complement {alphahull} | R Documentation |

## 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 |

`alpha` |
Value of |

### 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)`

.

### Value

`compl` |
Output matrix. For each row |

### 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

### 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)
## End(Not run)
```

*alphahull*version 2.5 Index]