Work in progress to get constrained and conforming Delaunay triangulation for shapes in R with CGAL. We have
- point-only triangulation in
tri_xy()- it’s fast - segment-input constraints in
insert_mesh()- only returns the vertices, which might include new points - constrained and conformal triangulations in-progress …
We need:
- export of new vertex pool and its edge- or triangle-based indices
- proper inputs of nested polygon shapes
- robust dealing with segment and polygon soups, so that numeric insertions are robust (and c.)
We can now hit this more easily thanks to cgal4h thanks to Ahmadou Dicko.
laridae came out of a need for constrained triangulation for a
topology-in-R project. That effort has moved on somewhat, proving the
case by using RTriangle and then bedding down the normalization model
in the hypertidy/silicate package.
R now has cgal4h providing the library infrastructure for CGAL and euclid providing numerically robust geometric primitives (it’s unclear if laridae or whatever it becomes will use euclid).
RTriangle is really fast, but it’s not as fast as CGAL. CGAL can also be used to update a triangulation, which means (I think) that we could build an unconstrained triangulation from all the coordinates, and then add in any segments, even unclosed linear paths. At any rate, being able to update a mesh has a lot of applications, especially for neighbouring shapes, and for on-demand (extent or zoom dependent level of detail) tasks.
The interest (which seems miniscule …) in constrained triangulations is discussed here along with the overall landscape in R.
https://github.com/r-spatial/discuss/issues/6
Installation
remotes::install_github("hypertidy/laridae")
## not universe ready yet
#install.packages("laridae", repos = "https://hypertidy.r-universe.dev")Triangulation
Triangulate with CGAL via
laridae. The function tri_xy
performs an exact Delaunay triangulation on all vertices, returning a
triplet-index for each triangle (zero-based in CGAL).
The variants tri_xy1() and tri_xy2() work slightly differently
illustrating CGAL usage in C++ (thanks to Mark Padgham). ‘xy1’ order is
not trivial … but we ignore that for now.
library(laridae)
x <- c(0, 0, 1)
y <- c(0, 1, 1)
plot(x, y, pch = "+")
(idx0 <- tri_xy(x, y))
#> [1] 3 2 1
(idx1 <- tri_xy1(x, y))
#> [1] 2 3 1
(idx2 <- tri_xy2(x, y))
#> [1] 3 2 1
polygon(cbind(x, y)[idx0, ])
x <- c(2.3,3.0,7.0,1.0,3.0,8.0)
y <- c(2.3,3.0,2.0,5.0,8.0,9.0)
idx <- tri_xy(x, y)
plot(x, y, pch = "+", cex = 1.5)
polygon(cbind(x, y)[rbind(matrix(idx, 3L), NA), ], lty = 3)
Some timings, to show we aren’t wildly off-base and that CGAL wins for raw unconstrained Delaunay triangulation.
set.seed(90)
x <- rnorm(1e3, sd = 4)
y <- rnorm(1e3, sd = 2)
library(laridae)
# plot a matrix xy as points
# and add the triangulation indexed
# by structural triplet row-identifiers
poly_index <- function(xy, index, ...) {
plot(xy, ...)
## assume index is 0,1,2,0,1,2,0,1,...
ii <- c(rbind(matrix(index, nrow = 3), NA_integer_))
## have forgetten why polypath fails, so just use polygon
polygon(xy[ii, 1], xy[ii, 2])
}
ps <- RTriangle::pslg(P = cbind(x, y))
microbenchmark::microbenchmark(
ind_t <- tri_xy(x, y),
ind_t1 <- tri_xy1(x, y),
ind_t2 <- tri_xy2(x, y),
RT <- RTriangle::triangulate(ps)
)
#> Unit: microseconds
#> expr min lq mean median uq
#> ind_t <- tri_xy(x, y) 958.795 1127.766 1184.248 1202.551 1244.693
#> ind_t1 <- tri_xy1(x, y) 964.334 1111.205 1180.227 1187.096 1239.847
#> ind_t2 <- tri_xy2(x, y) 968.167 1116.894 1177.740 1199.422 1242.877
#> RT <- RTriangle::triangulate(ps) 1677.312 1972.573 2076.233 2078.806 2194.787
#> max neval cld
#> 1412.635 100 a
#> 1489.043 100 a
#> 1408.256 100 a
#> 2371.038 100 b
length(ind_t)
#> [1] 5961
length(ind_t1)
#> [1] 5961
length(ind_t2)
#> [1] 5961
length(RT$T)
#> [1] 5961
p <- par(mfrow = c(2, 2), mar = rep(0, 4))
poly_index(cbind(x, y), ind_t, pch = ".")
