#8656: face_lattice does not seem to work for unbounded polyhedra
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Reporter: novoselt | Owner: mhampton
Type: defect | Status: new
Priority: major | Milestone: sage-4.4.2
Component: geometry | Keywords:
Author: | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Comment(by novoselt):
Well, what exactly is a face of a convex polyhedron in an affine space? I
take as the definition "an intersection of a supporting hyperplane with
the polyhedron." In this case rays cannot appear "by themselves," without
the vertex from which they are going.
I think that you actually agree with me ;-) Why did you throw away the
"edge at infinity?" I think, because it is "completely at infinity." But
the same is true for a standalone ray representing "a vertex at infinity"
- this face has no points in the affine space. On the other hand, "a half-
infinite" edge defined by a vertex and a ray lives in the affine space
except for one endpoint. So I would argue that we should throw away all
faces generated by infinite point only. Or at least do this by default and
have an option to include them, but then I think that ALL infinite faces
should be included.
Actually, just a couple of days ago I have implemented the algorithm used
for face_lattice following the reference in the documentation (and my
implementation seems to be faster, but it is not yet ready for inclusion).
This algorithm computes the Hasse diagram of an atomic and coatomic
lattice starting from incidences. In the case of a bounded full-
dimensional polyhedron vertices are atoms and facets are coatoms. If the
polytope includes rays and lines in addition to vertices and equations in
addition to inequalities, one should be more careful.
Andrey
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Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/8656#comment:3>
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