af1d31f6e6
In all but a few cases (for example, a single point on a plane), we should be able to us the radius-coradius boost symmetry to make the average co-radius—representing the "overall scale"—roughly one.
346 lines
11 KiB
Julia
346 lines
11 KiB
Julia
include("HittingSet.jl")
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module Engine
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export Construction, mprod, codimension, dimension
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import Subscripts
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using LinearAlgebra
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using AbstractAlgebra
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using Groebner
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using HomotopyContinuation: Variable, Expression, System
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using ..HittingSet
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# --- commutative algebra ---
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# as of version 0.36.6, AbstractAlgebra only supports ideals in multivariate
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# polynomial rings when the coefficients are integers. we use Groebner to extend
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# support to rationals and to finite fields of prime order
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Generic.reduce_gens(I::Generic.Ideal{U}) where {T <: FieldElement, U <: MPolyRingElem{T}} =
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Generic.Ideal{U}(base_ring(I), groebner(gens(I)))
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function codimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}}
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leading = [exponent_vector(f, 1) for f in gens(I)]
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targets = [Set(findall(.!iszero.(exp_vec))) for exp_vec in leading]
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length(HittingSet.solve(HittingSetProblem(targets), maxdepth))
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end
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dimension(I::Generic.Ideal{U}, maxdepth = Inf) where {T <: RingElement, U <: MPolyRingElem{T}} =
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length(gens(base_ring(I))) - codimension(I, maxdepth)
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# hat tip Sascha Timme
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# https://github.com/JuliaHomotopyContinuation/HomotopyContinuation.jl/issues/520#issuecomment-1317681521
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function Base.convert(::Type{Expression}, f::MPolyRingElem)
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variables = Variable.(symbols(parent(f)))
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f_data = zip(coefficients(f), exponent_vectors(f))
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sum(cf * prod(variables .^ exp_vec) for (cf, exp_vec) in f_data)
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end
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# create a ModelKit.System from an ideal in a multivariate polynomial ring. the
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# variable ordering is taken from the polynomial ring
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function System(I::Generic.Ideal)
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eqns = Expression.(gens(I))
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variables = Variable.(symbols(base_ring(I)))
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System(eqns, variables = variables)
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end
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## [to do] not needed right now
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# create a ModelKit.System from a list of elements of a multivariate polynomial
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# ring. the variable ordering is taken from the polynomial ring
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##function System(eqns::AbstractVector{MPolyRingElem})
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## if isempty(eqns)
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## return System([])
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## else
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## variables = Variable.(symbols(parent(f)))
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## return System(Expression.(eqns), variables = variables)
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## end
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##end
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# --- primitve elements ---
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abstract type Element{T} end
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mutable struct Point{T} <: Element{T}
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coords::Vector{MPolyRingElem{T}}
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vec::Union{Vector{MPolyRingElem{T}}, Nothing}
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rel::Nothing
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## [to do] constructor argument never needed?
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Point{T}(
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coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
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vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing
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) where T = new(coords, vec, nothing)
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end
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function buildvec!(pt::Point)
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coordring = parent(pt.coords[1])
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pt.vec = [one(coordring), dot(pt.coords, pt.coords), pt.coords...]
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end
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mutable struct Sphere{T} <: Element{T}
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coords::Vector{MPolyRingElem{T}}
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vec::Union{Vector{MPolyRingElem{T}}, Nothing}
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rel::Union{MPolyRingElem{T}, Nothing}
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## [to do] constructor argument never needed?
