stla / rcdt

R package for 2D constrained Delaunay triangulation

The ‘RCDT’ package - constrained 2D Delaunay triangulation

The pentagram

# vertices
R <- sqrt((5-sqrt(5))/10)     # outer circumradius
r <- sqrt((25-11*sqrt(5))/10) # circumradius of the inner pentagon
X <- R * vapply(0L:4L, function(i) cos(pi/180 * (90+72*i)), numeric(1L))
Y <- R * vapply(0L:4L, function(i) sin(pi/180 * (90+72*i)), numeric(1L))
x <- r * vapply(0L:4L, function(i) cos(pi/180 * (126+72*i)), numeric(1L))
y <- r * vapply(0L:4L, function(i) sin(pi/180 * (126+72*i)), numeric(1L))
vertices <- rbind(
  c(X[1L], Y[1L]),
  c(x[1L], y[1L]),
  c(X[2L], Y[2L]),
  c(x[2L], y[2L]),
  c(X[3L], Y[3L]),
  c(x[3L], y[3L]),
  c(X[4L], Y[4L]),
  c(x[4L], y[4L]),
  c(X[5L], Y[5L]),
  c(x[5L], y[5L])
)

# edge indices (pairs)
edges <- cbind(1L:10L, c(2L:10L, 1L))

# constrained Delaunay triangulation
library(RCDT)
del <- delaunay(vertices, edges)

# plot
opar <- par(mar = c(0, 0, 0, 0))
plotDelaunay(
  del, type = "n", asp = 1, fillcolor = "distinct", lwd_borders = 3,
  xlab = NA, ylab = NA, axes = FALSE
)
par(opar)

# area
delaunayArea(del)
## [1] 0.3102707
sqrt(650 - 290*sqrt(5)) / 4 # exact value
## [1] 0.3102707

An eight-pointed star

I found its vertices with the Julia library Luxor.

vertices <- rbind(
  c(2.121320343559643, 2.1213203435596424),
  c(0.5740251485476348, 1.38581929876693),
  c(0.0, 3.0),
  c(-0.5740251485476346, 1.38581929876693),
  c(-2.1213203435596424, 2.121320343559643),
  c(-1.38581929876693, 0.5740251485476349),
  c(-3.0, 0.0),
  c(-1.3858192987669302, -0.5740251485476345),
  c(-2.121320343559643, -2.1213203435596424),
  c(-0.5740251485476355, -1.3858192987669298),
  c(0.0, -3.0),
  c(0.574025148547635, -1.38581929876693),
  c(2.121320343559642, -2.121320343559643),
  c(1.3858192987669298, -0.5740251485476355),
  c(3.0, 0.0),
  c(1.38581929876693, 0.5740251485476349)
)

# edge indices
edges <- cbind(1L:16L, c(2L:16L, 1L))

library(RCDT)
del <- delaunay(vertices, edges)

opar <- par(mar = c(0, 0, 0, 0))
plotDelaunay(
  del, type = "n", asp = 1, fillcolor = "distinct", 
  col_borders = "navy", lty_edges = 2, lwd_borders = 3, lwd_edges = 2, 
  xlab = NA, ylab = NA, axes = FALSE
)
par(opar)

Triangulation of a polygon with a hole

n <- 100L # outer number of sides
angles1 <- seq(0, 2*pi, length.out = n + 1L)[-1L]
outer_points <- cbind(cos(angles1), sin(angles1))
m <- 10L  # inner number of sides
angles2 <- seq(0, 2*pi, length.out = m + 1L)[-1L]
inner_points <- 0.5 * cbind(cos(angles2), sin(angles2))
points <- rbind(outer_points, inner_points)
# constraint edges
indices <- 1L:n
edges_outer <- cbind(
  indices, c(indices[-1L], indices[1L])
)
indices <- n + 1L:m
edges_inner <- cbind(
  indices, c(indices[-1L], indices[1L])
)
edges <- rbind(edges_outer, edges_inner)
# constrained Delaunay triangulation
del <- delaunay(points, edges) 

# plot
opar <- par(mar = c(0, 0, 0, 0))
plotDelaunay(
  del, type = "n", asp = 1, lwd_borders = 3, col_borders = "black", 
  fillcolor = "random", col_edges = "yellow",
  axes = FALSE, xlab = NA, ylab = NA
)
par(opar)

One can also enter a vector of colors in the fillcolor argument. First, see the number of triangles:

del[["mesh"]]
##  mesh3d object with 110 vertices, 110 triangles.

There are 110 triangles. Let’s make a cyclic vector of 110 colors:

colors <- viridisLite::viridis(55)
colors <- c(colors, rev(colors))

And let’s plot now:

opar <- par(mar = c(0, 0, 0, 0))
plotDelaunay(
  del, type = "n", asp = 1, lwd_borders = 3, col_borders = "black", 
  fillcolor = colors, col_edges = "black", lwd_edges = 1.5,
  axes = FALSE, xlab = NA, ylab = NA
)
par(opar)

The colors are assigned to the triangles in the order they are given, but only after the triangles have been circularly ordered.

A funny curve

I found this curve here.

t_ <- seq(-pi, pi, length.out = 193L)[-1L]
r_ <- 0.1 + 5*sqrt(cos(6*t_)^2 + 0.7^2)
xy <- cbind(r_*cos(t_), r_*sin(t_))
edges1 <- cbind(1L:192L, c(2L:192L, 1L))
inner <- which(r_ == min(r_))
edges2 <- cbind(inner, c(tail(inner, -1L), inner[1L]))
del <- delaunay(xy, edges = rbind(edges1, edges2))

opar <- par(mar = c(0, 0, 0, 0))
plotDelaunay(
  del, type = "n", col_borders = "black", lwd_borders = 2, 
  fillcolor = "random", col_edges = "white", 
  axes = FALSE, xlab = NA, ylab = NA, asp = 1
)
polygon(xy[inner, ], col = "#ffff99")
par(opar)

License

The ‘RCDT’ package as a whole is distributed under GPL-3 (GNU GENERAL PUBLIC LICENSE version 3).

It uses the C++ library CDT which is permissively licensed under MPL-2.0. A copy of the ‘CDT’ license is provided in the file LICENSE.note, and the source code of this library can be found in the src folder.

rcdt's Issues

Switch to use `colorsGen`

Could we switch this to use your colorsGen too?

properties of the triangulation

Do you have plans to expose the area/angle/steiner points control? I'm not familiar enough with the code here, so just to point to the properties in {RTriangle} triangulation() that allow specifying maximum area and angle and other properties of the triangulation. Is that of interest for you?

BTW, this is SO EXCELLENT!

Thank you so much, I've been trying to get this going with CGAL but failed - previously I've used {RTriangle} in my {anglr} package.

If of interest here's a spatial example, using {silicate} to break down a polygon layer to edges. That SC0 function also works on line objects, and sp package types as well.

sc <- silicate::SC0(sf::read_sf(system.file("gpkg/nc.gpkg", package = "sf", mustWork = TRUE))[1, ])

library(RCDT)
d <- delaunay(as.matrix(sc$vertex[c("x_", "y_")]), 
              as.matrix(do.call(rbind, sc$object$topology_)[c(".vx0", ".vx1")]))

plotDelaunay(d)

voronoi tesselation

Dear authors,

Thanks so much for creating the outstanding package. As the triangulation has been implemented, could I ask if the fast algorithm can be genralized to voronoi tesselation as well?