alphahull (version 2.5)

complement: Complement of the alpha-convex hull

Description

This function calculates the complement of the $\alpha$-convex hull of a given sample for $\alpha >0$.

Usage

complement(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 $\alpha$.

Value

compl

Output matrix. For each row i, compl[i,] contains the information relative to an open ball or halfplane of the complement of the $\alpha$-convex hull, see Details.

Details

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

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

The complement of the $\alpha$-convex hull is calculated as a union of open balls and halfplanes that do not contain any point of the sample. See Edelsbrunnner et al. (1983) for a basic description of the algorithm. The construction of the complement is based on the Delaunay triangulation and Voronoi diagram of the sample, provided by the function delvor. The function complement returns a matrix compl. For each row i, compl[i,] contains the information relative to an open ball or halfplane of the complement. The first three columns are assigned to the characterization of the ball or halfplane i. The information relative to the edge of the Delaunay triangulation that generates the ball or halfplane i is contained in compl[i,4:16]. Thus, if the row i refers to an open ball, compl[i,1:3] contains the center and radius of the ball. Furthermore, compl[i,17:18] and compl[i,19] refer to the unitary vector $v$ and the angle $\theta$ that characterize the arc that joins the two sample points that define the ball i. If the row i refers to a halfplane, compl[i,1:3] determines its equation. For the halfplane $y>a+bx$, compl[i,1:3]=(a,b,-1). In the same way, for the halfplane $y<a+bx$, compl[i,1:3]=(a,b,-2), for the halfplane $x>a$, compl[i,1:3]=(a,0,-3) and for the halfplane $x<a$, compl[i,1:3]=(a,0,-4).

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.

See Also

delvor, ahull.

Examples

# NOT RUN {
# Random sample in the unit square
x <- matrix(runif(100), nc = 2)
# Value of alpha
alpha <- 0.2
# Complement of the alpha-convex hull
compl <- complement(x, alpha = alpha)
# }