From 31d5e7e864704895dbf0273c46185476e53e40f1 Mon Sep 17 00:00:00 2001
From: Aaron Fenyes <aaron.fenyes@fareycircles.ooo>
Date: Mon, 12 Feb 2024 22:48:16 -0500
Subject: [PATCH] Play with two points on two spheres

Guess conditions that make the scaling constraint impossible to satisfy.
---
 engine-proto/Engine.Algebraic.jl |  2 ++
 engine-proto/Engine.jl           | 43 +++++++++++++++++++++++---------
 2 files changed, 33 insertions(+), 12 deletions(-)

diff --git a/engine-proto/Engine.Algebraic.jl b/engine-proto/Engine.Algebraic.jl
index b9b790a..ca39967 100644
--- a/engine-proto/Engine.Algebraic.jl
+++ b/engine-proto/Engine.Algebraic.jl
@@ -184,6 +184,8 @@ function realize(ctx::Construction{T}) where T
   )
   
   # add relations to center, orient, and scale the construction
+  # [to do] the scaling constraint, as written, can be impossible to satisfy
+  #         when all of the spheres have to go through the origin
   if !isempty(ctx.points)
     append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3])
   end
diff --git a/engine-proto/Engine.jl b/engine-proto/Engine.jl
index 113c6a3..7ddf72d 100644
--- a/engine-proto/Engine.jl
+++ b/engine-proto/Engine.jl
@@ -39,7 +39,7 @@ CoeffType = Rational{Int64}
 ##freedom = Engine.dimension(ideal_ab_s)
 ##println("Two points on a sphere: $freedom degrees of freedom")
 
-spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
+##spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 ##tangencies = [
 ##  Engine.AlignsWithBy{CoeffType}(
 ##    spheres[n],
@@ -48,26 +48,45 @@ spheres = [Engine.Sphere{CoeffType}() for _ in 1:3]
 ##  )
 ##  for n in 1:3
 ##]
-tangencies = [
-  Engine.AlignsWithBy{CoeffType}(spheres[1], spheres[2], CoeffType(-1)),
-  Engine.AlignsWithBy{CoeffType}(spheres[2], spheres[3], CoeffType(-1//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)
+##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")
-println("Chain of three spheres: $freedom degrees of freedom")
+
+p = Engine.Point{CoeffType}()
+q = Engine.Point{CoeffType}()
+a = Engine.Sphere{CoeffType}()
+b = Engine.Sphere{CoeffType}()
+p_on_a = Engine.LiesOn{CoeffType}(p, a)
+p_on_b = Engine.LiesOn{CoeffType}(p, b)
+q_on_a = Engine.LiesOn{CoeffType}(q, a)
+q_on_b = Engine.LiesOn{CoeffType}(q, b)
+ctx_joined = Engine.Construction{CoeffType}(
+  elements = Set([p, q, a, b]),
+  relations= Set([p_on_a, p_on_b, q_on_a, q_on_b])
+)
+ideal_joined, eqns_joined = Engine.realize(ctx_joined)
+freedom = Engine.dimension(ideal_joined)
+println("Two points on two 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))
 Random.seed!(6071)
-n_planes = 16
+n_planes = 3
 samples = []
 for _ in 1:n_planes
   real_solns = solution.(Engine.Numerical.real_samples(system, freedom))