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---

# Straight-Line Flow

## Acting on surfaces by matrices.

```{code-cell}
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---
from flatsurf import translation_surfaces

s = translation_surfaces.veech_double_n_gon(5)
```

```{code-cell}
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s.plot()
```

Defines the tangent_bundle on the surface defined over the ``base_ring`` of s.

```{code-cell}
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TB = s.tangent_bundle()
```

```{code-cell}
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baricenter = sum(s.polygon(0).vertices()) / 5
```

Define the tangent vector based at the baricenter of polygon 0 aimed downward.

```{code-cell}
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v = TB(0, baricenter, (0, -1))
```

Convert to a straight-line trajectory. Trajectories are unions of segments in polygons.

```{code-cell}
---
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traj = v.straight_line_trajectory()
```

```{code-cell}
---
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s.plot() + traj.plot()
```

Flow into the next $100$ polygons or until the trajectory hits a vertex.

```{code-cell}
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---
traj.flow(100)
```

```{code-cell}
---
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---
s.plot() + traj.plot()
```

We can tell its type.

```{code-cell}
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traj.is_saddle_connection()
```

You can also test if a straight-line trajectory is closed or a forward/backward separatrix.

Lets do it again but in the slope one direction.

```{code-cell}
---
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v = TB(0, baricenter, (1, 1))
```

```{code-cell}
---
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traj = v.straight_line_trajectory()
```

```{code-cell}
---
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---
traj.flow(100)
```

```{code-cell}
---
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---
s.plot() + traj.plot()
```

We remark that it follows from work of Veech that the slope one direction is ergodic for the straight-line flow.