### A Pluto.jl notebook ###
# v0.19.13

using Markdown
using InteractiveUtils

# This Pluto notebook uses @bind for interactivity. When running this notebook outside of Pluto, the following 'mock version' of @bind gives bound variables a default value (instead of an error).
macro bind(def, element)
    return quote
        local iv = try
            Base.loaded_modules[Base.PkgId(Base.UUID("6e696c72-6542-2067-7265-42206c756150"), "AbstractPlutoDingetjes")].Bonds.initial_value
        catch
            b -> missing
        end
        local el = $(esc(element))
        global $(esc(def)) = Core.applicable(Base.get, el) ? Base.get(el) : iv(el)
        el
    end
end

# ╔═╡ 04e7ac42-4b1b-11ed-1cb3-e76ac86d4203
using Manifolds, RecursiveArrayTools, OrdinaryDiffEq,
    DiffEqCallbacks, BoundaryValueDiffEq, PlutoUI

# ╔═╡ 8159d5bc-2c3f-4657-80c4-661e9162b79d
md"""
# Working in Charts: the embedded Torus

There are two conceptually different approaches to working on a manifold: working in charts and chart-free representations.

The first one, widespread in differential geometry textbooks, is based on defining an atlas on the manifold and performing computations in selected charts. This approach, while generic, is not ideally suitable in all circumstances. For example, working in charts that do not cover the entire manifold causes issues with having to switch charts when operating on a manifold.

The second one is beneficial if there exists a representation of points and tangent vectors for a manifold which allows for efficient closed-form formulas for standard functions like the exponential map or Riemannian distance in this representation. These computations are then chart-free. `Manifolds.jl` supports both approaches, although the chart-free approach is the main focus of the library.

In this tutorial we focus on chart-based computation.
"""

# ╔═╡ b85a56ad-3b74-402d-8fbe-9f0d130253e6
md"""
The manifold we consider is the `M` is the torus in form of the [`EmbeddedTorus`](https://juliamanifolds.github.io/Manifolds.jl/latest/manifolds/torus.html#Manifolds.EmbeddedTorus), that is the representation defined as a surface of revolution of a circle of radius 2 around a circle of radius 3.
The atlas we will perform computations in is its [`DefaultTorusAtlas`](https://juliamanifolds.github.io/Manifolds.jl/latest/manifolds/torus.html#Manifolds.DefaultTorusAtlas) `A`, consisting of a family of charts indexed by two angles, that specify the base point of the chart.

We will draw geodesics time between `0` and `t_end`, and then sample the solution at multiples of `dt` and draw a line connecting sampled points.
"""

# ╔═╡ 27038d25-32b3-433d-ad47-f7ae330fef1c
M = Manifolds.EmbeddedTorus(3, 2)

# ╔═╡ e0c54d00-c698-4ea6-a710-37fc284ed83e
A = Manifolds.DefaultTorusAtlas()

# ╔═╡ f0d60996-ae46-4bfa-8408-b6e28cba1f6b
md"""
## Setup

We will first set up our plot with an empty torus.
`param_points` are points on the surface of the torus that will be used for basic surface shape in Makie.jl.
The torus will be colored according to its Gaussian curvature stored in `gcs`. We later want to have a color scale that has negative curvature blue, zero curvature white and positive curvature red so `gcs_mm` is the largest absolute value of the curvature that will be needed to properly set range of curvature values.
"""

# ╔═╡ 96fcb06e-2517-4c5b-b350-375d1ecb3258
md"""
 In the documentation this tutorial represents a static situation (loading precomputed images). In the Pluto notebook, you can set interactive to true to change the points and tangent vectors in the following example.
"""

# ╔═╡ 87272f4b-f898-4edb-8842-0464ffc4bf01
interactive = false;

# ╔═╡ 0fe4dd5d-9838-4236-8ef9-bfe9afb76ac7
md"""
We also cache the images to reduce the documentation generation time.
If you activate `rendering` the images are saved in non-interactive mode.
"""

# ╔═╡ 9efc27d4-caa0-4e9c-85e6-e9dbdbafbb9c
begin
    localpath = join(splitpath(@__FILE__)[1:(end - 1)], "/") # files folder
    image_prefix = localpath * "/working-in-charts"
    @info image_prefix
    render_images = false
end;

# ╔═╡ 40487cc6-707f-4ca3-b735-6d5f50b07f74
# Only load Makie if necessary
if (interactive || render_images)
    using GLMakie, Makie
    GLMakie.activate!()
end

