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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:

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) })

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