ahull {alphahull}R Documentation

alpha-convex hull calculation

Description

This function calculates the α-convex hull of a given sample of points in the plane for α>0.

Usage

ahull(x, y = NULL, alpha)

Arguments

x, y The x and y arguments provide the x and y coordinates of a set of points. Alternatively, a single argument x can be provided, see Details.
alpha Value of α.

Details

An attempt is made to interpret the arguments x and y in a way suitable for computing the α-convex hull. Any reasonable way of defining the coordinates is acceptable, see xy.coords.

The α-convex hull is defined for any finite number of points. However, since the algorithm is based on the Delaunay triangulation, at least three non-collinear points are required.

If y is NULL and x is an object of class "delvor", then the α-convex hull is computed with no need to invoke again the function delvor (it reduces the computational cost).

The complement of the α-convex hull can be written as the union of O(n) open balls and halfplanes, see complement. The boundary of the α-convex hull is formed by arcs of open balls of radius α (besides possible isolated sample points). The arcs are determined by the intersections of some of the balls that define the complement of the α-convex hull. The extremes of an arc are given by c+rA_theta v and c+rA_{-theta}v where c and r represent the center and radius of the arc, repectively, and A_theta v represents the clockwise rotation of angle theta of the unitary vector v.

Value

A list with the following components:

arcs For each arc in the boundary of the α-convex hull, the columns of the matrix arcs store the center c and radius r of the arc, the unitary vector v and the angle theta that define the arc, see Details. The matrix arcs also stores the sample points that lie on the boundary of the α-convex hull (rows where columns 3 to 6 are equal to zero).
length Length of the boundary of the α-convex hull, see lengthahull.
complement Output matrix from complement.
alpha Value of α.
x A 2-column matrix with the coordinates of the set of points.
ashape.obj Object of class "ashape" returned by the function ashape.

References

Edelsbrunner, H., Kirkpatrick, D.G. and Seidel, R. (1983). On the shape of a set of points in the plane. IEEE Transactions on Information Theory, 29(4), pp.551-559.

Rodriguez-Casal, R. (2007). Set estimation under convexity type assumptions. Annales de l'I.H.P.- Probabilites & Statistiques, 43, pp.763-774.

Pateiro-Lopez, B. (2008). Set estimation under convexity type restrictions. Phd. Thesis. Universidad de Santiago de Compostela. ISBN 978-84-9887-084-8.

See Also

plot.ahull.

Examples

# Random sample in the unit square
x<-matrix(runif(100),nc=2)
# Value of alpha
alpha<-0.2
# Alpha-convex hull
ahull.obj<-ahull(x,alpha=alpha)

[Package alphahull version 0.1 Index]