# ╔═╡ f2276b50-49f4-478f-9a96-0e374a37fe2f
"""
	torus_figure()

This function generates a simple plot of a torus and returns the new figure containing the plot.
"""
function torus_figure()
    fig = Figure(resolution = (1400, 1000), fontsize = 16)
    ax = LScene(fig[1, 1], show_axis = true)
    ϴs, φs = LinRange(-π, π, 50), LinRange(-π, π, 50)
    param_points = [Manifolds._torus_param(M, θ, φ) for θ in ϴs, φ in φs]
    X1, Y1, Z1 = [[p[i] for p in param_points] for i in 1:3]
    gcs = [gaussian_curvature(M, p) for p in param_points]
    gcs_mm = max(abs(minimum(gcs)), abs(maximum(gcs)))
    pltobj = surface!(
        ax,
        X1,
        Y1,
        Z1;
        shading = true,
        ambient = Vec3f(0.65, 0.65, 0.65),
        backlight = 1.0f0,
        color = gcs,
        colormap = Reverse(:RdBu),
        colorrange = (-gcs_mm, gcs_mm),
        transparency = true,
    )
    wireframe!(ax, X1, Y1, Z1; transparency = true, color = :gray, linewidth = 0.5)
    zoom!(ax.scene, cameracontrols(ax.scene), 0.98)
    Colorbar(fig[1, 2], pltobj, height = Relative(0.5), label = "Gaussian curvature")
    return ax, fig
end

# ╔═╡ c18609fe-b510-447f-9d2a-d1e6eb1da3c2
md"""
## (Interactive) Values for the geodesic.

`solve_for` is a helper function that solves a parallel transport along geodesic problem on the torus `M`.
`p0x` is the (ϴ, φ) parametrization of the point from which we will transport the vector.
We first calculate the coordinates in the embedding of `p0x` and store it as `p`, and then get the initial chart from atlas `A` appropriate for starting working at point `p`.
The vector we transport has coordinates `Y_transp` in the induced tangent space basis of chart `i_p0x`.
The function returns the full solution to the parallel transport problem, containing the sequence of charts that was used and solutions of differential equations computed using `OrdinaryDiffEq`.

`bvp_i` is needed later for a different purpose, it is the chart index we will use for solving the logarithmic map boundary value problem in.

Next we solve the vector transport problem `solve_for([θₚ, φₚ], [θₓ, φₓ], [θy, φy])`, sample the result at the selected time steps and store the result in `geo`. The solution includes the geodesic which we extract and convert to a sequence of points digestible by `Makie.jl`, `geo_ps`.
`[θₚ, φₚ]` is the parametrization in chart (0, 0) of the starting point of the geodesic.
The direction of the geodesic is determined by `[θₓ, φₓ]`, coordinates of the tangent vector at the starting point expressed in the induced basis of chart `i_p0x` (which depends on the initial point).
Finally, `[θy, φy]` are the coordinates of the tangent vector that will be transported along the geodesic, which are also expressed in same basis as `[θₓ, φₓ]`.

We won't draw the transported vector at every point as there would be too many arrows, which is why we select every 100th point only for that purpose with `pt_indices`. Then, `geo_ps_pt` contains points at which the transported vector is tangent to and `geo_Ys` the transported vector at that point, represented in the embedding.

The logarithmic map will be solved between points with parametrization `bvp_a1` and `bvp_a2` in chart `bvp_i`.
The result is assigned to variable `bvp_sol` and then sampled with time step 0.05. The result of this sampling is converted from parameters in chart `bvp_i` to point in the embedding and stored in `geo_r`.
"""

# ╔═╡ 2532be0e-6d48-4775-b78f-e48f04e5c3a3
function solve_for(p0x, X_p0x, Y_transp, T)
    p = [Manifolds._torus_param(M, p0x...)...]
    i_p0x = Manifolds.get_chart_index(M, A, p)
    p_exp = Manifolds.solve_chart_parallel_transport_ode(
        M,
        [0.0, 0.0],
        X_p0x,
        A,
        i_p0x,
        Y_transp;
        final_time = T,
    )
    return p_exp
end;

# ╔═╡ 9ee3fbaf-a579-480e-ae33-9d6eb8b07fbc
md"""
### Solving parallel Transport ODE
"""

# ╔═╡ 34cd1003-3790-4db1-bcf8-2e98cf9b98b6
if interactive
    md"""
    We have to determine the 

    * ``t_\mathrm{end}`` $(@bind t_end PlutoUI.Slider(range(0,100,1001), default=2, show_value=true))
    * ``\mathrm{d}t`` $(@bind dt PlutoUI.Slider(range(1e-3,0.5,500), default=1e-1, show_value=true))
    """
else
    t_end = 2.0
    dt = 1.0e-1
    md"""
    We set the ``t_\mathrm{end}`` to $(t_end) and ``\mathrm{d}t`` to $(dt)
    """
end

