#10148: Automorphism group of a Polyhedron
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Reporter: vbraun | Owner: mhampton
Type: enhancement | Status: needs_work
Priority: major | Milestone: sage-feature
Component: geometry | Keywords:
Author: Volker Braun | Upstream: N/A
Reviewer: | Merged:
Work_issues: |
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Changes (by novoselt):
* status: needs_review => needs_work
Comment:
1. I think that it is OK that a private method `_affine_coordinates` does
not check if the input is valid (i.e. you can pass a point which is not in
the affine space), but it is worth noting this feature in its
documentation.
2. There are some copy-paste typos in the math part of
`automorphism_group` docstring.
3. `assert(False)` looks really weird, even though I understand what does
it mean. Personally, I'd rather not have that line at all.
4. It would be nice to have an action by permutations on polyhedra. (This
can wait for another ticket.)
5. The most important point: the description of the computed group is not
entirely clear and I am not sure if you planned the behaviour in the
examples below. It seems to me that you need to switch to projective
embedding to apply the algorithm. It also seems to me that you need to get
rid of the lines first since otherwise the computed group becomes very
obscure, since even directions of line generators are not uniquely
determined.
6. If the computed group is not the group of symmetries in a clear way, I
don't think that the name `automorphism_group` is justified. It should be
either more descriptive or take a mandatory parameter specifying its type,
like `automorphism_group("restricted")` with `automorphism_group()`
raising an exception. I think that at least for bounded polyhedra
`automorphism_group` should mean GL(R,n)-symmetries.
Examples:
{{{
sage: points = map(vector, [(1,0), (0,1), (-2,-1)])
sage: p = Polyhedron(vertices=points)
sage: p.automorphism_group().order()
2
sage: p = Polyhedron(vertices=[pt - points[0] for pt in points])
sage: p.automorphism_group().order()
6
sage: p = Polyhedron(vertices=[pt - points[1] for pt in points])
sage: p.automorphism_group().order()
6
sage: p = Polyhedron(vertices=[pt - 2*points[1] for pt in points])
sage: p.automorphism_group().order()
1
}}}
I want to get the same number in each case. Personally, I like 2 as the
number of GL(Z,2)-symmetries, but as I understand the referenced paper, it
should be 6, the number of GL(R,2)-symmetries.
--
Ticket URL: <http://trac.sagemath.org/sage_trac/ticket/10148#comment:3>
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