Clean up example of three mutually tangent spheres

engine-proto/Engine.Algebraic.jl

@@ -198,7 +198,6 @@ function realize(ctx::Construction{T}) where T
   end
   
   (Generic.Ideal(coordring, eqns), eqns)
-  ## [test] (nothing, eqns)
 end
 
 end

engine-proto/Engine.jl

@@ -23,63 +23,19 @@ 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, 1][n])
+    CoeffType(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 = 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")
+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 ---
 
@@ -90,7 +46,7 @@ vbls = Variable.(symbols(coordring))
 system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
 norm2 = vec -> real(dot(conj.(vec), vec))
 rng = MersenneTwister(6071)
-n_planes = 36
+n_planes = 6
 samples = []
 for _ in 1:n_planes
   real_solns = solution.(Engine.Numerical.real_samples(system, freedom, rng = rng))
@@ -100,22 +56,7 @@ for _ in 1:n_planes
     end
   end
 end
-println("$(length(samples)) sample solutions:")
+println("Found $(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)