dyna3/engine-proto/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}

##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)^n
  )
  for n in 1:3
]
##tangencies = [
  ##Engine.LiesOn{CoeffType}(points[1], spheres[2]),
  ##Engine.LiesOn{CoeffType}(points[1], spheres[3]),
  ##Engine.LiesOn{CoeffType}(points[2], spheres[3]),
  ##Engine.LiesOn{CoeffType}(points[2], spheres[1]),
  ##Engine.LiesOn{CoeffType}(points[3], spheres[1]),
  ##Engine.LiesOn{CoeffType}(points[3], spheres[2])
##]
ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
##small_eqns_tan_sph = eqns_tan_sph
##small_eqns_tan_sph = [
##  eqns_tan_sph;
##  spheres[2].coords - [1, 0, 0, 0, 1];
##  spheres[3].coords - [1, 0, 0, 0, -1];
##]
##small_ideal_tan_sph = Generic.Ideal(base_ring(ideal_tan_sph), small_eqns_tan_sph)
freedom = Engine.dimension(ideal_tan_sph)
println("Three mutually tangent spheres, with two fixed: $freedom degrees of freedom")

##points = [Engine.Point{CoeffType}() for _ in 1:3]
##spheres = [Engine.Sphere{CoeffType}() for _ in 1:2]
##ctx_joined = Engine.Construction{CoeffType}(
##  elements = Set([points; spheres]),
##  relations= Set([
##    Engine.LiesOn{CoeffType}(pt, sph)
##    for pt in points for sph in spheres
##  ])
##)
##ideal_joined, eqns_joined = Engine.realize(ctx_joined)
##freedom = Engine.dimension(ideal_joined)
##println("$(length(points)) points on $(length(spheres)) 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(6701)
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("$(length(samples)) sample solutions:")
##for soln in samples
##  ## display([vbls round.(soln, digits = 6)]) ## [verbose]
##  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

# 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