Showed a complex vector bundle by blowing up the origin.

docs/src/newsreport.md

@@ -301,6 +301,8 @@ This is related to the fact that the weak interaction in the Standard Model is n
 
 ![34](./assets/newsreport/34.PNG)
 
+[An example of a complex line bundle](https://github.com/iamazadi/Porta.jl/blob/master/models/newsreport/gaugeconnections/fig1511complexlinebundle.jl)
+
 ![35](./assets/newsreport/35.PNG)
 
 ![36](./assets/newsreport/36.PNG)

models/newsreport/gaugeconnections/fig1511complexlinebundle.jl

@@ -0,0 +1,371 @@
+using FileIO
+using GLMakie
+using Porta
+
+
+figuresize = (4096, 2160)
+segments = 120
+frames_number = 360
+modelname = "fig1511complexlinebundle"
+x̂ = ℝ³([1.0; 0.0; 0.0])
+ŷ = ℝ³([0.0; 1.0; 0.0])
+ẑ = ℝ³([0.0; 0.0; 1.0])
+eyeposition = normalize(ℝ³(1.0, 1.0, 1.0)) * float(π)
+lookat = ℝ³(0.0, 0.0, 0.0)
+up = normalize(ℝ³(0.0, 0.0, 1.0))
+sphereradius = 1.0
+mask = load("data/basemap_mask.png")
+reference = load("data/basemap_color.png")
+attributespath = "data/naturalearth/geometry-attributes.csv"
+nodespath = "data/naturalearth/geometry-nodes.csv"
+boundary_names = Set()
+boundary_nodes = Vector{Vector{ℝ³}}()
+points = Vector{Vector{ℍ}}()
+indices = Dict()
+T, X, Y, Z = vec(normalize(ℝ⁴(1.0, 0.0, 1.0, 0.0)))
+u = 𝕍(T, X, Y, Z)
+q = ℍ(T, X, Y, Z)
+tolerance = 1e-3
+@assert(isnull(u, atol = tolerance), "u in not a null vector, $u.")
+@assert(isapprox(norm(q), 1, atol = tolerance), "q in not a unit quaternion, $(norm(q)).")
+ϵ = 0.1
+timesign = 1
+T = float(timesign)
+gauge1 = 0.0
+gauge2 = π / 2
+gauge3 = float(π)
+gauge4 = 3π / 2
+gauge5 = 2π
+latitudescale = 1 / 2
+longitudescale = 1 / 4
+chart = (-π * latitudescale / 2, π * latitudescale / 2, -π * longitudescale, π * longitudescale)
+M = Identity(4)
+markersize = 0.05
+linewidth = 20
+arrowsize = Vec3f(0.06, 0.08, 0.1)
+arrowlinewidth = 0.04
+fontsize = 0.25
+zero = Point3f(0.0, 0.0, 0.0)
+lspace1 = range(-π, stop = float(π), length = segments)
+lspace2 = range(-π / 2, stop = π / 2, length = segments)
+colorants = [:red, :green, :blue, :orange, :gold]
+colormap1 = :reds
+colormap2 = :Greens
+colormap3 = :Blues
+colormap4 = :Oranges
+colormap5 = :gold
+
+makefigure() = Figure(size = figuresize)
+fig = with_theme(makefigure, theme_black())
+pl = PointLight(Point3f(0), RGBf(0.0862, 0.0862, 0.0862))
+al = AmbientLight(RGBf(0.9, 0.9, 0.9))
+lscene = LScene(fig[1, 1], show_axis=true, scenekw = (lights = [pl, al], clear=true, backgroundcolor = :white))
+
+## Load the Natural Earth data
+countries = loadcountries(attributespath, nodespath)
+while length(boundary_names) < 10
+    push!(boundary_names, rand(countries["name"]))
+end
+for i in eachindex(countries["name"])
+    for name in boundary_names
+        if countries["name"][i] == name
+            push!(boundary_nodes, countries["nodes"][i])
+            println(name)
+            indices[name] = length(boundary_nodes)
+        end
+    end
+end
+for i in eachindex(boundary_nodes)
+    _points = Vector{ℍ}()
+    for node in boundary_nodes[i]
+        r, θ, ϕ = convert_to_geographic(node)
+        push!(_points, q * ℍ(exp(ϕ * longitudescale * K(1) + θ * latitudescale * K(2))))
+    end
+    push!(points, _points)
+end
+basemap1 = Basemap(lscene, q, gauge1, M, chart, segments, mask, transparency = true)
+basemap2 = Basemap(lscene, q, gauge2, M, chart, segments, mask, transparency = true)
+basemap3 = Basemap(lscene, q, gauge3, M, chart, segments, mask, transparency = true)
+basemap4 = Basemap(lscene, q, gauge4, M, chart, segments, mask, transparency = true)
+
+whirls1 = []
+whirls2 = []
+whirls3 = []
+whirls4 = []
+for i in eachindex(boundary_nodes)
+    color1 = getcolor(boundary_nodes[i], reference, 0.1)
+    color2 = getcolor(boundary_nodes[i], reference, 0.2)
