Added two models to show fiber symmetries with curvature on a 2-dimen…

…sional base manifold.

docs/src/newsreport.md

@@ -325,8 +325,12 @@ This is related to the fact that the weak interaction in the Standard Model is n
 
 ![40](./assets/newsreport/40.PNG)
 
+[A complex 1-dimensional vector space as the fiber space, where the *stretch* corresponds to multiplication by a real number.](https://github.com/iamazadi/Porta.jl/blob/master/models/newsreport/gaugeconnections/fig1520stretchbyrealnumbers.jl)
+
 ![41](./assets/newsreport/41.PNG)
 
+[A complex 1-dimensional vector space as the fiber space, where the *stretch* corresponds to multiplication by a complex number.](https://github.com/iamazadi/Porta.jl/blob/master/models/newsreport/gaugeconnections/fig1521stretchbycomplexnumbers.jl)
+
 Hence, by the uniqueness of integral curves (which is a theorem about the uniqueness of solutions to odrinary differential equations) we have ``\phi_X(s) \cdot \phi_X(t) = \phi_X(s + t) \ \forall t \in I \cap (t_{min} - s, t_{max} - s)``.
 This implies the claim by uniqueness of solutions of ordinary differential equations.
 The unique solution of this differential equation for ``\gamma(t)`` is ``\gamma(t) = e^{tr(X)t}``.

