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A new paper on the α-convex hull appeared in the Journal of Statistical Software today (http://www.jstatsoft.org/v34/i05/paper). The α-convex hull is an interesting problem which caught my attention long time ago but I didn’t know a solution then. R has a function chull() which can generate (indices of) the convex hull for a series of points. Now we can use the R package alphahull to compute the α-convex hull. For those who are not familiar with the α-convex hull, the animation below might be a good illustration for the difference between a convex hull and an α-convex hull. Note how the parameter α affects the shape of the hull:

alpha-convex hull with different alpha's

The above animation can be reproduced with the code below (uncomment the lines to create a GIF animation with the animation package):

set.seed(123)
theta = runif(n <- 300, 0, 2 * pi)
r = sqrt(runif(n, 0.25^2, 0.5^2))
x = cbind(0.5 + r * cos(theta), 0.5 + r * sin(theta))

## library(animation)
## saveMovie({
library(alphahull)
par(mar = rep(0, 4), xaxt = "n", yaxt = "n", bg = "black", col = "white")
for (alpha in seq(0.25, 0, -0.01)) {
plot(ahull(x, alpha = alpha), pch = 20, col = "white", panel.last = text(0.5,
0.5, sprintf("alpha = %.2f", alpha), cex = 2))
}
## }, moviename = "alpha-convex-hull")


Again, you may interactively play with the convex hull using the gWidgets package:

## install.packages('gWidgetsRGtk2') first if not installed
library(gWidgetsRGtk2)
options(guiToolkit = "RGtk2")

g = glayout(container = gwindow("alpha-convex hull demo"))
g[1, 1:2, expand = TRUE] = ggraphics(container = g)
g[2, 1] = "alpha"
g[2, 2, expand = TRUE] = gslider(from = 0, to = 0.3, value = 0.2, by = 0.01,
container = g, handler = function(h, ...) {
par(mar = rep(0, 4), xaxt = "n", yaxt = "n")
plot(ahull(x, alpha = svalue(h\$obj)), pch = 20, ann = FALSE)
})