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repop.jl
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repop.jl
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export convexhull, convexhull!, conichull, polar
const HAny{T} = Union{HRep{T}, HRepElement{T}}
const VAny{T} = Union{VRep{T}, VRepElement{T}}
"""
intersect(P1::HRep, P2::HRep)
Takes the intersection of `P1` and `P2` ``\\{\\, x : x \\in P_1, x \\in P_2 \\,\\}``.
It is very efficient between two H-representations or between two polyhedron for which the H-representation has already been computed.
However, if `P1` (resp. `P2`) is a polyhedron for which the H-representation has not been computed yet, it will trigger a representation conversion which is costly.
See the [Polyhedral Computation FAQ](http://www.cs.mcgill.ca/~fukuda/soft/polyfaq/node25.html) for a discussion on this operation.
The type of the result will be chosen closer to the type of `P1`. For instance, if `P1` is a polyhedron (resp. H-representation) and `P2` is a H-representation (resp. polyhedron), `intersect(P1, P2)` will be a polyhedron (resp. H-representation).
If `P1` and `P2` are both polyhedra (resp. H-representation), the resulting polyhedron type (resp. H-representation type) will be computed according to the type of `P1`.
The coefficient type however, will be promoted as required taking both the coefficient type of `P1` and `P2` into account.
"""
function Base.intersect(p1::HRep, p2::HRep, ps::HRep...)
p = (p1, p2, ps...)
T = promote_coefficient_type(p)
similar(p, hmap((i, x) -> convert(similar_type(typeof(x), T), x), FullDim(p[1]), T, p...)...)
end
Base.intersect(p::HRep, el::HRepElement) = p ∩ intersect(el)
Base.intersect(el::HRepElement, p::HRep) = p ∩ el
Base.intersect(h::HRepElement) = hrep([h])
Base.intersect(hp1::HyperPlane, hp2::HyperPlane, hps::HyperPlane...) = hrep([hp1, hp2, hps...])
Base.intersect(hs1::HalfSpace, hs2::HalfSpace, hss::HalfSpace...) = hrep([hs1, hs2, hss...])
Base.intersect(h1::HyperPlane{T}, h2::HalfSpace{T}) where {T} = hrep([h1], [h2])
function Base.intersect(h1::HyperPlane, h2::HalfSpace)
T = promote_type(coefficient_type(h1), coefficient_type(h2))
f(h) = convert(similar_type(typeof(h), T), h)
intersect(f(h1), f(h2))
end
Base.intersect(h1::HalfSpace, h2::HyperPlane) = h2 ∩ h1
Base.intersect(p1::HAny, p2::HAny, ps::HAny...) = intersect(p1 ∩ p2, ps...)
#function Base.intersect(p::HAny...)
# T = promote_type(coefficient_type.(p)...)
# f(p) = convert(similar_type(typeof(p), T), p)
# intersect(f.(p)...)
#end
"""
intersect!(p::HRep, h::Union{HRepresentation, HRepElement})
Same as [`intersect`](@ref) except that `p` is modified to be equal to the intersection.
"""
function Base.intersect!(p::HRep, ::Union{HRepresentation, HRepElement})
error("intersect! not implemented for $(typeof(p)). It probably does not support in-place modification, try `intersect` (without the `!`) instead.")
end
function Base.intersect!(p::Polyhedron, h::Union{HRepresentation, HRepElement})
resethrep!(p, hrep(p) ∩ h)
end
"""
convexhull(P1::VRep, P2::VRep)
Takes the convex hull of `P1` and `P2` ``\\{\\, \\lambda x + (1-\\lambda) y : x \\in P_1, y \\in P_2 \\,\\}``.
It is very efficient between two V-representations or between two polyhedron for which the V-representation has already been computed.
However, if `P1` (resp. `P2`) is a polyhedron for which the V-representation has not been computed yet, it will trigger a representation conversion which is costly.