## can't work as order is not aligned, but still fun
poly_index(cbind(x, y), ind_t1, pch = ".")
poly_index(cbind(x, y), ind_t2, pch = ".")
plot(RT)
par(p)Constrained triangulation
Currently laridae only has vertex-output and “reporting” of the result. I can’t yet see how to
- get the vertex pool (it may have expanded given mesh properties, overlapping segments, etc.)
- get the triangle index
The only other implementation in R is in RTriangle, so we use that for comparison.
sfx <- silicate::inlandwaters
sc <- silicate::SC(sfx)
X <- sc$vertex$x_
Y <- sc$vertex$y_
i0 <- match(sc$edge$.vx0, sc$vertex$vertex_)
i1 <- match(sc$edge$.vx1, sc$vertex$vertex_)
system.time(insert_constraint(X, Y, i0 , i1))
#> Number of vertices before: 30835
#> Number of vertices after: 31079
#> user system elapsed
#> 0.368 0.000 0.367
system.time(segment_constraint(sc))
#> The number of resulting constrained edges is: 30843
#> user system elapsed
#> 0.392 0.000 0.393
## insert_mesh actually returns the vertices (which might include new ones)
system.time(xy_out <- insert_mesh(X, Y, i0 , i1))
#> Number of vertices before: 30835
#> Number of vertices after: 31079
#> user system elapsed
#> 0.075 0.000 0.075
plot(xy_out, pch = ".")
## compare RTriangle, it's fast if we don't include pslg() time
ps <- RTriangle::pslg(cbind(X, Y), S = cbind(i0, i1))
system.time({
tr <- RTriangle::triangulate(ps, D = TRUE)
})
#> user system elapsed
#> 0.079 0.000 0.079
plot(tr$P, pch= ".")
segments(tr$P[tr$E[,1],1], tr$P[tr$E[,1],2],
tr$P[tr$E[,2],1], tr$P[tr$E[,2],2])
str(tr)
#> List of 12
#> $ P : num [1:31778, 1:2] -681074 -680885 -680821 -680474 -680376 ...
#> $ PB: int [1:31778, 1] 1 1 1 1 1 1 1 1 1 1 ...
#> $ PA: num[1:31778, 0 ]
#> $ T : int [1:43051, 1:3] 4455 4455 4510 4542 4510 4410 4376 4380 4397 4390 ...
#> $ S : int [1:31786, 1:2] 24305 24316 24328 24328 24326 24321 24343 24354 24349 24349 ...
#> $ SB: int [1:31786, 1] 0 0 0 0 0 0 0 0 0 0 ...
#> $ E : int [1:74679, 1:2] 4455 4456 4492 4492 4542 4510 4492 4542 4562 4492 ...
#> $ EB: int [1:74679, 1] 1 1 0 0 0 1 0 1 0 1 ...
#> $ VP: num [1:43051, 1:2] 51155 52903 52733 54473 52163 ...
#> $ VE: int [1:74679, 1:2] 1 1 1 2 2 3 3 4 4 5 ...
#> $ VN: num [1:43051, 1:2] -2148 -124 0 0 0 ...
#> $ VA: num[1:43051, 0 ]
#> - attr(*, "class")= chr "triangulation"History
Was originally called cgalgris.
Code of Conduct
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