# Alpha complex

## alpha_complex_persistence

This program computes the persistent homology with coefficient field Z/pZ of the dD alpha complex built from a dD point cloud. The output diagram contains one bar per line, written with the convention:

```
p dim birth death
```

where `dim`

is the dimension of the homological feature, `birth`

and `death`

are respectively the birth and death of the feature,
and `p`

is the characteristic of the field *Z/pZ* used for homology coefficients (`p`

must be a prime number).

**Usage**

```
alpha_complex_persistence [options] <input OFF file>
```

where
`<input OFF file>`

is the path to the input point cloud in nOFF ASCII format.

**Allowed options**

`-h [ --help ]`

Produce help message`-o [ --output-file ]`

Name of file in which the persistence diagram is written. Default print in standard output.`-r [ --max-alpha-square-value ]`

(default = inf) Maximal alpha square value for the Alpha complex construction.`-p [ --field-charac ]`

(default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.`-m [ --min-persistence ]`

(default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

**Example**

```
alpha_complex_persistence -r 32 -p 2 -m 0.45 ../../data/points/tore3D_300.off
```

N.B.:

- Filtration values are alpha square values.

## alpha_complex_3d_persistence

This program computes the persistent homology with coefficient field Z/pZ of the 3D alpha complex built from a 3D point cloud. The output diagram contains one bar per line, written with the convention:

```
p dim birth death
```

where `dim`

is the dimension of the homological feature, `birth`

and `death`

are respectively the birth and death of the feature, and `p`

is the characteristic of the field *Z/pZ* used for homology coefficients (`p`

must be a prime number).

**Usage**

```
alpha_complex_3d_persistence [options] <input OFF file>
```

where `<input OFF file>`

is the path to the input point cloud in nOFF ASCII format.

**Allowed options**

`-h [ --help ]`

Produce help message`-o [ --output-file ]`

Name of file in which the persistence diagram is written. Default print in standard output.`-p [ --field-charac ]`

(default=11) Characteristic p of the coefficient field Z/pZ for computing homology.`-m [ --min-persistence ]`

(default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

**Example**

```
alpha_complex_3d_persistence ../../data/points/tore3D_300.off -p 2 -m 0.45
```

N.B.:

`alpha_complex_3d_persistence`

only accepts OFF files in dimension 3.- Filtration values are alpha square values.

## exact_alpha_complex_3d_persistence

Same as `alpha_complex_3d_persistence`

, but using exact computation.
It is slower, but it is necessary when points are on a grid for instance.

## weighted_alpha_complex_3d_persistence

Same as `alpha_complex_3d_persistence`

, but using weighted points.

**Usage**

```
weighted_alpha_complex_3d_persistence [options] <input OFF file> <weights input file>
```

where

`<input OFF file>`

is the path to the input point cloud in nOFF ASCII format.`<input weights file>`

is the path to the file containing the weights of the points (one value per line).

**Allowed options**

`-h [ --help ]`

Produce help message`-o [ --output-file ]`

Name of file in which the persistence diagram is written. Default print in standard output.`-p [ --field-charac ]`

(default=11) Characteristic p of the coefficient field Z/pZ for computing homology.`-m [ --min-persistence ]`

(default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.

**Example**

```
weighted_alpha_complex_3d_persistence ../../data/points/tore3D_300.off ../../data/points/tore3D_300.weights -p 2 -m 0.45
```

N.B.:

- Weights values are explained on CGAL Alpha shape and Regular triangulation documentation.
- Filtration values are alpha square values.

## periodic_alpha_complex_3d_persistence

Same as `alpha_complex_3d_persistence`

, but using periodic alpha shape 3d.
Refer to the CGAL’s 3D Periodic Triangulations User Manual for more details.

**Usage**

```
periodic_alpha_complex_3d_persistence [options] <input OFF file> <cuboid file>
```

where

`<input OFF file>`

is the path to the input point cloud in nOFF ASCII format.`<cuboid file>`

is the path to the file describing the periodic domain. It must be in the format described here.

**Allowed options**

`-h [ --help ]`

Produce help message`-o [ --output-file ]`

Name of file in which the persistence diagram is written. Default print in standard output.`-p [ --field-charac ]`

(default=11) Characteristic p of the coefficient field Z/pZ for computing homology.`-m [ --min-persistence ]`

(default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals

**Example**

```
periodic_alpha_complex_3d_persistence ../../data/points/grid_10_10_10_in_0_1.off ../../data/points/iso_cuboid_3_in_0_1.txt -p 3 -m 1.0
```

N.B.:

- Cuboid file must be in the format described here.
- Filtration values are alpha square values.