Write examples directly in light cone basis

engine-proto/ConstructionViewer.jl

@@ -207,14 +207,16 @@ end
 
 # ~~~ sandbox setup ~~~
 
-# in the default view, e4 + e5 is the point at infinity
+elements = begin
-elements = Engine.nullmix * sqrt(0.5) * BigFloat[
+  const a = sqrt(BigFloat(3)/2)
-  1  1 -1 -1  0;
+  sqrt(0.5) * BigFloat[
-  1 -1  1 -1  0;
+    1    1   -1   -1    0
-  1 -1 -1  1  0;
+    1   -1    1   -1    0
-  0  0  0  0 -sqrt(6);
+    1   -1   -1    1    0
-  1  1  1  1  2
+   -0.5 -0.5 -0.5 -0.5 -a-1
+    0.5  0.5  0.5  0.5 -a+1
   ]
+end
 
 # show construction
 viewer = Viewer.ConstructionViewer()

engine-proto/gram-test/Engine.jl

@@ -29,7 +29,7 @@ function rand_on_shell(rng::AbstractRNG, shell::T) where T <: Number
   space_part = rand_on_sphere(rng, T, 4)
   rapidity = randn(rng, T)
   sig = sign(shell)
-  [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
+  nullmix * [sconh(rapidity, sig)*space_part; sconh(rapidity, -sig)]
 end
 
 rand_on_shell(rng::AbstractRNG, shells::Array{T}) where T <: Number =
@@ -39,19 +39,16 @@ rand_on_shell(shells::Array{<:Number}) = rand_on_shell(Random.default_rng(), she
 
 # === elements ===
 
-## [temp] in light cone coordinates
 point(pos) = [pos; 1; dot(pos, pos)]
 
-## [temp] in standard coordinates
+plane(normal, offset) = [normal; 0; offset]
-plane(normal, offset) = [normal; offset; offset]
 
-## [temp] in standard coordinates
 function sphere(center, radius)
   dist_sq = dot(center, center)
   [
     center / radius;
-    0.5 * ((dist_sq - 1) / radius - radius);
+    -0.5 / radius;
-    0.5 * ((dist_sq + 1) / radius - radius)
+    0.5 * (dist_sq / radius - radius)
   ]
 end
 

engine-proto/gram-test/circles-in-triangle.jl

@@ -55,7 +55,7 @@ gram = sparse(J, K, values)
 ## guess = Engine.rand_on_shell(fill(BigFloat(-1), 8))
 
 # set initial guess
-guess = Engine.nullmix * hcat(
+guess = hcat(
   Engine.plane(BigFloat[0, 0, 1], BigFloat(0)),
   Engine.sphere(BigFloat[0, 0, 0], BigFloat(1//2)),
   Engine.plane(BigFloat[1, 0, 0], BigFloat(1)),
@@ -64,7 +64,7 @@ guess = Engine.nullmix * hcat(
   Engine.sphere(BigFloat[-1, 0, 0], BigFloat(1//5)),
   Engine.sphere(BigFloat[cos(-pi/3), sin(-pi/3), 0], BigFloat(1//5)),
   Engine.sphere(BigFloat[cos(pi/3), sin(pi/3), 0], BigFloat(1//5)),
-  BigFloat[0, 0, 0, 1, 1]
+  BigFloat[0, 0, 0, 0, 1]
 )
 frozen = [CartesianIndex(j, 9) for j in 4:5]
 #=

engine-proto/gram-test/overlapping-pyramids.jl

@@ -36,16 +36,19 @@ gram[1, 6] = gram[6, 1]
 # in this initial guess, the mutual tangency condition is satisfied for spheres
 # 1 through 5
 Random.seed!(50793)
-guess = Engine.nullmix * hcat(
+guess = begin
+  const a = sqrt(BigFloat(3)/2)
+  hcat(
     sqrt(1/BigFloat(2)) * BigFloat[
-    1  1 -1 -1  0;
+      1    1   -1   -1    0
-    1 -1  1 -1  0;
+      1   -1    1   -1    0
-    1 -1 -1  1  0;
+      1   -1   -1    1    0
-    0  0  0  0 -sqrt(BigFloat(6));
+     -0.5 -0.5 -0.5 -0.5 -a-1
-    1  1  1  1  2;
+      0.5  0.5  0.5  0.5 -a+1
     ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
     Engine.rand_on_shell(fill(BigFloat(-1), 2))
   )
+end
 
 # complete the gram matrix
 #=

engine-proto/gram-test/sphere-in-tetrahedron.jl

@@ -41,15 +41,15 @@ gram = sparse(J, K, values)
 
 # set initial guess
 Random.seed!(99230)
-guess = Engine.nullmix * hcat(
+guess = hcat(
   sqrt(1/BigFloat(3)) * BigFloat[
     1  1 -1 -1  0
     1 -1  1 -1  0
     1 -1 -1  1  0
-    1  1  1  1 -2
+    0  0  0  0 -1.5
-    1  1  1  1  1
+    1  1  1  1 -0.5
   ] + 0.2*Engine.rand_on_shell(fill(BigFloat(-1), 5)),
-  BigFloat[0, 0, 0, 1, 1]
+  BigFloat[0, 0, 0, 0, 1]
 )
 frozen = [CartesianIndex(j, 6) for j in 1:5]