# Piecewise-Linear Vineyard¶

Given a simplicial complex , and a sequence of values on each one of its vertices, one may construct a homotopy of PL functions on the complex that interpolates linearly between the values. For any given time in the homotopy, we get a function from the simplicial complex to the real line, and we can compute its persistence diagram. Stacking all such diagrams together we get a persistence vineyard [CEM06]. An example that computes such a vineyard is in `examples/pl-functions/pl-vineyard.cpp`.

Once compiled, it takes three files as the input:

```pl-vineyard complex values output-prefix
```

`complex` lists the simplices of the simplicial complex , one per-line:

```0
1
0 1
2
0 2
...
```

`values` lists the vertex values of the consequtive “frames” of the homotopy. Each line is a sequence of as many numbers as there are vertices in the complex. It describes a single frame. pl-vineayrd constructs the homotopy over the interval , where is the number of frames. As an example of `values` input:

```3.3   6    2
1.2   3    10
7.5   2.1  0
```

This input means: . Similarly, ; .

The vineyard is saved to the files prefixed with `output-prefix`, followed by the dimension and extension, e.g. `myfunction1.vin` or `myfunction1.edg`, depending on the format. The two formats are vines and edges. The former saves one vine per line, listed as a stream of triplets BIRTH DEATH TIME:

```4 5 0 3.4 5.6 0.4 3 6 1 ...
```

The edge format represents the vine as a sequence of edges, each given as a start and end point. So the above vine would appear as:

```4 5 0
3.4 5.6 0.4
3.4 5.6 0.4
3 6 1
...
```

pl-vineyard takes additional options:

-s, --skip-infinite

Do not output infinite vines.

-v, --save-vines

Output vines, instead of the default edge format.

-e, --explicit-events

Go through the events one by one (useful for the debugging).