dyna3/engine-proto/alg-test/Engine.jl

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
using GLMakie

CoeffType = Rational{Int64}

spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
tangencies = [
  Engine.AlignsWithBy{CoeffType}(
    spheres[n],
    spheres[mod1(n+1, length(spheres))],
    CoeffType(1)
  )
  for n in 1:3
]
ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
freedom = Engine.dimension(ideal_tan_sph)
println("Three mutually tangent spheres: $freedom degrees of freedom")

# --- test rational cut ---

coordring = base_ring(ideal_tan_sph)
vbls = Variable.(symbols(coordring))

# test a random witness set
system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
norm2 = vec -> real(dot(conj.(vec), vec))
rng = MersenneTwister(6071)
n_planes = 6
samples = []
for _ in 1:n_planes
  real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
  for soln in real_solns
    if all(norm2(soln - samp) > 1e-4*length(gens(coordring)) for samp in samples)
      push!(samples, soln)
    end
  end
end
println("Found $(length(samples)) sample solutions")

# show a sample solution
function show_solution(ctx, vals)
  # evaluate elements
  real_vals = real.(vals)
  disp_points = [Engine.Numerical.evaluate(pt, real_vals) for pt in ctx.points]
  disp_spheres = [Engine.Numerical.evaluate(sph, real_vals) for sph in ctx.spheres]
  
  # create scene
  scene = Scene()
  cam3d!(scene)
  scatter!(scene, disp_points, color = :green)
  for sph in disp_spheres
    mesh!(scene, sph, color = :gray)
  end
  scene
end