131 lines
4.3 KiB
Julia
131 lines
4.3 KiB
Julia
include("HittingSet.jl")
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module Engine
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include("Engine.Algebraic.jl")
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include("Engine.Numerical.jl")
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using .Algebraic
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using .Numerical
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export Construction, mprod, codimension, dimension
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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 = 3
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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 |