#18454: New `random_cone()` function
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Reporter: mjo | Owner:
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-6.8
Component: geometry | Resolution:
Keywords: | Merged in:
Authors: Michael Orlitzky | Reviewers:
Report Upstream: N/A | Work issues:
Branch: | Commit:
u/mjo/ticket/18454 | 55da704f2f101cfdc25ba1c0b35f38072cb0e7f3
Dependencies: | Stopgaps:
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Comment (by novoselt):
Replying to [comment:6 mjo]:
> Replying to [comment:5 novoselt]:
> > Well, it would be helpful to know in what sense the cone is "random".
If it is a cone on a random set of rays, then dimension+1 is the only
sensible upper bound to me - such a cone always can be constructed.
> In any dimension you can do better, up to 2*dimension -- just take the
standard basis and add negative multiples of them. The resulting come will
have 2*dimension generating rays and be equal to the entire space.
"Better" in what sense? Take (or "generate randomly") rays (1,0), (0,1),
(-1,-1) in the plane and you can no longer add rays which are independent
of these three. "Random generation" of four rays in opposite pairs is much
less likely to happen. So if you keep generating rays removing those which
are dependent with already constructed ones until you reach a predefined
maximum, you cannot guarantee that more than 3 are possible.
> > In R^2^ it may be possible to force construction of cones with more
than 4 rays - given that the code can handle 4 instead of 3 for the whole
space, probably it can live with 100.
> Can you really get more than four in R^2^?
I meant that you can have a cone in the plane with a hundred rays that you
decided to call generating ones - Sage constructors may allow you to do it
for cones with linear subspaces, although I don't know why one would want
it.
> > But if we are talking about the minimal number of extremal rays, then
for R^1,2^ maximum is the dimension and for all others we can do anything
- generate a random n-gon in a plane and lift it.
>
> Right, so here you definitely wouldn't want to limit the number of rays
to dimension+1.
Nor to twice the dimension!
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Ticket URL: <http://trac.sagemath.org/ticket/18454#comment:7>
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