+    color3 = getcolor(boundary_nodes[i], reference, 0.3)
+    color4 = getcolor(boundary_nodes[i], reference, 0.4)
+    whirl1 = Whirl(lscene, points[i], gauge1, gauge2, M, segments, color1, transparency = true)
+    whirl2 = Whirl(lscene, points[i], gauge2, gauge3, M, segments, color2, transparency = true)
+    whirl3 = Whirl(lscene, points[i], gauge3, gauge4, M, segments, color3, transparency = true)
+    whirl4 = Whirl(lscene, points[i], gauge4, gauge5, M, segments, color4, transparency = true)
+    push!(whirls1, whirl1)
+    push!(whirls2, whirl2)
+    push!(whirls3, whirl3)
+    push!(whirls4, whirl4)
+end
+
+origin = Observable(zero)
+tangenttail1 = Observable(zero)
+tangenttail2 = Observable(zero)
+tangenttail3 = Observable(zero)
+tangenttail4 = Observable(zero)
+tangenttail5 = Observable(zero)
+tangenthead1 = Observable(Point3f(x̂))
+tangenthead2 = Observable(Point3f(x̂))
+tangenthead3 = Observable(Point3f(x̂))
+tangenthead4 = Observable(Point3f(x̂))
+tangenthead5 = Observable(Point3f(x̂))
+pointa1 = Observable(zero)
+pointa2 = Observable(zero)
+pointa3 = Observable(zero)
+pointa4 = Observable(zero)
+pointa5 = Observable(zero)
+pointb1 = Observable(zero)
+pointb2 = Observable(zero)
+pointb3 = Observable(zero)
+pointb4 = Observable(zero)
+pointb5 = Observable(zero)
+pointc1 = Observable(zero)
+pointc2 = Observable(zero)
+pointc3 = Observable(zero)
+pointc4 = Observable(zero)
+pointc5 = Observable(zero)
+pointd1 = Observable(zero)
+pointd2 = Observable(zero)
+pointd3 = Observable(zero)
+pointd4 = Observable(zero)
+pointd5 = Observable(zero)
+
+meshscatter!(lscene, origin, markersize = markersize, color = :black)
+meshscatter!(lscene, pointa1, markersize = markersize, color = colorants[1])
+meshscatter!(lscene, pointb1, markersize = markersize, color = colorants[1])
+meshscatter!(lscene, pointc1, markersize = markersize, color = colorants[1])
+meshscatter!(lscene, pointd1, markersize = markersize, color = colorants[1])
+meshscatter!(lscene, pointa2, markersize = markersize, color = colorants[2], transparency = true)
+meshscatter!(lscene, pointb2, markersize = markersize, color = colorants[2], transparency = true)
+meshscatter!(lscene, pointc2, markersize = markersize, color = colorants[2], transparency = true)
+meshscatter!(lscene, pointd2, markersize = markersize, color = colorants[2], transparency = true)
+meshscatter!(lscene, pointa3, markersize = markersize, color = colorants[3], transparency = true)
+meshscatter!(lscene, pointb3, markersize = markersize, color = colorants[3], transparency = true)
+meshscatter!(lscene, pointc3, markersize = markersize, color = colorants[3], transparency = true)
+meshscatter!(lscene, pointd3, markersize = markersize, color = colorants[3], transparency = true)
+meshscatter!(lscene, pointa4, markersize = markersize, color = colorants[4], transparency = true)
+meshscatter!(lscene, pointb4, markersize = markersize, color = colorants[4], transparency = true)
+meshscatter!(lscene, pointc4, markersize = markersize, color = colorants[4], transparency = true)
+meshscatter!(lscene, pointd4, markersize = markersize, color = colorants[4], transparency = true)
+meshscatter!(lscene, pointa5, markersize = markersize, color = colorants[5], transparency = true)
+meshscatter!(lscene, pointb5, markersize = markersize, color = colorants[5], transparency = true)
+meshscatter!(lscene, pointc5, markersize = markersize, color = colorants[5], transparency = true)
+meshscatter!(lscene, pointd5, markersize = markersize, color = colorants[5], transparency = true)
+meshscatter!(lscene, tangenttail1, markersize = markersize, color = colorants[1])
+meshscatter!(lscene, tangenttail2, markersize = markersize, color = colorants[2], transparency = true)
+meshscatter!(lscene, tangenttail3, markersize = markersize, color = colorants[3], transparency = true)