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Sphere{T}(
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coords::Vector{MPolyRingElem{T}} = MPolyRingElem{T}[],
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vec::Union{Vector{MPolyRingElem{T}}, Nothing} = nothing,
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rel::Union{MPolyRingElem{T}, Nothing} = nothing
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) where T = new(coords, vec, rel)
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end
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function buildvec!(sph::Sphere)
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coordring = parent(sph.coords[1])
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sph.vec = sph.coords
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sph.rel = mprod(sph.coords, sph.coords) + one(coordring)
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end
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const coordnames = IdDict{Symbol, Vector{Union{Symbol, Nothing}}}(
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nameof(Point) => [nothing, nothing, :xₚ, :yₚ, :zₚ],
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nameof(Sphere) => [:rₛ, :sₛ, :xₛ, :yₛ, :zₛ]
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)
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coordname(elt::Element, index) = coordnames[nameof(typeof(elt))][index]
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function pushcoordname!(coordnamelist, indexed_elt::Tuple{Any, Element}, coordindex)
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eltindex, elt = indexed_elt
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name = coordname(elt, coordindex)
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if !isnothing(name)
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subscript = Subscripts.sub(string(eltindex))
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push!(coordnamelist, Symbol(name, subscript))
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end
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end
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function takecoord!(coordlist, indexed_elt::Tuple{Any, Element}, coordindex)
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elt = indexed_elt[2]
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if !isnothing(coordname(elt, coordindex))
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push!(elt.coords, popfirst!(coordlist))
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end
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end
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# --- primitive relations ---
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abstract type Relation{T} end
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mprod(v, w) = (v[1]*w[2] + w[1]*v[2]) / 2 - dot(v[3:end], w[3:end])
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# elements: point, sphere
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struct LiesOn{T} <: Relation{T}
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elements::Vector{Element{T}}
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LiesOn{T}(pt::Point{T}, sph::Sphere{T}) where T = new{T}([pt, sph])
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end
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equation(rel::LiesOn) = mprod(rel.elements[1].vec, rel.elements[2].vec)
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# elements: sphere, sphere
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struct AlignsWithBy{T} <: Relation{T}
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elements::Vector{Element{T}}
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cos_angle::T
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AlignsWithBy{T}(sph1::Sphere{T}, sph2::Sphere{T}, cos_angle::T) where T = new{T}([sph1, sph2], cos_angle)
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end
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equation(rel::AlignsWithBy) = mprod(rel.elements[1].vec, rel.elements[2].vec) - rel.cos_angle
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# --- constructions ---
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mutable struct Construction{T}
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points::Set{Point{T}}
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spheres::Set{Sphere{T}}
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relations::Set{Relation{T}}
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function Construction{T}(; elements = Set{Element{T}}(), relations = Set{Relation{T}}()) where T
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allelements = union(elements, (rel.elements for rel in relations)...)
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new{T}(
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filter(elt -> isa(elt, Point), allelements),
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filter(elt -> isa(elt, Sphere), allelements),
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relations
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)
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end
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end
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function Base.push!(ctx::Construction{T}, elt::Point{T}) where T
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push!(ctx.points, elt)
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end
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function Base.push!(ctx::Construction{T}, elt::Sphere{T}) where T
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push!(ctx.spheres, elt)
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end
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function Base.push!(ctx::Construction{T}, rel::Relation{T}) where T
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push!(ctx.relations, rel)
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for elt in rel.elements
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push!(ctx, elt)
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end
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end
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function realize(ctx::Construction{T}) where T
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# collect coordinate names
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coordnamelist = Symbol[]
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eltenum = enumerate(Iterators.flatten((ctx.spheres, ctx.points)))
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for coordindex in 1:5
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for indexed_elt in eltenum
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pushcoordname!(coordnamelist, indexed_elt, coordindex)
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end
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end
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# construct coordinate ring
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coordring, coordqueue = polynomial_ring(parent_type(T)(), coordnamelist, ordering = :degrevlex)
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# retrieve coordinates
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for (_, elt) in eltenum
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empty!(elt.coords)
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end
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for coordindex in 1:5
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for indexed_elt in eltenum
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takecoord!(coordqueue, indexed_elt, coordindex)
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end
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end
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# construct coordinate vectors
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for (_, elt) in eltenum
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buildvec!(elt)
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end
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# turn relations into equations
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eqns = vcat(
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equation.(ctx.relations),
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[elt.rel for (_, elt) in eltenum if !isnothing(elt.rel)]
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)
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# add relations to center, orient, and scale the construction
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if !isempty(ctx.points)
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append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3])
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end
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if !isempty(ctx.spheres)
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append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
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end
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n_elts = length(ctx.points) + length(ctx.spheres)
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if n_elts > 0
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push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
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end
