Say how many sample solutions we found

Choose hyperplanes that go through the trivial solution.

engine-proto/Engine.jl

@@ -225,6 +225,8 @@ end
 
 # ~~~ sandbox setup ~~~
 
+using Random
+using Distributions
 using AbstractAlgebra
 using HomotopyContinuation
 
@@ -243,7 +245,8 @@ Engine.push!(ctx, b)
 Engine.push!(ctx, s)
 Engine.push!(ctx, b_on_s)
 ideal_ab_s, eqns_ab_s = Engine.realize(ctx)
-println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of freedom")
+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 = [
@@ -260,33 +263,54 @@ println("Two points on a sphere: ", Engine.dimension(ideal_ab_s), " degrees of f
 
 # --- test rational cut ---
 
-cut = [
-  sum(vcat([1, 1, 1] .* a.coords, [1, 1, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
-  sum(vcat([2, 1, 1] .* a.coords, [1, 2, 1, 1, 1] .* (s.coords - [0, 0, 0, 0, 1])))
-  sum(vcat([1, 2, 0] .* a.coords, [1, 1, 0, 1, 2] .* (s.coords - [0, 0, 0, 0, 1])))
-]
-cut_ideal_ab_s = Generic.Ideal(base_ring(ideal_ab_s), [gens(ideal_ab_s); cut])
-cut_dim = Engine.dimension(cut_ideal_ab_s)
-println("Two points on a sphere, after cut: ", cut_dim, " degrees of freedom")
-if cut_dim == 0
-  vbls = Variable.(symbols(base_ring(ideal_ab_s)))
-  cut_system = 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
+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 corresponding witness set
-  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]
-  cut_subspace = LinearSubspace(cut_matrix, [1, 1, 2])
-  witness = witness_set(System(eqns_ab_s, variables = vbls), cut_subspace)
-  println("witness solutions:")
-  for wtns in solutions(witness)
-    display(wtns)
+# test a random witness set
+system = CompiledSystem(System(eqns_ab_s, variables = vbls))
+max_slope = 2
+binom = Binomial(2max_slope, 1/2)
+Random.seed!(6071)
+samples = []
+for _ in 1:3
+  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
+  ##]
+  sph_z_ind = indexin([sph.coords[5] for sph in ctx.spheres], gens(coordring))
+  cut_offset = [sum(cf[sph_z_ind]) for cf in eachrow(cut_matrix)]
+  println("sphere z variables: ", vbls[sph_z_ind])
+  display(cut_matrix)
+  display(cut_offset)
+  cut_subspace = LinearSubspace(cut_matrix, cut_offset)
+  wtns = witness_set(system, cut_subspace)
+  append!(samples, solution.(filter(isreal, results(wtns))))
+end
+println("$(length(samples)) sample solutions:")
+for soln in samples
+  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