#18442: Implement the barycentric subdivision of the boundary of a polytope
-------------------------------------+-------------------------------------
Reporter: jipilab | Owner:
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-6.8
Component: geometry | Resolution:
Keywords: polytope, | Merged in:
barycentric subdivision | Reviewers:
Authors: Jean-Philippe | Work issues:
Labbé | Commit:
Report Upstream: N/A | 996a8a4205d9f959af8e4841e39d16f66a06fe92
Branch: | Stopgaps:
public/ticket/18442 |
Dependencies: |
-------------------------------------+-------------------------------------
Old description:
> The barycentric subdivision of a compact polytope P is obtained as
> follows:
>
> A complete flag of faces {{{F_0,F_1,...,F_{d-1},F_{d}}}} is a nested
> sequence of faces starting with a vertex {{{F_0}}} and ending with the
> full {{{d}}}-dimensional polytope {{{P}}} such that the dimension of
> {{{F_i}}} is {{{i}}}.
>
> To each complete flag of faces we can assign a simplex whose vertices are
> the geometric barycenter of the {{{d+1}}} faces in the complete flag.
> This gives a subdivision of the polytope into simplices. The number of
> simplices is equal to the number of complete chains in the face lattice.
>
> Similarly, it is possible to define the barycentric subdivision of the
> boundary of a compact polytope by taking only the complete flag of faces
> up to {{{d-1}}}-dimensional faces. The simplicial complex realizing this
> barycentric subdivision can be realized convexly by pulling the
> barycenters to the outside of the polytope with a convenient factor.
>
> The result of such a process is a polytope whose facets correspond to
> complete flags of faces of the original polytope.
>
> This method computes the barycentric subdivision of the boundary of a
> compact polyhedron.
>
> See also
> https://en.wikipedia.org/wiki/Barycentric_subdivision#Barycentric_subdivision_of_a_convex_polytope
New description:
The boundary of a polytope is a CW-complex for which we compute a
geometric realization of its barycentric subdivision as follows:
A complete flag of faces {{{F_0,F_1,...,F_{d-1},F_{d}}}} is a nested
sequence of faces starting with a vertex {{{F_0}}} and ending with the
full {{{d}}}-dimensional polytope {{{P}}} such that the dimension of
{{{F_i}}} is {{{i}}}.
To each complete flag of faces we can assign a simplex whose vertices are
the geometric barycenter of the {{{d+1}}} faces in the complete flag. This
gives a subdivision of the polytope into simplices. The number of
simplices is equal to the number of complete chains in the face lattice.
Similarly, it is possible to define the barycentric subdivision of the
boundary of a compact polytope by taking only the complete flag of faces
up to {{{d-1}}}-dimensional faces. The simplicial complex realizing this
barycentric subdivision can be realized convexly by pulling the
barycenters to the outside of the polytope with a convenient factor.
The result of such a process is a polytope whose facets correspond to
complete flags of faces of the original polytope.
This method computes a geometric realization of the barycentric
subdivision of the boundary of a compact polyhedron.
See also
https://en.wikipedia.org/wiki/Barycentric_subdivision#Barycentric_subdivision_of_a_convex_polytope
--
Comment (by jipilab):
Replying to [comment:27 dimpase]:
> Replying to [comment:26 jipilab]:
> > Yes. This is the definition of a polytope: a bounded polyhedron.
>
> you might also find definitions of polyheda, as finite unions of convex
polyhedra.
> That is, convexity is not always assumed.
This is true, for example in rigidity theory. As far as I know, the
Polyhedron class in Sage deals only with convex polyhedra. In this case,
it is assumed that it is convex.
>
> >
> > Basically the notion of polytope is subsumed in Sage by the class
Polyhedron.
> >
> > Theorically a polyhedron (with H representation) is always decomposed
as the sum of a polyhedral cone and a convex polytope, therefore polytopes
are considered in the class polyhedron simply where the cone is a point,
and the way to detect them is by checking compactness.
> >
> > The description uses polyhedron simply because polytope is more or
less absent from the nomenclature in sage apart from the database of
polytopes.
> >
> hmm, and what do you compute then? The boundary of a polyhedron is not
convex...
The boundary of a polytope is a CW-complex for which we compute a
geometric realization of its barycentric subdivision.
--
Ticket URL: <http://trac.sagemath.org/ticket/18442#comment:28>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
--
You received this message because you are subscribed to the Google Groups
"sage-trac" group.
To unsubscribe from this group and stop receiving emails from it, send an email
to [email protected].
To post to this group, send email to [email protected].
Visit this group at http://groups.google.com/group/sage-trac.
For more options, visit https://groups.google.com/d/optout.