# ╔═╡ 6db745f9-afa4-4b5a-8871-b58585165c9e
md" And we parametrise the start point and direction"

# ╔═╡ fea185a1-bfae-4bd8-9b2a-f889e54f3f70
if interactive
    md"""
    * start point θₚ $(@bind θₚ PlutoUI.Slider(range(-π,π,200), default=π/10, show_value=false)) | φₚ $(@bind φₚ PlutoUI.Slider(range(-π,π,200), default=-π/4))
    * start direction θₓ $(@bind θₓ PlutoUI.Slider(range(-π,π,200), default=π/2)) | φₓ $(@bind φₓ PlutoUI.Slider(range(-π,π,200), default=0.7))
    * a tangent vector θy $(@bind θy PlutoUI.Slider(range(-π,π,200), default=0.2)) | φy $(@bind φy PlutoUI.Slider(range(-π,π,200), default=-0.1)) to be transported
    """
else
    θₚ = π / 10
    φₚ = -π / 4
    θₓ = π / 2
    φₓ = 0.7
    θy = 0.2
    φy = -0.1
    #
    s = "( $(θₚ), $(φₚ))"
    s2 = "( $(θₓ), $(φₓ))"
    s3 = "$(θy), $(φy))"
    md"""
    We set the start point to
    * (θₚ, φₚ) = $(s)
    * (θₓ, φₓ) = $(s2)
    * (θy, φy) = $(s3).
    """
end

# ╔═╡ c1660206-d21a-4812-9dd3-bda91b633c0b
begin
    geo = solve_for([θₚ, φₚ], [θₓ, φₓ], [θy, φy], t_end)(0.0:dt:t_end)
    geo_ps = [Point3f(s[1]) for s in geo]
    pt_indices = 1:div(length(geo), 10):length(geo)
    geo_ps_pt = [Point3f(s[1]) for s in geo[pt_indices]]
    geo_Ys = [Point3f(s[3]) for s in geo[pt_indices]]
end;

# ╔═╡ a30fa94f-5669-4265-a541-03d16dbd5745
begin
    if render_images || interactive # generate images in these two cases
        ax1, fig1 = torus_figure()
        arrows!(ax1, geo_ps_pt, geo_Ys, linewidth = 0.05, color = :blue)
        lines!(geo_ps; linewidth = 4.0, color = :green)
        (!interactive) && Makie.save(image_prefix * "working-in-charts-transport.png", fig1)
        fig1
    else
        PlutoUI.LocalResource(image_prefix * "/working-in-charts-transport.png")
    end
end

# ╔═╡ a941fd19-faf5-49d0-8f68-ae2fbe45130d
md" ### Solving the logarithmic map ODE"

# ╔═╡ 922461b0-55a0-447b-b59d-cfff7b448858
if interactive

    md"""
    We set
    * start point θ₁ $(@bind θ₁ PlutoUI.Slider(range(-π,π,200), default=π/2)) |  φ₁ $(@bind φ₁ PlutoUI.Slider(range(-π,π,200), default=-1.0))
    * end point θ₂ $(@bind θ₂ PlutoUI.Slider(range(-π,π,200), default=-π/8)) | φ₂ $(@bind φ₂ PlutoUI.Slider(range(-π,π,200), default=π/2))
    """
else
    θ₁ = π / 2
    φ₁ = -1.0
    θ₂ = -π / 8
    φ₂ = π / 2
    s4 = "($(θ₁), $(φ₁))"
    s5 = "($(θ₂), $(φ₂))"
    md"""
    We set
    * (θ₁, φ₁) = $(s4)
    * (θ₂, φ₂) = $(s5)
    """
end

# ╔═╡ 2abef887-32a8-40ba-89e1-b6adbfe9174f
begin
    bvp_i = (0, 0)
    bvp_a1 = [θ₁, φ₁]
    bvp_a2 = [θ₂, φ₂]
    bvp_sol = Manifolds.solve_chart_log_bvp(M, bvp_a1, bvp_a2, A, bvp_i)
    geo_r = [Point3f(get_point(M, A, bvp_i, p[1:2])) for p in bvp_sol(0.0:0.05:1.0)]
end;

# ╔═╡ eea2fbc7-1ba8-49f8-916c-743984abe15d
begin
    if render_images || interactive # generate images in these two cases
        ax2, fig2 = torus_figure()
        lines!(geo_r; linewidth = 4.0, color = :green)
        (!interactive) && Makie.save(image_prefix * "/working-in-charts-geodesic.png", fig2)
        fig2
    else
        PlutoUI.LocalResource(image_prefix * "/working-in-charts-geodesic.png")
    end
end