+meshscatter!(lscene, tangenttail4, markersize = markersize, color = colorants[4], transparency = true)
+meshscatter!(lscene, tangenttail5, markersize = markersize, color = colorants[5], transparency = true)
+
+linepointsa1 = @lift([$pointa1, $tangenttail1])
+linepointsb1 = @lift([$pointb1, $tangenttail1])
+linepointsc1 = @lift([$pointc1, $tangenttail1])
+linepointsd1 = @lift([$pointd1, $tangenttail1])
+
+linepointsa2 = @lift([$pointa2, $tangenttail2])
+linepointsb2 = @lift([$pointb2, $tangenttail2])
+linepointsc2 = @lift([$pointc2, $tangenttail2])
+linepointsd2 = @lift([$pointd2, $tangenttail2])
+
+linepointsa3 = @lift([$pointa3, $tangenttail3])
+linepointsb3 = @lift([$pointb3, $tangenttail3])
+linepointsc3 = @lift([$pointc3, $tangenttail3])
+linepointsd3 = @lift([$pointd3, $tangenttail3])
+
+linepointsa4 = @lift([$pointa4, $tangenttail4])
+linepointsb4 = @lift([$pointb4, $tangenttail4])
+linepointsc4 = @lift([$pointc4, $tangenttail4])
+linepointsd4 = @lift([$pointd4, $tangenttail4])
+
+linepointsa5 = @lift([$pointa5, $tangenttail5])
+linepointsb5 = @lift([$pointb5, $tangenttail5])
+linepointsc5 = @lift([$pointc5, $tangenttail5])
+linepointsd5 = @lift([$pointd5, $tangenttail5])
+
+linecolors = Observable(collect(1:segments))
+lines!(lscene, linepointsa1, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap1)
+lines!(lscene, linepointsb1, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap1)
+lines!(lscene, linepointsc1, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap1)
+lines!(lscene, linepointsd1, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap1)
+
+lines!(lscene, linepointsa2, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+lines!(lscene, linepointsb2, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+lines!(lscene, linepointsc2, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+lines!(lscene, linepointsd2, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+
+lines!(lscene, linepointsa3, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+lines!(lscene, linepointsb3, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+lines!(lscene, linepointsc3, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+lines!(lscene, linepointsd3, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+
+lines!(lscene, linepointsa4, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+lines!(lscene, linepointsb4, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+lines!(lscene, linepointsc4, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+lines!(lscene, linepointsd4, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+
+lines!(lscene, linepointsa5, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+lines!(lscene, linepointsb5, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+lines!(lscene, linepointsc5, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+lines!(lscene, linepointsd5, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+
+fiber1 = Observable([Point3f(ℝ³(real(exp(im * α)), imag(exp(im * α)), 0.0)) for α in range(0, stop = 2π, length = segments)])
+fiber2 = Observable([Point3f(ℝ³(real(exp(im * α)), imag(exp(im * α)), 0.0)) for α in range(0, stop = 2π, length = segments)])
+fiber3 = Observable([Point3f(ℝ³(real(exp(im * α)), imag(exp(im * α)), 0.0)) for α in range(0, stop = 2π, length = segments)])
+fiber4 = Observable([Point3f(ℝ³(real(exp(im * α)), imag(exp(im * α)), 0.0)) for α in range(0, stop = 2π, length = segments)])
+fiber5 = Observable([Point3f(ℝ³(real(exp(im * α)), imag(exp(im * α)), 0.0)) for α in range(0, stop = 2π, length = segments)])
+lines!(lscene, fiber1, color = linecolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = colormap1)
