Stow algebraic engine prototype

We're using the Gram matrix engine for the next stage of development,
so the algebraic engine shouldn't be at the top level anymore.

engine-proto/alg-test/Engine.jl

@@ -0,0 +1,76 @@
+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