Polytopes: Initial documentation #468
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@benlorenz said that the docs will build for a draft pr, right now it is a hassle to build the docs locally. |
…ded PolyhedralGeometry category to the side bar
docs/src/PolyhedralGeometry/pg.md
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| - Theobald + Joswig | ||
| - Ziegler |
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Note that you can use bibtex references (see docs/oscar_references.bib; to cite something, use [bibkey](@cite).
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| Polyhedra are convex bodies. When a polyhedron $P \subset \mathbb{R}^n$ is bounded, it is called a polytope and the fundamental theorem of polytopes states that it may be written as the convex hull of finitely many points in $\mathbb{R}^n$. That is $$P = \textrm{conv}(p_1,\ldots,p_N), p_i \in \mathbb{R}^n.$$ Writing $P$ in this way is called its $V$-representation. | ||
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| Since many polyhedral computations are done through ```polymake```, the structure ```Polyhedron``` contains a pointer ```pm_polytope``` to the corresponding ```polymake``` object. Thus, all functionality of ```Polymake.jl``` is accessible by calling a ```Polymake.jl``` function on this pointer. |
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Is there a reason to use triple backticks here instead of single backticks?
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| Since many polyhedral computations are done through ```polymake```, the structure ```Polyhedron``` contains a pointer ```pm_polytope``` to the corresponding ```polymake``` object. Thus, all functionality of ```Polymake.jl``` is accessible by calling a ```Polymake.jl``` function on this pointer. | ||
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| **Warning**: Polyhedra in ```polymake``` and ```Polymake.jl``` use homogeneous coordinates. The polyhedra in ```Oscar``` use affine coordinates. |
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Perhaps this should be a "warning admonition"? See https://juliadocs.github.io/Documenter.jl/stable/showcase/#Warning-admonition
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| A subset $P \subseteq \mathbb{R}^n$ is called a (metric) polyhedron if it can be written as the intersection of finitely many closed affine halfspaces in $\mathbb{R}^n$. | ||
| That is, there exists a matrix $A$ and a vector $b$ such that | ||
| $$P = P(A,b) = \{ x \in \mathbb{R}^n \mid Ax \leq b\}.$$ |
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Maybe its a bot late, but we should think about xA leq b as well - after all, matrices are supposed to be row matrices...
| @@ -87,11 +107,34 @@ Base.length(iter::VertexPointIterator) = nvertices(iter.p) | |||
| """ | |||
| vertices(P::Polyhedron, [,as::Type{T} = Points]) | |||
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have you considered vertices(::Type{bla}, P) as an alternative to (P, as = bla)? Julia odes this for round, floor, ...
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Agreed, and we also do this elsewhere in Oscar (e.g. for order of groups, where one specify the desired integer type for the return value). So the caller would write vertices(Points, P). And you could still allow vertices(P) by providing a convenience method to set the default: vertices(P::Polyhedron) = vertices(Points, P)
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Here the first argument is not the type of the return value, so how does this fit with the functions you mention?
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On Thu, Jun 17, 2021 at 03:18:56AM -0700, thofma wrote:
@thofma commented on this pull request.
> @@ -87,11 +107,34 @@ Base.length(iter::VertexPointIterator) = nvertices(iter.p)
"""
vertices(P::Polyhedron, [,as::Type{T} = Points])
Here the first argument is not the type of the return value, so how does this fit with the functions you mention?
Its the spririt: the 1st argument determines the return type
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#468 (comment)
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This pr is a draft of the documentation of the Polytopes part of Oscar.jl.