+lines!(lscene, fiber2, color = linecolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+lines!(lscene, fiber3, color = linecolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+lines!(lscene, fiber4, color = linecolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+lines!(lscene, fiber5, color = linecolors, linewidth = 2linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+basepath1 = Observable(Point3f[])
+basepath2 = Observable(Point3f[])
+basepath3 = Observable(Point3f[])
+basepath4 = Observable(Point3f[])
+basepath5 = Observable(Point3f[])
+pathcolors = collect(1:frames_number)
+lines!(lscene, basepath1, color = pathcolors, linewidth = linewidth / 2, colorrange = (1, frames_number), colormap = colormap1)
+lines!(lscene, basepath2, color = pathcolors, linewidth = linewidth / 2, colorrange = (1, frames_number), colormap = colormap2, transparency = true)
+lines!(lscene, basepath3, color = pathcolors, linewidth = linewidth / 2, colorrange = (1, frames_number), colormap = colormap3, transparency = true)
+lines!(lscene, basepath4, color = pathcolors, linewidth = linewidth / 2, colorrange = (1, frames_number), colormap = colormap4, transparency = true)
+lines!(lscene, basepath5, color = pathcolors, linewidth = linewidth / 2, colorrange = (1, frames_number), colormap = colormap5, transparency = true)
+tangentcircle1 = Observable(Point3f[])
+tangentcircle2 = Observable(Point3f[])
+tangentcircle3 = Observable(Point3f[])
+tangentcircle4 = Observable(Point3f[])
+tangentcircle5 = Observable(Point3f[])
+lines!(lscene, tangentcircle1, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap1)
+lines!(lscene, tangentcircle2, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap2, transparency = true)
+lines!(lscene, tangentcircle3, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap3, transparency = true)
+lines!(lscene, tangentcircle4, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap4, transparency = true)
+lines!(lscene, tangentcircle5, color = linecolors, linewidth = linewidth, colorrange = (1, segments), colormap = colormap5, transparency = true)
+
+ps = @lift([$origin, $origin, $origin, $origin, $origin, $origin, $origin, $origin, $origin, $origin, $tangenttail1, $tangenttail2, $tangenttail3, $tangenttail4, $tangenttail5])
+ns = @lift([$pointa1, $pointc1, $pointa2, $pointc2, $pointa3, $pointc3, $pointa4, $pointc4, $pointa5, $pointc5, $tangenthead1, $tangenthead2, $tangenthead3, $tangenthead4, $tangenthead5])
+colorants1 = [colorants[1], colorants[1], colorants[2], colorants[2], colorants[3], colorants[3], colorants[4], colorants[4], colorants[5], colorants[5], colorants...]
+arrows!(lscene,
+    ps, ns, fxaa = true, # turn on anti-aliasing
+    color = colorants1,
+    linewidth = arrowlinewidth, arrowsize = arrowsize,
+    align = :origin
+)
+titles = ["O", "q₁", "-q₁", "q₂", "-q₂", "q₃", "-q₃", "q₄", "-q₄", "q₅", "-q₅", "π(q₁)", "π(q₂)", "π(q₃)", "π(q₄)", "π(q₅)"]
+rotation = gettextrotation(lscene)
+text!(lscene,
+    @lift(map(x -> Point3f(vec((isnan(x) ? ẑ : x))), [$origin, $pointa1, $pointc1, $pointa2, $pointc2, $pointa3, $pointc3, $pointa4, $pointc4, $pointa5, $pointc5, $tangenttail1, $tangenttail2, $tangenttail3, $tangenttail4, $tangenttail5])),
+    text = titles,
+    color = [:black, colorants1...],
+    rotation = rotation,
+    align = (:left, :baseline),
+    fontsize = fontsize,
+    markerspace = :data
+)
+
+
+animate(frame::Int) = begin
+    progress = Float64(frame / frames_number)
+    println("Frame: $frame, Progress: $progress")
+    ψ = progress * 2π
+    θ, ϕ = cos(4ψ) * π / 2, sin(6ψ) * π
+    _q = q * ℍ(exp(ϕ * longitudescale * K(1) + θ * latitudescale * K(2)))
+    a, b, c, d = vec(_q)
+    w = a + im * b
+    z = c + im * d
+    @assert(isapprox(abs(w)^2 + abs(z)^2, 1), "The point $_q is not in S³, in other words: |w|² + |z|² ≠ 1.")
