# Alpha_complex #

## `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 b d`

where `dim` is the dimension of the homological feature, `b` and `d` 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 <input OFF file> <p> <min_persistence>` 
where
`<input OFF file>` is the path to the input point cloud in OFF format.
`<p>` is the characteristic p of the coefficient field *Z/pZ* for computing homology. It must be a stricly positive integer.
`<min_persistence>` is the 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 2 0.45`

outputs:
```
Simplex_tree dim: 3
2  0 0 inf 
2  1 0.0682162 1.0001 
2  1 0.0934117 1.00003 
2  2 0.56444 1.03938 
```

Here we retrieve expected Betti numbers on a tore 3D:
```
Betti numbers[0] = 1
Betti numbers[1] = 2
Betti numbers[2] = 1
```

N.B.: 
* `alpha_complex_3d_persistence` only accepts OFF files in dimension 3.
* Filtration values are alpha square values.




## `periodic_alpha_complex_3d_persistence` ##
This program computes the persistent homology with coefficient field Z/pZ of the 3D periodic alpha complex built from a 3D point cloud. The output diagram contains one bar per line, written with the convention:

`p dim b d`

where `dim` is the dimension of the homological feature, `b` and `d` 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**
`periodic_alpha_complex_3d_persistence <input OFF file> <p> <min_persistence>`
where
`<input OFF file>` is the path to the input point cloud in OFF format.
`<p>` is the characteristic p of the coefficient field *Z/pZ* for computing homology. It must be a stricly positive integer.
`<min_persistence>` is the 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 3 1.0`

outputs:
```
Periodic Delaunay computed.
Simplex_tree dim: 3
3  0 0 inf 
3  1 0.0025 inf 
3  1 0.0025 inf 
3  1 0.0025 inf 
3  2 0.005 inf 
3  2 0.005 inf 
3  2 0.005 inf 
3  3 0.0075 inf 
```

Here we retrieve expected Betti numbers on an 3D iso-oriented cuboids:
```
Betti numbers[0] = 1
Betti numbers[1] = 3
Betti numbers[2] = 3
Betti numbers[3] = 1
```

N.B.: 
* `periodic_alpha_complex_3d_persistence` only accepts OFF files in dimension 3.
* In this example, the periodic cube is hard coded to { x = [0,1]; y = [0,1]; z = [0,1] }
* Filtration values are alpha square values.





## `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 b d`

where `dim` is the dimension of the homological feature, `b` and `d` 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] <OFF input file>`

**Allowed options**

* `-h [ --help ]` Produce help message
* `-o [ --output-file ]` Name of file in which the persistence diagram is written. By default, print in std::cout.
* `-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`

outputs:
```
Alpha complex is of dimension 3 - 9273 simplices - 300 vertices.
Simplex_tree dim: 3
2  0 0 inf 
2  1 0.0682162 1.0001 
2  1 0.0934117 1.00003 
2  2 0.56444 1.03938 
```

Here we retrieve expected Betti numbers on a tore 3D:
```
Betti numbers[0] = 1
Betti numbers[1] = 2
Betti numbers[2] = 1
```

N.B.: 
* `alpha_complex_persistence` only accepts OFF files in dimension d.
* Filtration values are alpha square values.