in above.
I focussed on the edges of the polygons and wrote a parser function bleiglas::read_polygon_edges()
that can transform the complex Voro++ output for this specific output case to a tidy data.table with six columns: the coordinates (x, y, z) of the start (a) and end point (b) of each polygon edge. A data.table is a tabular R data structure very similar to the standard data.frame. Read more about it here.
polygon_edges <- bleiglas::read_polygon_edges(raw_voro_output)
read_polygon_edges
automatically reverses the rescaling introduced in tesselate
with the unit_scaling
attribute.
Data: polygon_edges
polygon_edges
We can plot these polygon edges (black) together with the input sample points (red) in 3D.
rgl::axes3d()
rgl::points3d(c14$x, c14$y, c14$z, color = "red")
rgl::aspect3d(1, 1, 1)
rgl::segments3d(
x = as.vector(t(polygon_edges[,c(1,4)])),
y = as.vector(t(polygon_edges[,c(2,5)])),
z = as.vector(t(polygon_edges[,c(3,6)]))
)
rgl::view3d(userMatrix = view_matrix, zoom = 0.9)
rgl::axes3d() rgl::points3d(c14$x, c14$y, c14$z, color = "red") rgl::aspect3d(1, 1, 1) rgl::segments3d( x = as.vector(t(polygon_edges[,c(1,4)])), y = as.vector(t(polygon_edges[,c(2,5)])), z = as.vector(t(polygon_edges[,c(3,6)])) ) rgl::view3d(userMatrix = view_matrix, zoom = 0.9)
This 3D plot, even if rotatable using mouse input, is of rather limited value since it's very hard to read. I therefore wrote bleiglas::cut_polygons()
that can cut the 3D polygons at different levels of the z-axis. As the function assumes that x and y represent geographic coordinates, the cuts produce sets of spatial 2D polygons for different values of z -- in our example different points in time. The parameter cuts
takes a numeric vector of cutting points on the z axis. bleiglas::cut_polygons()
yields a rather raw format for specifying polygons. Another function, bleiglas::cut_polygons_to_sf()
, transforms it to sf
. Here crs
defines the spatial coordinate reference system of x and y to project the resulting 2D polygons correctly.
cut_surfaces <- bleiglas::cut_polygons( polygon_edges, cuts = c(2500, 2000, 1500) ) %>% bleiglas::cut_polygons_to_sf(crs = 4088)
Data: cut_surfaces
cut_surfaces
With this data we can plot a matrix of maps that show the cut surfaces.
cut_surfaces %>%
ggplot() +
geom_sf(
aes(fill = z),
color = "white",
lwd = 0.2
) +
geom_sf_text(aes(label = id)) +
facet_wrap(~z) +
theme(
axis.text = element_blank(),
axis.ticks = element_blank()
)
cut_surfaces %>% ggplot() + geom_sf( aes(fill = z), color = "white", lwd = 0.2 ) + facet_wrap(~z) + theme( axis.text = element_blank(), axis.ticks = element_blank() )
wzxhzdk:17
wzxhzdk:18
As all input dates come from Cameroon it makes sense to cut the polygon surfaces to the outline of this administrative unit.
cameroon_border <- rnaturalearth::ne_countries(scale = "medium", returnclass = "sf") %>%
dplyr::filter(name == "Cameroon") %>%
sf::st_transform(4088)
cut_surfaces_cropped <- cut_surfaces %>% sf::st_intersection(cameroon_border)
cut_surfaces_cropped %>%
ggplot() +
geom_sf(
aes(fill = z),
color = "white",
lwd = 0.2
) +
facet_wrap(~z) +
theme(
axis.text = element_blank(),
axis.ticks = element_blank()
)
Finally, we can also visualise any point-wise information in our input data as a feature of the tessellation polygons.
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