The type of the result will be chosen closer to the type of `P1`. For instance, if `P1` is a polyhedron (resp. V-representation) and `P2` is a V-representation (resp. polyhedron), `convexhull(P1, P2)` will be a polyhedron (resp. V-representation).
If `P1` and `P2` are both polyhedra (resp. V-representation), the resulting polyhedron type (resp. V-representation type) will be computed according to the type of `P1`.
The coefficient type however, will be promoted as required taking both the coefficient type of `P1` and `P2` into account.
"""
function convexhull(p1::VRep, p2::VRep, ps::VRep...)
p = (p1, p2, ps...)
T = promote_coefficient_type(p)
similar(p, vmap((i, x) -> convert(similar_type(typeof(x), T), x), FullDim(p[1]), T, p...)...)
end
convexhull(p::VRep, el::VRepElement) = convexhull(p, convexhull(el))
convexhull(el::VRepElement, p::VRep) = convexhull(p, el)
convexhull(v::VRepElement) = vrep([v])
convexhull(p1::AbstractVector, p2::AbstractVector, ps::AbstractVector...) = vrep([p1, p2, ps...])
convexhull(l1::Line, l2::Line, ls::Line...) = vrep([l1, l2, ls...])
convexhull(r1::Ray, r2::Ray, rs::Ray...) = vrep([r1, r2, rs...])
convexhull(p::AbstractVector{T}, r::Union{Line{T}, Ray{T}}) where {T} = vrep([p], [r])
function convexhull(p::AbstractVector, r::Union{Line, Ray})
T = promote_type(coefficient_type(p), coefficient_type(r))
f(x) = convert(similar_type(typeof(x), T), x)
return convexhull(f(p), f(r))
end
convexhull(r::Union{Line, Ray}, p::AbstractVector) = convexhull(p, r)
convexhull(l::Line, r::Ray) = vrep([l], [r])
convexhull(r::Ray, l::Line) = convexhull(l, r)
convexhull(p1::VAny, p2::VAny, ps::VAny...) = convexhull(convexhull(p1, p2), ps...)
#function convexhull(p::VAny...)
# T = promote_type(coefficient_type.(p)...)
# f(p) = convert(similar_type(typeof(p), T), p)
# convexhull(f.(p)...)
#end
"""
convexhull!(p1::VRep, p2::VRep)
Same as [`convexhull`](@ref) except that `p1` is modified to be equal to the convex hull.
"""
function convexhull!(p::VRep, ine::VRepresentation)
error("convexhull! not implemented for $(typeof(p)). It probably does not support in-place modification, try `convexhull` (without the `!`) instead.")
end
function convexhull!(p::Polyhedron, v::VRepresentation)
resetvrep!(p, convexhull(vrep(p), v))
end
# conify: same than conichull except that conify(::VRepElement) returns a VRepElement and not a V-representation
conify(v::VRep) = vrep(lines(v), [collect(rays(v)); Ray.(collect(points(v)))])
conify(v::VCone) = v
conify(p::AbstractVector) = Ray(p)
conify(r::Union{Line, Ray}) = r
conichull(p...) = convexhull(conify.(p)...)
function sumpoints(::FullDim, ::Type{T}, p1, p2) where {T}
_tout(p) = convert(similar_type(typeof(p), T), p)
ps = [_tout(po1 + po2) for po1 in points(p1) for po2 in points(p2)]
tuple(ps)
end
sumpoints(::FullDim, ::Type{T}, p1::Rep, p2::VCone) where {T} = change_coefficient_type.(preps(p1), T)
sumpoints(::FullDim, ::Type{T}, p1::VCone, p2::Rep) where {T} = change_coefficient_type.(preps(p2), T)
"""
+(p1::VRep, p2::VRep)
Minkowski sum between `p1` and `p2` using the V-representation.
If the V-representation is not computed for `p1` or `p2`, it is computed.
+(p::Rep, el::Union{Line, Ray})
+(el::Union{Line, Ray}, p::Rep)
Same as `p + vrep([el])`.
