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 |

`alpha` |
Value of |

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_θ v* and *c+rA_{-θ}v* where *c* and *r* represent the center and radius of the arc, repectively, and *A_θ v* represents the clockwise rotation of angle *θ* of the unitary vector *v*. Joining the end points of adjacent arcs we can define polygons that help us to determine the area of the estimator , see `areaahull`

.

A list with the following components:

`arcs` |
For each arc in the boundary of the |

`xahull` |
A 2-column matrix with the coordinates of the original set of points besides possible new end points of the arcs in the boundary of the |

`length` |
Length of the boundary of the |

`complement` |
Output matrix from |

`alpha` |
Value of |

`ashape.obj` |
Object of class |

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.

## Not run: # 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) plot(ahull.obj) # Uniform sample of size n=300 in the annulus B(c,0.5)\B(c,0.25), # with c=(0.5,0.5). n <- 300 theta<-runif(n,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)) # Value of alpha alpha <- 0.1 # Alpha-convex hull ahull.obj <- ahull(x, alpha = alpha) # The arcs defining the boundary of the alpha-convex hull are ordered plot(x) for (i in 1:dim(ahull.obj$arcs)[1]){ arc(ahull.obj$arcs[i,1:2],ahull.obj$arcs[i,3],ahull.obj$arcs[i,4:5], ahull.obj$arcs[i,6],col=2) Sys.sleep(0.5) } # Random sample from a uniform distribution on a Koch snowflake # with initial side length 1 and 3 iterations x <- rkoch(2000, side = 1, niter = 3) # Value of alpha alpha <- 0.05 # Alpha-convex hull ahull.obj <- ahull(x, alpha = alpha) plot(ahull.obj) ## End(Not run)

[Package *alphahull* version 2.2 Index]