Clean up example of three mutually tangent spheres

It seems like there are real solutions if and only if the product of the
cosines is positive.

engine-proto/Engine.Algebraic.jl

@@ -120,11 +120,11 @@ equation(rel::AlignsWithBy) = mprod(rel.elements[1].vec, rel.elements[2].vec) -
 # --- constructions ---
 
 mutable struct Construction{T}
-  points::Set{Point{T}}
-  spheres::Set{Sphere{T}}
-  relations::Set{Relation{T}}
+  points::Vector{Point{T}}
+  spheres::Vector{Sphere{T}}
+  relations::Vector{Relation{T}}
   
-  function Construction{T}(; elements = Set{Element{T}}(), relations = Set{Relation{T}}()) where T
+  function Construction{T}(; elements = Vector{Element{T}}(), relations = Vector{Relation{T}}()) where T
     allelements = union(elements, (rel.elements for rel in relations)...)
     new{T}(
       filter(elt -> isa(elt, Point), allelements),
@@ -197,8 +197,7 @@ function realize(ctx::Construction{T}) where T
     push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
   end
   
-  ## [test] (Generic.Ideal(coordring, eqns), eqns)
-  (nothing, eqns)
+  (Generic.Ideal(coordring, eqns), eqns)
 end
 
 end

engine-proto/Engine.Numerical.jl

@@ -1,5 +1,6 @@
 module Numerical
 
+using Random: default_rng
 using LinearAlgebra
 using AbstractAlgebra
 using HomotopyContinuation:
@@ -28,16 +29,16 @@ end
 
 # --- sampling ---
 
-function real_samples(F::AbstractSystem, dim)
+function real_samples(F::AbstractSystem, dim; rng = default_rng())
   # choose a random real hyperplane of codimension `dim` by intersecting
   # hyperplanes whose normal vectors are uniformly distributed over the unit
   # sphere
   # [to do] guard against the unlikely event that one of the normals is zero
   normals = transpose(hcat(
-    (normalize(randn(nvariables(F))) for _ in 1:dim)...
+    (normalize(randn(rng, nvariables(F))) for _ in 1:dim)...
   ))
   cut = LinearSubspace(normals, fill(0., dim))
-  filter(isreal, results(witness_set(F, cut)))
+  filter(isreal, results(witness_set(F, cut, seed = 0x1974abba)))
 end
 
 AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) =

engine-proto/Engine.jl

@@ -23,92 +23,40 @@ 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//1)
-##  )
-##  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 = Set(spheres), relations = Set(tangencies))
-##ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
-##freedom = Engine.dimension(ideal_tan_sph)
-##println("Three mutually tangent spheres: $freedom degrees of freedom")
-
-points = [Engine.Point{CoeffType}() for _ in 1:3]
-spheres = [Engine.Sphere{CoeffType}() for _ in 1:2]
-ctx_joined = Engine.Construction{CoeffType}(
-  elements = Set([points; spheres]),
-  relations= Set([
-    Engine.LiesOn{CoeffType}(pt, sph)
-    for pt in points for sph in spheres
-  ])
-)
-ideal_joined, eqns_joined = Engine.realize(ctx_joined)
-freedom = Engine.dimension(ideal_joined)
-println("$(length(points)) points on $(length(spheres)) spheres: $freedom degrees of freedom")
+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_joined)
+coordring = base_ring(ideal_tan_sph)
 vbls = Variable.(symbols(coordring))
 
 # test a random witness set
-system = CompiledSystem(System(eqns_joined, variables = vbls))
+system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
 norm2 = vec -> real(dot(conj.(vec), vec))
-Random.seed!(6071)
-n_planes = 3
+rng = MersenneTwister(6071)
+n_planes = 6
 samples = []
 for _ in 1:n_planes
-  real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
+  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
+println("Found $(length(samples)) sample solutions")
 
 # show a sample solution
 function show_solution(ctx, vals)