135 lines
4.2 KiB
Julia
135 lines
4.2 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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using GLMakie
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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)^n
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)
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for n in 1:3
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]
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##tangencies = [
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##Engine.LiesOn{CoeffType}(points[1], spheres[2]),
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##Engine.LiesOn{CoeffType}(points[1], spheres[3]),
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##Engine.LiesOn{CoeffType}(points[2], spheres[3]),
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##Engine.LiesOn{CoeffType}(points[2], spheres[1]),
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##Engine.LiesOn{CoeffType}(points[3], spheres[1]),
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##Engine.LiesOn{CoeffType}(points[3], spheres[2])
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##]
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ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
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ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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##small_eqns_tan_sph = eqns_tan_sph
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small_eqns_tan_sph = [
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eqns_tan_sph;
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spheres[2].coords - [1, 0, 0, 0, 1];
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spheres[3].coords - [1, 0, 0, 0, -1];
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]
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small_ideal_tan_sph = Generic.Ideal(base_ring(ideal_tan_sph), small_eqns_tan_sph)
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freedom = Engine.dimension(small_ideal_tan_sph)
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println("Three mutually tangent spheres, with two fixed: $freedom degrees of freedom")
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##points = [Engine.Point{CoeffType}() for _ in 1:3]
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##spheres = [Engine.Sphere{CoeffType}() for _ in 1:2]
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##ctx_joined = Engine.Construction{CoeffType}(
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## elements = Set([points; spheres]),
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## relations= Set([
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## Engine.LiesOn{CoeffType}(pt, sph)
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## for pt in points for sph in spheres
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## ])
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##)
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##ideal_joined, eqns_joined = Engine.realize(ctx_joined)
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##freedom = Engine.dimension(ideal_joined)
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##println("$(length(points)) points on $(length(spheres)) spheres: $freedom degrees of freedom")
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# --- test rational cut ---
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coordring = base_ring(small_ideal_tan_sph)
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vbls = Variable.(symbols(coordring))
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# test a random witness set
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system = CompiledSystem(System(small_eqns_tan_sph, variables = vbls))
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norm2 = vec -> real(dot(conj.(vec), vec))
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Random.seed!(6071)
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n_planes = 36
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samples = []
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for _ in 1:n_planes
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real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
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for soln in real_solns
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if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
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push!(samples, soln)
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end
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end
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end
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println("$(length(samples)) sample solutions:")
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##for soln in samples
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## ## display([vbls round.(soln, digits = 6)]) ## [verbose]
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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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# show a sample solution
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function show_solution(ctx, vals)
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# evaluate elements
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real_vals = real.(vals)
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disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
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disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
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# create scene
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scene = Scene()
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cam3d!(scene)
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scatter!(scene, disp_points, color = :green)
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for sph in disp_spheres
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mesh!(scene, sph, color = :gray)
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end
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scene
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end |