Find real solutions for three mutually tangent spheres
I'm not sure why the solver wasn't working before. It might've been just an unlucky random number draw.
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@ -186,16 +186,16 @@ function realize(ctx::Construction{T}) where T
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# add relations to center, orient, and scale the construction
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# [to do] the scaling constraint, as written, can be impossible to satisfy
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# when all of the spheres have to go through the origin
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##if !isempty(ctx.points)
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## append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3])
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##end
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##if !isempty(ctx.spheres)
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## append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
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##end
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##n_elts = length(ctx.points) + length(ctx.spheres)
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##if n_elts > 0
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## push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
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##end
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if !isempty(ctx.points)
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append!(eqns, [sum(pt.coords[k] for pt in ctx.points) for k in 1:3])
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end
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if !isempty(ctx.spheres)
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append!(eqns, [sum(sph.coords[k] for sph in ctx.spheres) for k in 3:4])
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end
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n_elts = length(ctx.points) + length(ctx.spheres)
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if n_elts > 0
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push!(eqns, sum(elt.vec[2] for elt in Iterators.flatten((ctx.points, ctx.spheres))) - n_elts)
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end
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(Generic.Ideal(coordring, eqns), eqns)
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## [test] (nothing, eqns)
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@ -59,13 +59,13 @@ tangencies = [
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ctx_tan_sph = Engine.Construction{CoeffType}(elements = spheres, relations = tangencies)
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ideal_tan_sph, eqns_tan_sph = Engine.realize(ctx_tan_sph)
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##small_eqns_tan_sph = eqns_tan_sph
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small_eqns_tan_sph = [
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eqns_tan_sph;
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spheres[2].coords - [1, 0, 0, 0, 1];
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spheres[3].coords - [1, 0, 0, 0, -1];
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]
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small_ideal_tan_sph = Generic.Ideal(base_ring(ideal_tan_sph), small_eqns_tan_sph)
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freedom = Engine.dimension(small_ideal_tan_sph)
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##small_eqns_tan_sph = [
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## eqns_tan_sph;
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## spheres[2].coords - [1, 0, 0, 0, 1];
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## spheres[3].coords - [1, 0, 0, 0, -1];
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##]
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##small_ideal_tan_sph = Generic.Ideal(base_ring(ideal_tan_sph), small_eqns_tan_sph)
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freedom = Engine.dimension(ideal_tan_sph)
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println("Three mutually tangent spheres, with two fixed: $freedom degrees of freedom")
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##points = [Engine.Point{CoeffType}() for _ in 1:3]
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@ -83,11 +83,11 @@ println("Three mutually tangent spheres, with two fixed: $freedom degrees of fre
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# --- test rational cut ---
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coordring = base_ring(small_ideal_tan_sph)
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coordring = base_ring(ideal_tan_sph)
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vbls = Variable.(symbols(coordring))
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# test a random witness set
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system = CompiledSystem(System(small_eqns_tan_sph, variables = vbls))
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system = CompiledSystem(System(eqns_tan_sph, variables = vbls))
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norm2 = vec -> real(dot(conj.(vec), vec))
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rng = MersenneTwister(6071)
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n_planes = 3
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