"""
function Base.:+(p1::VRep{T1}, p2::VRep{T2}) where {T1, T2}
T = typeof(zero(T1) + zero(T2))
similar((p1, p2), FullDim(p1), T, sumpoints(FullDim(p1), T, p1, p2)..., change_coefficient_type.(rreps(p1, p2), T)...)
end
Base.:+(p::Rep, el::Union{Line, Ray}) = p + vrep([el])
Base.:+(el::Union{Line, Ray}, p::Rep) = p + el
# p1 has priority
function usehrep(p1::Polyhedron, p2::Polyhedron)
hrepiscomputed(p1) && (!vrepiscomputed(p1) || hrepiscomputed(p2))
end
function hcartesianproduct(p1::HRep, p2::HRep)
d = sum_fulldim(FullDim(p1), FullDim(p2))
T = promote_coefficient_type((p1, p2))
f = (i, x) -> zeropad(x, i == 1 ? FullDim(p2) : neg_fulldim(FullDim(p1)))
function dimension_map(i)
if i <= fulldim(p1)
return (1, i)
else
return (2, i - fulldim(p1))
end
end
similar((p1, p2), d, T, hmap(f, d, T, p1, p2)...;
dimension_map = dimension_map)
end
function vcartesianproduct(p1::VRep, p2::VRep)
d = sum_fulldim(FullDim(p1), FullDim(p2))
T = promote_coefficient_type((p1, p2))
# Always type of first arg
f1 = (i, x) -> zeropad(x, FullDim(p2))
f2 = (i, x) -> zeropad(x, neg_fulldim(FullDim(p1)))
q1 = similar(p1, d, T, vmap(f1, d, T, p1)...)
q2 = similar(p2, d, T, vmap(f2, d, T, p2)...)
q1 + q2
end
cartesianproduct(p1::HRep, p2::HRep) = hcartesianproduct(p1, p2)
cartesianproduct(p1::VRep, p2::VRep) = vcartesianproduct(p1, p2)
function cartesianproduct(p1::Polyhedron, p2::Polyhedron)
if usehrep(p1, p2)
hcartesianproduct(p1, p2)
else
vcartesianproduct(p1, p2)
end
end
"""
*(p1::Rep, p2::Rep)
Cartesian product between the polyhedra `p1` and `p2`.
"""
Base.:(*)(p1::Rep, p2::Rep) = cartesianproduct(p1, p2)
"""
\\(P::Union{AbstractMatrix, UniformScaling}, p::HRep)
Transform the polyhedron represented by ``p`` into ``P^{-1} p`` by transforming each halfspace ``\\langle a, x \\rangle \\le \\beta`` into ``\\langle P^\\top a, x \\rangle \\le \\beta`` and each hyperplane ``\\langle a, x \\rangle = \\beta`` into ``\\langle P^\\top a, x \\rangle = \\beta``.
"""
Base.:(\)(P::Union{AbstractMatrix, UniformScaling}, rep::HRep) = rep / P'
function linear_preimage_transpose(P, p::HRep{Tin}, d) where Tin
f = (i, h) -> h / P
T = MA.promote_sum_mul(Tin, eltype(P))
return similar(p, d, T, hmap(f, d, T, p)...,
dimension_map = i -> nothing)
end
"""
/(p::HRep, P::Union{AbstractMatrix, UniformScaling})
Transform the polyhedron represented by ``p`` into ``P^{-T} p`` by transforming each halfspace ``\\langle a, x \\rangle \\le \\beta`` into ``\\langle P a, x \\rangle \\le \\beta`` and each hyperplane ``\\langle a, x \\rangle = \\beta`` into ``\\langle P a, x \\rangle = \\beta``.
