cxhullEdges {cxhull}

Vertices and edges of convex hull

Description

Computes the vertices and the edges of the convex hull of a set of points.

Usage

cxhullEdges(points, adjacencies = FALSE, orderEdges = FALSE)

Arguments

points

numeric matrix, one point per row; it must contain at least three columns (the two-dimensional case is not implemented yet)

adjacencies

Boolean, whether to return the vertex adjacencies

orderEdges

Boolean, whether to order the edges in the output

Value

A list with two fields: vertices and edges. The vertices field is a list which provides an id for each vertex and its coordinates. If adjacencies=TRUE, it provides in addition the ids of the adjacent vertices for each vertex. The edges fields is an integer matrix with two columns. Each row provides the two ids of the vertices of the corresponding edge.

Examples

library(cxhull)
# let's try with the hexacosichoron (see `?hexacosichoron`)
#   it is convex so its convex hull is itself
VE <- cxhullEdges(hexacosichoron)
edges <- VE[["edges"]]
random_edge <- edges[sample.int(720L, 1L), ]
A <- hexacosichoron[random_edge[1L], ]
B <- hexacosichoron[random_edge[2L], ]
sqrt(c(crossprod(A - B))) # this is 2/phi
# Now let's project the polytope to the H4 Coxeter plane 
phi <- (1 + sqrt(5)) / 2
u1 <- c(
  0, 
  2*phi*sin(pi/30), 
  0, 
  1
)
u2 <- c(
  2*phi*sin(pi/15), 
  0, 
  2*sin(2*pi/15), 
  0
)
u1 <- u1 / sqrt(c(crossprod(u1)))
u2 <- u2 / sqrt(c(crossprod(u2)))
# projections to the Coxeter plane
proj <- function(v){
  c(c(crossprod(v, u1)), c(crossprod(v, u2)))
}
points <- t(apply(hexacosichoron, 1L, proj))
# we will assign a color to each edge  
#   according to the norms of its two vertices
norms2 <- round(apply(points, 1L, crossprod), 1L)
( tbl <- table(norms2) )
#> 0.4 1.6 2.4 3.6 
#>  30  30  30  30 
values <- as.numeric(names(tbl))
grd <- as.matrix(expand.grid(values, values)) 
grd <- grd[grd[, 1L] <= grd[, 2L], ]
pairs <- apply(grd, 1L, paste0, collapse = "-")
colors <- hcl.colors(nrow(grd), palette = "Hawaii", rev = TRUE)
if(require("colorspace")) {
  colors <- colorspace::darken(colors, amount = 0.3)
}
names(colors) <- pairs
# plot ####
opar <- par(mar = c(0, 0, 0, 0))
plot(
  points[!duplicated(points), ], pch = 19, cex = 0.3, asp = 1, 
  axes = FALSE, xlab = NA, ylab = NA
)
for(i in 1L:nrow(edges)){
  twopoints <- points[edges[i, ], ]
  nrms2 <- round(sort(apply(twopoints, 1L, crossprod)), 1L)
  pair <- paste0(nrms2, collapse = "-")
  lines(twopoints, lwd = 0.5, col = colors[pair])
}
par(opar)