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---

# The GL(2,R) Action, the Veech Group, Delaunay Decomposition

## 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()
```

```{code-cell}
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m = matrix([[2, 1], [1, 1]])
```

You can act on surfaces with the $GL(2,R)$ action

```{code-cell}
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ss = m * s
ss
```

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

To "renormalize" you can improve the presentation using the Delaunay decomposition.

```{code-cell}
---
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sss = ss.delaunay_decomposition()
sss
```

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

## The Veech group

Set $s$ to be the double pentagon again.

```{code-cell}
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s = translation_surfaces.veech_double_n_gon(5)
```

The surface has a horizontal cylinder decomposition all of whose moduli are given as below

```{code-cell}
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p = s.polygon(0)
modulus = (p.vertex(3)[1] - p.vertex(2)[1]) / (p.vertex(2)[0] - p.vertex(4)[0])
AA(modulus)
```

```{code-cell}
---
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m = matrix(s.base_ring(), [[1, 1 / modulus], [0, 1]])
show(m)
```

```{code-cell}
---
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---
show(matrix(AA, m))
```

The following can be used to check that $m$ is in the Veech group of $s$.

```{code-cell}
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s.canonicalize() == (m * s).canonicalize()
```

## Infinite surfaces

Infinite surfaces support multiplication by matrices and computing the Delaunay decomposition. (Computation is done "lazily")

```{code-cell}
---
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s = translation_surfaces.chamanara(1 / 2)
```

```{code-cell}
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---
s.plot(edge_labels=False, polygon_labels=False)
```

```{code-cell}
---
jupyter:
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---
ss = s.delaunay_decomposition()
```

```{code-cell}
---
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gs = ss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
```

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

```{code-cell}
---
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---
m = matrix([[2, 0], [0, 1 / 2]])
```

```{code-cell}
---
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ms = m * s
```

```{code-cell}
---
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gs = ms.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
```

```{code-cell}
---
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---
mss = ms.delaunay_decomposition()
```

```{code-cell}
---
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gs = mss.graphical_surface(edge_labels=False, polygon_labels=False)
gs.make_all_visible(limit=20)
gs.plot()
```

You can tell from the above picture that $m$ is in the Veech group.