diff --git a/engine-proto/Engine.Algebraic.jl b/engine-proto/Engine.Algebraic.jl
index a9b6667..380cee1 100644
--- a/engine-proto/Engine.Algebraic.jl
+++ b/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::Vector{Point{T}}
-  spheres::Vector{Sphere{T}}
-  relations::Vector{Relation{T}}
+  points::Set{Point{T}}
+  spheres::Set{Sphere{T}}
+  relations::Set{Relation{T}}
   
-  function Construction{T}(; elements = Vector{Element{T}}(), relations = Vector{Relation{T}}()) where T
+  function Construction{T}(; elements = Set{Element{T}}(), relations = Set{Relation{T}}()) where T
     allelements = union(elements, (rel.elements for rel in relations)...)
     new{T}(
       filter(elt -> isa(elt, Point), allelements),
@@ -197,7 +197,8 @@ 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
   
-  (Generic.Ideal(coordring, eqns), eqns)
+  ## [test] (Generic.Ideal(coordring, eqns), eqns)
+  (nothing, eqns)
 end
 
 end
\ No newline at end of file
diff --git a/engine-proto/Engine.Numerical.jl b/engine-proto/Engine.Numerical.jl
index d1e14bd..48fb682 100644
--- a/engine-proto/Engine.Numerical.jl
+++ b/engine-proto/Engine.Numerical.jl
@@ -1,6 +1,5 @@
 module Numerical
 
-using Random: default_rng
 using LinearAlgebra
 using AbstractAlgebra
 using HomotopyContinuation:
@@ -29,16 +28,16 @@ end
 
 # --- sampling ---
 
-function real_samples(F::AbstractSystem, dim; rng = default_rng())
+function real_samples(F::AbstractSystem, dim)
   # 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(rng, nvariables(F))) for _ in 1:dim)...
+    (normalize(randn(nvariables(F))) for _ in 1:dim)...
   ))
   cut = LinearSubspace(normals, fill(0., dim))
-  filter(isreal, results(witness_set(F, cut, seed = 0x1974abba)))
+  filter(isreal, results(witness_set(F, cut)))
 end
 
 AbstractAlgebra.evaluate(pt::Point, vals::Vector{<:RingElement}) =
diff --git a/engine-proto/Engine.jl b/engine-proto/Engine.jl
index f6f92c5..49011c6 100644
--- a/engine-proto/Engine.jl
+++ b/engine-proto/Engine.jl
@@ -23,40 +23,92 @@ 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")
+##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")
 
 # --- test rational cut ---
 
-coordring = base_ring(ideal_tan_sph)
+coordring = base_ring(ideal_joined)
 vbls = Variable.(symbols(coordring))
 
 # test a random witness set
-system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
+system = CompiledSystem(System(eqns_joined, variables = vbls))
 norm2 = vec -> real(dot(conj.(vec), vec))
-rng = MersenneTwister(6071)
-n_planes = 6
+Random.seed!(6071)
+n_planes = 3
 samples = []
 for _ in 1:n_planes
-  real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
+  real_solns = solution.(Engine.Numerical.real_samples(system, freedom))
   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 $(length(samples)) sample solutions")
+println("$(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)