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(Generic.Ideal(coordring, eqns), eqns)
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end
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end
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# ~~~ sandbox setup ~~~
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using Random
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using Distributions
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using LinearAlgebra
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using AbstractAlgebra
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using HomotopyContinuation
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CoeffType = Rational{Int64}
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a = Engine.Point{CoeffType}()
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s = Engine.Sphere{CoeffType}()
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a_on_s = Engine.LiesOn{CoeffType}(a, s)
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ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
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##ideal_a_s = Engine.realize(ctx)
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##println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freedom")
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b = Engine.Point{CoeffType}()
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b_on_s = Engine.LiesOn{CoeffType}(b, s)
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Engine.push!(ctx, b)
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Engine.push!(ctx, s)
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Engine.push!(ctx, b_on_s)
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ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
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freedom = Engine.dimension(ideal_ab_s)
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println("Two points on a sphere: ", freedom, " degrees of freedom")
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##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
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##tangencies = [
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## Engine.AlignsWithBy{CoeffType}(
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## spheres[n],
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## spheres[mod1(n+1, length(spheres))],
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## CoeffType(-1//1)
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## )
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## for n in 1:3
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##]
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##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
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##ideal_tan_sph = Engine.realize(ctx_tan_sph)
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##println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freedom")
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# --- test rational cut ---
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coordring = base_ring(ideal_ab_s)
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vbls = Variable.(symbols(coordring))
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##cut_system = CompiledSystem(System([eqns_ab_s; cut], variables = vbls))
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##cut_result = HomotopyContinuation.solve(cut_system)
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##println("non-singular solutions:")
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##for soln in solutions(cut_result)
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## display(soln)
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##end
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##println("singular solutions:")
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##for sing in singular(cut_result)
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## display(sing.solution)
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##end
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# test a random witness set
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system = CompiledSystem(System(eqns_ab_s, variables = vbls))
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sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
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println("sphere z variables: ", vbls[sph_z_ind])
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trivial_soln = fill(0, length(gens(coordring)))
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trivial_soln[sph_z_ind] .= 1
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println("trivial solutions: $trivial_soln")
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norm2 = vec -> real(dot(conj.(vec), vec))
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is_nontrivial = soln -> norm2(abs.(real.(soln)) - trivial_soln) > 1e-4*length(gens(coordring))
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max_slope = 5
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binom = Binomial(2max_slope, 1/2)
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Random.seed!(6071)
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n_planes = 36
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for through_trivial in [false, true]
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samples = []
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for _ in 1:n_planes
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cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
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##cut_matrix = [
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## 1 1 1 1 0 1 1 0 1 1 0;
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## 1 2 1 2 0 1 1 0 1 1 0;
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## 1 1 0 1 0 1 2 0 2 0 0
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##]
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## [verbose] display(cut_matrix)
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if through_trivial
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cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
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## [verbose] display(cut_offset)
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cut_subspace = LinearSubspace(cut_matrix, cut_offset)
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else
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cut_subspace = LinearSubspace(cut_matrix, fill(0, freedom))
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end
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wtns = witness_set(system, cut_subspace)
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real_solns = solution.(filter(isreal, results(wtns)))
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nontrivial_solns = filter(is_nontrivial, real_solns)
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println("$(length(real_solns) - length(nontrivial_solns)) trivial solutions found")
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for soln in nontrivial_solns
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##[test] for soln in filter(is_nontrivial, solution.(filter(isreal, results(wtns))))
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if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
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push!(samples, soln)
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end
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end
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end
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if through_trivial
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println("--- planes through trivial solution ---")
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else
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println("--- planes through origin ---")
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end
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println("$(length(samples)) sample solutions, not including the trivial one:")
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for soln in samples
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## [verbose] display([vbls round.(soln, digits = 6)])
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k_sq = abs2(soln[1])
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if abs2(soln[end-2]) > 1e-12
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if k_sq < 1e-12
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println(" center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))")
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else
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sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
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println(" center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))")
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end
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else
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sum_sq = sum(soln[[4, 7, 10]] .^ 2)
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println(" center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))")
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end
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end
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end |