# 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?

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I suggest to look into obj_repr() source code to see how it does the job.