Engine prototype #13
@ -186,19 +186,19 @@ function realize(ctx::Construction{T}) where T
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# add relations to center, orient, and scale the construction
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# add relations to center, orient, and scale the construction
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# [to do] the scaling constraint, as written, can be impossible to satisfy
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# [to do] the scaling constraint, as written, can be impossible to satisfy
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# when all of the spheres have to go through the origin
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# when all of the spheres have to go through the origin
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if !isempty(ctx.points)
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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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## 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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##end
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if !isempty(ctx.spheres)
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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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## 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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##end
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n_elts = length(ctx.points) + length(ctx.spheres)
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##n_elts = length(ctx.points) + length(ctx.spheres)
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if n_elts > 0
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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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## 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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##end
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## [test] (Generic.Ideal(coordring, eqns), eqns)
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(Generic.Ideal(coordring, eqns), eqns)
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(nothing, eqns)
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## [test] (nothing, eqns)
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end
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end
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end
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end
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@ -39,15 +39,15 @@ CoeffType = Rational{Int64}
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##freedom = Engine.dimension(ideal_ab_s)
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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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##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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spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
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##tangencies = [
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tangencies = [
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## Engine.AlignsWithBy{CoeffType}(
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Engine.AlignsWithBy{CoeffType}(
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## spheres[n],
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spheres[n],
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## spheres[mod1(n+1, length(spheres))],
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spheres[mod1(n+1, length(spheres))],
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## CoeffType(-1//1)
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CoeffType(-1)^n
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## )
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)
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## for n in 1:3
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for n in 1:3
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##]
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]
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##tangencies = [
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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[2]),
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##Engine.LiesOn{CoeffType}(points[1], spheres[3]),
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##Engine.LiesOn{CoeffType}(points[1], spheres[3]),
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@ -56,34 +56,41 @@ CoeffType = Rational{Int64}
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##Engine.LiesOn{CoeffType}(points[3], 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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##Engine.LiesOn{CoeffType}(points[3], spheres[2])
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##]
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##]
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##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
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ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
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##ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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##freedom = Engine.dimension(ideal_tan_sph)
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##small_eqns_tan_sph = eqns_tan_sph
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##println("Three mutually tangent spheres: $freedom degrees of freedom")
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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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##points = [Engine.Point{CoeffType}() for _ in 1:3]
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spheres = [Engine.Sphere{CoeffType}() for _ in 1:2]
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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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##ctx_joined = Engine.Construction{CoeffType}(
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elements = Set([points; spheres]),
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## elements = Set([points; spheres]),
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relations= Set([
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## relations= Set([
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Engine.LiesOn{CoeffType}(pt, sph)
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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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## for pt in points for sph in spheres
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])
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## ])
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)
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##)
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ideal_joined, eqns_joined = Engine.realize(ctx_joined)
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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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##freedom = Engine.dimension(ideal_joined)
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println("$(length(points)) points on $(length(spheres)) spheres: $freedom degrees of freedom")
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##println("$(length(points)) points on $(length(spheres)) spheres: $freedom degrees of freedom")
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# --- test rational cut ---
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# --- test rational cut ---
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coordring = base_ring(ideal_joined)
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coordring = base_ring(small_ideal_tan_sph)
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vbls = Variable.(symbols(coordring))
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vbls = Variable.(symbols(coordring))
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# test a random witness set
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# test a random witness set
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system = CompiledSystem(System(eqns_joined, variables = vbls))
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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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norm2 = vec -> real(dot(conj.(vec), vec))
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Random.seed!(6071)
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Random.seed!(6071)
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n_planes = 3
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n_planes = 36
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samples = []
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samples = []
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for _ in 1:n_planes
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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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real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
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@ -94,21 +101,21 @@ for _ in 1:n_planes
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end
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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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println("$(length(samples)) sample solutions:")
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for soln in samples
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##for soln in samples
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## display([vbls round.(soln, digits = 6)]) ## [verbose]
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## ## display([vbls round.(soln, digits = 6)]) ## [verbose]
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k_sq = abs2(soln[1])
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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 abs2(soln[end-2]) > 1e-12
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if k_sq < 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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## 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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## else
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sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
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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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## 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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## end
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else
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## else
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sum_sq = sum(soln[[4, 7, 10]] .^ 2)
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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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## 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
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##end
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# show a sample solution
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# show a sample solution
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function show_solution(ctx, vals)
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function show_solution(ctx, vals)
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