# 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
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.
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)
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.
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