ahull {alphahull} | R Documentation |
This function calculates the α-convex hull of a given sample of points in the plane for α>0.
ahull(x, y = NULL, alpha)
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. |
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.
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 . |
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.
# 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)