# Oriented vertices of Polyhedron

Hi, I wonder what is the most convenient way to extract the ordered/oriented vertices of a Polyhedron/PolyhedronFace.

Specifically, say that we have a cube:

```
sage: cb = Polyhedron(vertices=[(0,0,0), (0,0,1), (0,1,0), (0,1,1), (1,0,0), (1,0,1), (1,1,0), (1,1,1)])
```

which consists of 6 facets:

```
sage: cb.facets()
(A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices,
A 2-dimensional face of a Polyhedron in ZZ^3 defined as the convex hull of 4 vertices)
```

I can easily know the indices of vertices (in `cb.Vrepresentation`

) that form each facets, e.g., for the first facet:

```
sage: cb.facets()[0].ambient_V_indices()
(0, 1, 4, 5)
```

However, the four vertices are not properly ordered (linked) as I inspect the adjacency matrix:

```
sage: cb.vertex_adjacency_matrix()
[0 1 1 0 1 0 0 0]
[1 0 0 1 0 1 0 0]
[1 0 0 1 0 0 1 0]
[0 1 1 0 0 0 0 1]
[1 0 0 0 0 1 1 0]
[0 1 0 0 1 0 0 1]
[0 0 1 0 1 0 0 1]
[0 0 0 1 0 1 1 0]
```

(from the adjacent matrix the ordered indices of the four vertices should be 0->1->5->4, or reversed as 4->5->1->0).

The proper orientation of the vertices is needed for B-rep. Though I can extract the proper ordering with `vertex_adjacency_matrix()`

, how to make sure they're consistently CCW orientated (seen from outside) still confuses me.

I'm kinda sure these are well managed in Sage, because the `obj_repr()`

method in plot3d seems working well providing a consistent facet-vertex representation:

```
sage: cb_render = cb.render_solid()
sage: cb_render.obj_repr(cb_render.default_render_params())
['g obj_1',
'usemtl texture7',
['v 0 0 0',
'v 0 0 1',
'v 0 1 0',
'v 0 1 1',
'v 1 0 0',
'v 1 0 1',
'v 1 1 0',
'v 1 1 1'],
['f 8 4 2 6',
'f 7 3 4 8',
'f 8 6 5 7',
'f 3 1 2 4',
'f 5 1 3 7',
'f 6 2 1 5'],
['f 6 2 4 8',
'f 8 4 3 7',
'f 7 5 6 8',
'f 4 2 1 3',
'f 7 3 1 5',
'f 5 1 2 6']]
```

(notice that the vertex index in OBJ representation starts from 1 instead of 0; the line `f 6 2 1 5`

corresponds to the ordering `4->5->1->0`

that I want)

Is there a native method for Polyhedron or PolyhdronFace that I can directly use for this?

I suggest to look into

`obj_repr()`

source code to see how it does the job.Am I understanding correctly that you are referring only to 3-dimensional polytopes? (Because I woudn't even understand the question for higher dimensions).