"""
function Base.:(/)(p::HRep, P::AbstractMatrix)
if size(P, 2) != fulldim(p)
throw(DimensionMismatch("The number of rows of P must match the dimension of the H-representation"))
end
# FIXME For a matrix P of StaticArrays, `size(P, 1)` should be type stable
return linear_preimage_transpose(P, p, size(P, 1))
end
function Base.:(/)(p::HRep, P::UniformScaling)
return linear_preimage_transpose(P, p, FullDim(p))
end
function linear_image(P, p::VRep{Tin}, d) where Tin
f = (i, v) -> P * v
T = MA.promote_sum_mul(Tin, eltype(P))
return similar(p, d, T, vmap(f, d, T, p)...)
end
"""
*(P::Union{AbstractMatrix, UniformScaling}, p::VRep)
Transform the polyhedron represented by ``p`` into ``P p`` by transforming each element of the V-representation (points, symmetric points, rays and lines) `x` into ``P x``.
"""
function Base.:(*)(P::AbstractMatrix, p::VRep)
if size(P, 2) != fulldim(p)
throw(DimensionMismatch("The number of rows of P must match the dimension of the V-representation"))
end
# FIXME For a matrix P of StaticArrays, `size(P, 1)` should be type stable
return linear_image(P, p, size(P, 1))
end
function Base.:(*)(P::UniformScaling, p::VRep)
return linear_image(P, p, FullDim(p))
end
"""
*(α::Number, p::Rep)
Transform the polyhedron represented by ``p`` into ``\\alpha p`` by transforming
each element of the V-representation (points, symmetric points, rays and lines)
`x` into ``\\alpha x``.
"""
function Base.:(*)(α::Number, p::Polyhedron{T}) where T
if vrepiscomputed(p) || iszero(α)
return (α * LinearAlgebra.I) * p
else
# If `α` is `Int` and `T` is `Rational{BigInt}`,
# `inv(α)` would be `Float64` and then we'll have `BigFloat`.
return p / (inv(convert(T, α)) * LinearAlgebra.I)
end
end
function Base.:(*)(α::Number, p::VRepresentation)
return (α * LinearAlgebra.I) * p
end
function Base.:(*)(α::Number, p::HRepresentation{T}) where {T}
return p / (inv(convert(T, α)) * LinearAlgebra.I)
end
"""
polar(rep::Representation)
Return the polar of the polyhedron `rep` assumed to contain
the origin.
The polar of a convex set `S` is defined as the set of `y`
such that `⟨x, y⟩ ≤ 1` for all `x in S`.
Note that the polar of a V-representation is a H-representation
and vice versa.
"""
function polar end
function polar(vr::VRepresentation{T}) where T
points_halfspaces = [HalfSpace(x, one(T)) for x in points(vr)]
rays_halfspaces = [HalfSpace(coord(r), zero(T)) for r in rays(vr)]
lines_hyperplanes = [HyperPlane(coord(r), zero(T)) for r in lines(vr)]
return hrep(lines_hyperplanes, [points_halfspaces; rays_halfspaces])
end
_polar_error(h) = error("Cannot take the polar of a H-representation with `$h` as it does not contain the origin.")
function _polar(h::HyperPlane)
if !isapproxzero(h.β)
_polar_error(h)
end
return Line(h.a)
end
function polar(hr::HRepresentation{T}) where T
U = MA.promote_operation(/, T, T)
V = similar_type(vectortype(typeof(hr)), U)
points = V[]
rays = Ray{U, V}[]
for h in halfspaces(hr)
if isapproxzero(h.β)
push!(rays, Ray(h.a))
elseif h.β > 0
push!(points, h.a / h.β)
else
_polar_error(h)
end
end
lines = Line{U, V}[_polar(h) for h in hyperplanes(hr)]
if isempty(points)
push!(points, origin(V, fulldim(hr)))
end
return vrep(points, lines, rays, d = FullDim(hr))
end
function polar(p::Polyhedron)
if hrepiscomputed(p) # TODO we should compute the polar of both rep if both are computed
return polyhedron(polar(hrep(p)), library(p))
else
return polyhedron(polar(vrep(p)), library(p))
end
end