+    κ = SpinVector(w, z, timesign)
+    κ′ = SpinVector(Complex(κ) - 1.0 / √2 * ϵ / κ.a[2], timesign)
+    κprojection = project(normalize(ℍ(vec( 𝕍( κ)))))
+    κ′projection = project(normalize(ℍ(vec( 𝕍( κ′)))))
+    tangenthead5[] = tangenthead4[]
+    tangenthead4[] = tangenthead3[]
+    tangenthead3[] = tangenthead2[]
+    tangenthead2[] = tangenthead1[]
+    tangenthead1[] = Point3f(normalize(κ′projection - κprojection))
+    tangenttail5[] = tangenttail4[]
+    tangenttail4[] = tangenttail3[]
+    tangenttail3[] = tangenttail2[]
+    tangenttail2[] = tangenttail1[]
+    pointa5[] = pointa4[]
+    pointa4[] = pointa3[]
+    pointa3[] = pointa2[]
+    pointa2[] = pointa1[]
+    pointa1[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge1))))))
+    pointb5[] = pointb4[]
+    pointb4[] = pointb3[]
+    pointb3[] = pointb2[]
+    pointb2[] = pointb1[]
+    pointb1[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge2))))))
+    pointc5[] = pointc4[]
+    pointc4[] = pointc3[]
+    pointc3[] = pointc2[]
+    pointc2[] = pointc1[]
+    pointc1[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge3))))))
+    pointd5[] = pointd4[]
+    pointd4[] = pointd3[]
+    pointd3[] = pointd2[]
+    pointd2[] = pointd1[]
+    pointd1[] = Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * gauge4))))))
+    fiber5[] = fiber4[]
+    fiber4[] = fiber3[]
+    fiber3[] = fiber2[]
+    fiber2[] = fiber1[]
+    fiber1[] = [Point3f(project(normalize(M * (_q * ℍ(exp(K(3) * α)))))) for α in range(0, stop = 2π, length = segments)]
+    tangentcircle5[] = tangentcircle4[]
+    tangentcircle4[] = tangentcircle3[]
+    tangentcircle3[] = tangentcircle2[]
+    tangentcircle2[] = tangentcircle1[]
+    tangentcircle1[] = Point3f[]
+    basepath5[] = basepath4[]
+    basepath4[] = basepath3[]
+    basepath3[] = basepath2[]
+    basepath2[] = basepath1[]
+    for i in 1:segments
+        α = exp(im * i / segments * 2π)
+        κ = SpinVector(α * w, α * z, timesign)
+        κ′ = SpinVector(Complex(κ) - 1.0 / √2 * ϵ / κ.a[2], timesign)
+        pa = ℝ³(hopfmap(normalize(ℍ(vec(κ)))))
+        pb = ℝ³(hopfmap(normalize(ℍ(vec(κ′)))))
+        if i == 1
+            tangenttail1[] = Point3f(pa)
+            push!(basepath1[], tangenttail1[])
+        end
+        push!(tangentcircle1[], Point3f(pa + normalize(pb - pa)))
+    end
+    notify(basepath1)
+    notify(basepath2)
+    notify(basepath3)
+    notify(basepath4)
+    notify(basepath5)
+    notify(tangentcircle1)
+    notify(tangentcircle2)
+    notify(tangentcircle3)
+    notify(tangentcircle4)
+    notify(tangentcircle5)
+    global lookat =  ℝ³(tangenttail1[] + (tangenttail1[] + tangenthead1[]) + pointa1[] + pointb1[] + pointc1[] + pointd1[]) * (1 / 6)
+    # global eyeposition = -normalize(ℝ³(tangenttail1[])) * float(π)
+    updatecamera!(lscene, eyeposition, lookat, up)
+end
+
+
+animate(1)
+
+for i in 1:5
+    animate(1)
+end
+basepath1[] = Point3f[]
+basepath2[] = Point3f[]
+basepath3[] = Point3f[]
+basepath4[] = Point3f[]
+basepath5[] = Point3f[]
+record(fig, joinpath("gallery", "$modelname.mp4"), 1:frames_number) do frame
+    animate(frame)
+end