models/newsreport/gaugeconnections/fig1520stretchbyrealnumbers.jl

@@ -0,0 +1,228 @@
+using FileIO
+using GLMakie
+using Porta
+
+
+scalarfield(x::Float64, y::Float64) = vec(convert_to_cartesian([1.0; (x + π / 2) / 2; y]))[3]
+
+
+f(θ::Float64, ϕ::Float64, k::Float64) = begin
+    z = θ + ϕ * im
+    A = im * k * z
+    ℝ³([θ; ϕ; k * scalarfield(real(z - A), imag(z - A))])
+end
+
+
+figuresize = (4096, 2160)
+segments = 100
+frames_number = 360
+modelname = "fig1520stretchbyrealnumbers"
+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, 0.0, 0.5)) * float(2π)
+lookat = ℝ³(0.0, 0.0, 0.0)
+up = normalize(ℝ³(0.0, 0.0, 1.0))
+linewidth = 20
+distance = float(2π)
+radius = x̂ * float(π)
+attributespath = "data/naturalearth/geometry-attributes.csv"
+nodespath = "data/naturalearth/geometry-nodes.csv"
+boundary_names = Set{String}()
+boundary_nodes = Vector{Vector{ℝ³}}()
+indices = Dict{String,Int}()
+reference = load("data/basemap_color.png")
+mask = load("data/basemap_mask.png")
+markersize = 0.05
+arrowsize = Vec3f(0.06, 0.08, 0.1)
+arrowlinewidth = 0.04
+arrowscale = 0.5
+k = 1.0
+ϵ = 1e-3
+totalstages = 6
+colorrange = frames_number / totalstages
+spawn = true
+interpolation = 1.0
+strain_intensity = 1.0
+
+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) < 30
+    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
+
+lspace1 = range(-π, stop = float(π), length = segments)
+lspace2 = range(-π / 2, stop = π / 2, length = segments)
+section1 = [ℝ³([-θ; ϕ; 0.0]) for θ in lspace2, ϕ in lspace1]
+section2 = [f(-θ, ϕ, k) + k * ẑ for θ in lspace2, ϕ in lspace1]
+section3 = [f(-θ, ϕ, k) + 2k * ẑ for θ in lspace2, ϕ in lspace1]
+section4 = [f(-θ, ϕ, k) + 3k * ẑ for θ in lspace2, ϕ in lspace1]
+section5 = [f(-θ, ϕ, k) + 4k * ẑ for θ in lspace2, ϕ in lspace1]
+section6 = [f(-θ, ϕ, k) + 5k * ẑ for θ in lspace2, ϕ in lspace1]
+sectionobservable1 = buildsurface(lscene, section1, mask, transparency = true)
+sectionobservable2 = buildsurface(lscene, section2, mask, transparency = true)
+sectionobservable3 = buildsurface(lscene, section3, mask, transparency = true)
+sectionobservable4 = buildsurface(lscene, section4, mask, transparency = true)
+sectionobservable5 = buildsurface(lscene, section5, mask, transparency = true)
+sectionobservable6 = buildsurface(lscene, section6, mask, transparency = true)
+
+fibersobservables = []
+lspace = range(0.0, stop = distance, length = segments)
+for index in 1:length(boundary_names)
+    boundary = convert_to_geographic.(boundary_nodes[index])
+    fiber = [ℝ³(vec(boundary[i])[2:3]..., _distance) for i in eachindex(boundary), _distance in lspace]
+    color = fill(getcolor(boundary_nodes[index], reference, 0.5), length(boundary), segments)
+    push!(fibersobservables, buildsurface(lscene, fiber, color, transparency = true))
+end
+
+path = Observable(Point3f[])
+basepath = Observable(Point3f[])
+colors = Observable(Int[])
+lines!(lscene, path, color = colors, linewidth = linewidth, colorrange = (1, colorrange), colormap = :plasma)
+lines!(lscene, basepath, color = colors, linewidth = linewidth, colorrange = (1, colorrange), colormap = :plasma)
+
+uptail = Observable(Point3f(x̂))
+upheadx = Observable(Point3f(x̂))
+upheady = Observable(Point3f(ŷ))
+basetail = Observable(Point3f(x̂))
+baseheadx = Observable(Point3f(x̂))
+baseheady = Observable(Point3f(ŷ))
+ps = @lift([$uptail, $uptail, $basetail, $basetail])
+ns = @lift([$upheadx, $upheady, $baseheadx, $baseheady])
+colorants = [:red, :green]
+arrows!(lscene,
+    ps, ns, fxaa = true, # turn on anti-aliasing
+    color = [colorants..., colorants...],
+    linewidth = arrowlinewidth, arrowsize = arrowsize,
+    align = :origin, transparency = false
+)
+meshscatter!(lscene, uptail, markersize = markersize, color = :black)
+meshscatter!(lscene, basetail, markersize = markersize, color = :black)
+
+spawnframe = []
+spawnps = []
+spawnns = []
+spawnupball = []
+spawnbaseball = []
+
+
+calculateframe(frame::Int, k::Float64) = begin
+    progress = Float64(frame / frames_number)
+    stage = min(totalstages - 1, Int(floor(totalstages * progress))) + 1
+    stageprogress = totalstages * (progress - (stage - 1) * 1.0 / totalstages)
+    α = progress * 2π * totalstages
+    p = exp(im * α)
+    x = real(p)
+    y = imag(p)
+    _basepoint = ℝ³(x, y, 0.0)
