134 lines
4.4 KiB
Julia
134 lines
4.4 KiB
Julia
include("HittingSet.jl")
|
|
|
|
module Engine
|
|
|
|
include("Engine.Algebraic.jl")
|
|
include("Engine.Numerical.jl")
|
|
|
|
using .Algebraic
|
|
using .Numerical
|
|
|
|
export Construction, mprod, codimension, dimension
|
|
|
|
end
|
|
|
|
# ~~~ sandbox setup ~~~
|
|
|
|
using Random
|
|
using Distributions
|
|
using LinearAlgebra
|
|
using AbstractAlgebra
|
|
using HomotopyContinuation
|
|
|
|
CoeffType = Rational{Int64}
|
|
|
|
a = Engine.Point{CoeffType}()
|
|
s = Engine.Sphere{CoeffType}()
|
|
a_on_s = Engine.LiesOn{CoeffType}(a, s)
|
|
ctx = Engine.Construction{CoeffType}(elements = Set([a]), relations= Set([a_on_s]))
|
|
##ideal_a_s = Engine.realize(ctx)
|
|
##println("A point on a sphere: ", Engine.dimension(ideal_a_s), " degrees of freedom")
|
|
|
|
b = Engine.Point{CoeffType}()
|
|
b_on_s = Engine.LiesOn{CoeffType}(b, s)
|
|
Engine.push!(ctx, b)
|
|
Engine.push!(ctx, s)
|
|
Engine.push!(ctx, b_on_s)
|
|
ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
|
|
freedom = Engine.dimension(ideal_ab_s)
|
|
println("Two points on a sphere: ", freedom, " degrees of freedom")
|
|
|
|
##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
|
|
##tangencies = [
|
|
## Engine.AlignsWithBy{CoeffType}(
|
|
## spheres[n],
|
|
## spheres[mod1(n+1, length(spheres))],
|
|
## CoeffType(-1//1)
|
|
## )
|
|
## for n in 1:3
|
|
##]
|
|
##ctx_tan_sph = Engine.Construction{CoeffType}(elements = Set(spheres), relations = Set(tangencies))
|
|
##ideal_tan_sph = Engine.realize(ctx_tan_sph)
|
|
##println("Three mutually tangent spheres: ", Engine.dimension(ideal_tan_sph), " degrees of freedom")
|
|
|
|
# --- test rational cut ---
|
|
|
|
coordring = base_ring(ideal_ab_s)
|
|
vbls = Variable.(symbols(coordring))
|
|
##cut_system = CompiledSystem(System([eqns_ab_s; cut], variables = vbls))
|
|
##cut_result = HomotopyContinuation.solve(cut_system)
|
|
##println("non-singular solutions:")
|
|
##for soln in solutions(cut_result)
|
|
## display(soln)
|
|
##end
|
|
##println("singular solutions:")
|
|
##for sing in singular(cut_result)
|
|
## display(sing.solution)
|
|
##end
|
|
|
|
# test a random witness set
|
|
system = CompiledSystem(System(eqns_ab_s, variables = vbls))
|
|
sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
|
|
println("sphere z variables: ", vbls[sph_z_ind])
|
|
trivial_soln = fill(0, length(gens(coordring)))
|
|
trivial_soln[sph_z_ind] .= 1
|
|
println("trivial solutions: $trivial_soln")
|
|
norm2 = vec -> real(dot(conj.(vec), vec))
|
|
is_nontrivial = soln -> norm2(abs.(real.(soln)) - trivial_soln) > 1e-4*length(gens(coordring))
|
|
##max_slope = 5
|
|
##binom = Binomial(2max_slope, 1/2)
|
|
Random.seed!(6071)
|
|
n_planes = 3
|
|
for through_trivial in [false, true]
|
|
samples = []
|
|
for _ in 1:n_planes
|
|
cut_matrix = transpose(hcat(
|
|
(normalize(randn(length(gens(coordring)))) for _ in 1:freedom)...
|
|
))
|
|
##cut_matrix = rand(binom, freedom, length(gens(coordring))) .- max_slope
|
|
##cut_matrix = [
|
|
## 1 1 1 1 0 1 1 0 1 1 0;
|
|
## 1 2 1 2 0 1 1 0 1 1 0;
|
|
## 1 1 0 1 0 1 2 0 2 0 0
|
|
##]
|
|
display(cut_matrix) ## [verbose]
|
|
if through_trivial
|
|
cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
|
|
display(cut_offset) ## [verbose]
|
|
cut_subspace = LinearSubspace(cut_matrix, cut_offset)
|
|
else
|
|
cut_subspace = LinearSubspace(cut_matrix, fill(0., freedom))
|
|
end
|
|
wtns = witness_set(system, cut_subspace)
|
|
real_solns = solution.(filter(isreal, results(wtns)))
|
|
nontrivial_solns = filter(is_nontrivial, real_solns)
|
|
println("$(length(real_solns) - length(nontrivial_solns)) trivial solutions found")
|
|
for soln in nontrivial_solns
|
|
##[test] for soln in filter(is_nontrivial, solution.(filter(isreal, results(wtns))))
|
|
if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
|
|
push!(samples, soln)
|
|
end
|
|
end
|
|
end
|
|
if through_trivial
|
|
println("--- planes through trivial solution ---")
|
|
else
|
|
println("--- planes through origin ---")
|
|
end
|
|
println("$(length(samples)) sample solutions, not including the trivial one:")
|
|
for soln in samples
|
|
## [verbose] display([vbls round.(soln, digits = 6)])
|
|
k_sq = abs2(soln[1])
|
|
if abs2(soln[end-2]) > 1e-12
|
|
if k_sq < 1e-12
|
|
println(" center at infinity: z coordinates $(round(soln[end], digits = 6)) and $(round(soln[end-1], digits = 6))")
|
|
else
|
|
sum_sq = soln[4]^2 + soln[7]^2 + soln[end-2]^2 / k_sq
|
|
println(" center on z axis: r² = $(round(1/k_sq, digits = 6)), x² + y² + h² = $(round(sum_sq, digits = 6))")
|
|
end
|
|
else
|
|
sum_sq = sum(soln[[4, 7, 10]] .^ 2)
|
|
println(" center at origin: r² = $(round(1/k_sq, digits = 6)); x² + y² + z² = $(round(sum_sq, digits = 6))")
|
|
end
|
|
end
|
|
end |