Write examples directly in light cone basis
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@ -207,14 +207,16 @@ end
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# ~~~ sandbox setup ~~~
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# in the default view, e4 + e5 is the point at infinity
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elements = Engine.nullmix * sqrt(0.5) * BigFloat[
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1 1 -1 -1 0;
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1 -1 1 -1 0;
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1 -1 -1 1 0;
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0 0 0 0 -sqrt(6);
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1 1 1 1 2
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]
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elements = begin
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const a = sqrt(BigFloat(3)/2)
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sqrt(0.5) * BigFloat[
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1 1 -1 -1 0
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1 -1 1 -1 0
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1 -1 -1 1 0
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-0.5 -0.5 -0.5 -0.5 -a-1
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0.5 0.5 0.5 0.5 -a+1
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]
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end
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# show construction
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viewer = Viewer.ConstructionViewer()
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@ -29,7 +29,7 @@ function rand_on_shell(rng::AbstractRNG, shell::T) where T <: Number
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space_part = rand_on_sphere(rng, T, 4)
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rapidity = randn(rng, T)
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sig = sign(shell)
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[sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
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nullmix * [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
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end
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rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number =
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@ -39,19 +39,16 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
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# === elements ===
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## [temp] in light cone coordinates
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point(pos) = [pos; 1; dot(pos, pos)]
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## [temp] in standard coordinates
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plane(normal, offset) = [normal; offset; offset]
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plane(normal, offset) = [normal; 0; offset]
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## [temp] in standard coordinates
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function sphere(center, radius)
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dist_sq = dot(center, center)
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[
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center / radius;
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0.5 * ((dist_sq - 1) / radius - radius);
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0.5 * ((dist_sq + 1) / radius - radius)
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-0.5 / radius;
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0.5 * (dist_sq / radius - radius)
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]
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end
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@ -55,7 +55,7 @@ gram = sparse(J, K, values)
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## guess = Engine.rand_on_shell(fill(BigFloat(-1), 8))
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# set initial guess
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guess = Engine.nullmix * hcat(
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guess = hcat(
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Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
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Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)),
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Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
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@ -64,7 +64,7 @@ guess = Engine.nullmix * hcat(
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Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)),
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Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)),
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Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)),
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BigFloat[0, 0, 0, 1, 1]
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BigFloat[0, 0, 0, 0, 1]
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)
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frozen = [CartesianIndex(j, 9) for j in 4:5]
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#=
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@ -36,16 +36,19 @@ gram[1, 6] = gram[6, 1]
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# in this initial guess, the mutual tangency condition is satisfied for spheres
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# 1 through 5
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Random.seed!(50793)
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guess = Engine.nullmix * hcat(
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sqrt(1/BigFloat(2)) * BigFloat[
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1 1 -1 -1 0;
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1 -1 1 -1 0;
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1 -1 -1 1 0;
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0 0 0 0 -sqrt(BigFloat(6));
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1 1 1 1 2;
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] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
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Engine.rand_on_shell(fill(BigFloat(-1), 2))
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)
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guess = begin
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const a = sqrt(BigFloat(3)/2)
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hcat(
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sqrt(1/BigFloat(2)) * BigFloat[
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1 1 -1 -1 0
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1 -1 1 -1 0
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1 -1 -1 1 0
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-0.5 -0.5 -0.5 -0.5 -a-1
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0.5 0.5 0.5 0.5 -a+1
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] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
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Engine.rand_on_shell(fill(BigFloat(-1), 2))
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)
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end
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# complete the gram matrix
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#=
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@ -41,15 +41,15 @@ gram = sparse(J, K, values)
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# set initial guess
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Random.seed!(99230)
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guess = Engine.nullmix * hcat(
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guess = hcat(
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sqrt(1/BigFloat(3)) * BigFloat[
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1 1 -1 -1 0
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1 -1 1 -1 0
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1 -1 -1 1 0
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1 1 1 1 -2
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1 1 1 1 1
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0 0 0 0 -1.5
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1 1 1 1 -0.5
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] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
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BigFloat[0, 0, 0, 1, 1]
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BigFloat[0, 0, 0, 0, 1]
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)
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frozen = [CartesianIndex(j, 6) for j in 1:5]
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