+    height = k * (stage - 1) + stageprogress * vec(f(x, y, k))[3]
+    _uppoint = f(x, y, k) + ℝ³(0.0, 0.0, height)
+    _uptail = Point3f(_uppoint)
+    _basetail = Point3f(_basepoint)
+    _baseheadx = Point3f(x̂)
+    _baseheady = Point3f(ŷ)
+    _upheadx = Point3f((f(x + ϵ, y, k) - f(x, y, k)) * (1 / ϵ))
+    _upheady = Point3f((f(x, y + ϵ, k) - f(x, y, k)) * (1 / ϵ))
+    _basetail, _baseheadx, _baseheady, _uptail, _upheadx, _upheady, height
+end
+
+
+animate(frame::Int) = begin
+    progress = Float64(frame / frames_number)
+    stage = min(totalstages - 1, Int(floor(totalstages * progress))) + 1
+    stageprogress = totalstages * (progress - (stage - 1) * 1.0 / totalstages)
+    println("Frame: $frame, Stage: $stage, Total Stages: $totalstages, Progress: $stageprogress")
+    if stage < totalstages
+        global k = abs(cos(progress * 2π))
+        section1 = [ℝ³([-θ; ϕ; 0.0]) for θ in lspace2, ϕ in lspace1]
+        section2 = [f(-θ, ϕ, k) + k * ẑ for θ in lspace2, ϕ in lspace1]
+        section3 = [f(-θ, ϕ, k) + 2k * ẑ for θ in lspace2, ϕ in lspace1]
+        section4 = [f(-θ, ϕ, k) + 3k * ẑ for θ in lspace2, ϕ in lspace1]
+        section5 = [f(-θ, ϕ, k) + 4k * ẑ for θ in lspace2, ϕ in lspace1]
+        section6 = [f(-θ, ϕ, k) + 5k * ẑ for θ in lspace2, ϕ in lspace1]
+        updatesurface!(section1, sectionobservable1)
+        updatesurface!(section2, sectionobservable2)
+        updatesurface!(section3, sectionobservable3)
+        updatesurface!(section4, sectionobservable4)
+        updatesurface!(section5, sectionobservable5)
+        updatesurface!(section6, sectionobservable6)
+
+        basetail[], baseheadx[], baseheady[], uptail[], upheadx[], upheady[], height = calculateframe(frame, k)
+        spawn = true
+        if spawn && frame % 5 == 0
+            _ps = Observable([uptail[], uptail[], basetail[], basetail[]])
+            _ns = Observable([upheadx[], upheady[], baseheadx[], baseheady[]])
+            colorants = [:red, :green]
+            arrows!(lscene,
+                _ps, _ns, fxaa = true, # turn on anti-aliasing
+                color = [colorants..., colorants...],
+                linewidth = arrowlinewidth * arrowscale, arrowsize = arrowsize .* arrowscale,
+                align = :origin, transparency = true
+            )
+            upball = Observable(uptail[])
+            baseball = Observable(basetail[])
+            meshscatter!(lscene, upball, markersize = markersize * arrowscale, color = :black)
+            meshscatter!(lscene, baseball, markersize = markersize * arrowscale, color = :black)
+            push!(spawnps, _ps)
+            push!(spawnns, _ns)
+            push!(spawnupball, upball)
+            push!(spawnbaseball, baseball)
+            push!(spawnframe, frame)
+        end
+        push!(basepath[], basetail[])
+        push!(path[], uptail[])
+        push!(colors[], frame % colorrange)
+        for index in 1:frame - 1
+            _basetail, _baseheadx, _baseheady, _uptail, _upheadx, _upheady, _height = calculateframe(index, k)
+            basepath[][index] = _basetail
+            path[][index] = _uptail
+        end
+        for index in eachindex(spawnframe)
+            _basetail, _baseheadx, _baseheady, _uptail, _upheadx, _upheady, _height = calculateframe(spawnframe[index], k)
+            spawnps[index][] = [_uptail, _uptail, _basetail, _basetail]
+            spawnns[index][] = [_upheadx, _upheady, _baseheadx, _baseheady]
+            spawnbaseball[index][] = _basetail
+            spawnupball[index][] = _uptail
+        end
+        notify(basepath)
+        notify(path)
+        notify(colors)
+        lookat = ℝ³(uptail[])
+        updatecamera!(lscene, eyeposition + ℝ³(0.0, 0.0, height), lookat, up)
+    else
+        lookat = (1 - stageprogress) * ℝ³(0.0, 0.0, vec(ℝ³(uptail[]))[3])
+        updatecamera!(lscene, -2ẑ * stageprogress + rotate(eyeposition + (1 - stageprogress) * ℝ³(0.0, 0.0, vec(ℝ³(uptail[]))[3]), ℍ(stageprogress * 2π, ẑ)), lookat, up)
+    end
+end
+
+
+# animate(1)
+
+# path[] = Point3f[]
+# basepath[] = Point3f[]
+# colors[] = Int[]
+# for i in 1:frames_number
+#     animate(i)
+# end
+
+path[] = Point3f[]
+basepath[] = Point3f[]
+colors[] = Int[]
+record(fig, joinpath("gallery", "$modelname.mp4"), 1:frames_number) do frame
+    animate(frame)
+end