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[[deps.StatsBase]]
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[[deps.StatsFuns]]
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[[deps.StrideArraysCore]]
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[[deps.StructArrays]]
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[[deps.SuiteSparse]]
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[[deps.SuiteSparse_jll]]
deps = ["Artifacts", "Libdl", "Pkg", "libblastrampoline_jll"]
uuid = "bea87d4a-7f5b-5778-9afe-8cc45184846c"
version = "5.10.1+0"

[[deps.TOML]]
deps = ["Dates"]
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[[deps.TableTraits]]
deps = ["IteratorInterfaceExtensions"]
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[[deps.Tables]]
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[[deps.Tar]]
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[[deps.TensorCore]]
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[[deps.Test]]
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[[deps.ThreadingUtilities]]
deps = ["ManualMemory"]
git-tree-sha1 = "f8629df51cab659d70d2e5618a430b4d3f37f2c3"
uuid = "8290d209-cae3-49c0-8002-c8c24d57dab5"
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[[deps.TiffImages]]
deps = ["ColorTypes", "DataStructures", "DocStringExtensions", "FileIO", "FixedPointNumbers", "IndirectArrays", "Inflate", "Mmap", "OffsetArrays", "PkgVersion", "ProgressMeter", "UUIDs"]
git-tree-sha1 = "70e6d2da9210371c927176cb7a56d41ef1260db7"
uuid = "731e570b-9d59-4bfa-96dc-6df516fadf69"
version = "0.6.1"

[[deps.TranscodingStreams]]
deps = ["Random", "Test"]
git-tree-sha1 = "8a75929dcd3c38611db2f8d08546decb514fcadf"
uuid = "3bb67fe8-82b1-5028-8e26-92a6c54297fa"
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[[deps.TriangularSolve]]
deps = ["CloseOpenIntervals", "IfElse", "LayoutPointers", "LinearAlgebra", "LoopVectorization", "Polyester", "SnoopPrecompile", "Static", "VectorizationBase"]
git-tree-sha1 = "fdddcf6b2c7751cd97de69c18157aacc18fbc660"
uuid = "d5829a12-d9aa-46ab-831f-fb7c9ab06edf"
version = "0.1.14"

[[deps.Tricks]]
git-tree-sha1 = "6bac775f2d42a611cdfcd1fb217ee719630c4175"
uuid = "410a4b4d-49e4-4fbc-ab6d-cb71b17b3775"
version = "0.1.6"

[[deps.TriplotBase]]
git-tree-sha1 = "4d4ed7f294cda19382ff7de4c137d24d16adc89b"
uuid = "981d1d27-644d-49a2-9326-4793e63143c3"
version = "0.1.0"

[[deps.URIs]]
git-tree-sha1 = "e59ecc5a41b000fa94423a578d29290c7266fc10"
uuid = "5c2747f8-b7ea-4ff2-ba2e-563bfd36b1d4"
version = "1.4.0"

[[deps.UUIDs]]
deps = ["Random", "SHA"]
uuid = "cf7118a7-6976-5b1a-9a39-7adc72f591a4"

[[deps.UnPack]]
git-tree-sha1 = "387c1f73762231e86e0c9c5443ce3b4a0a9a0c2b"
uuid = "3a884ed6-31ef-47d7-9d2a-63182c4928ed"
version = "1.0.2"

[[deps.Unicode]]
uuid = "4ec0a83e-493e-50e2-b9ac-8f72acf5a8f5"

[[deps.UnicodeFun]]
deps = ["REPL"]
git-tree-sha1 = "53915e50200959667e78a92a418594b428dffddf"
uuid = "1cfade01-22cf-5700-b092-accc4b62d6e1"
version = "0.4.1"

[[deps.VectorizationBase]]
deps = ["ArrayInterface", "CPUSummary", "HostCPUFeatures", "IfElse", "LayoutPointers", "Libdl", "LinearAlgebra", "SIMDTypes", "Static"]
git-tree-sha1 = "866e77ea9c675306652f5b5b9010ccbccc684c79"
uuid = "3d5dd08c-fd9d-11e8-17fa-ed2836048c2f"
version = "0.21.53"

[[deps.VertexSafeGraphs]]
deps = ["Graphs"]
git-tree-sha1 = "8351f8d73d7e880bfc042a8b6922684ebeafb35c"
uuid = "19fa3120-7c27-5ec5-8db8-b0b0aa330d6f"
version = "0.2.0"

[[deps.WoodburyMatrices]]
deps = ["LinearAlgebra", "SparseArrays"]
git-tree-sha1 = "de67fa59e33ad156a590055375a30b23c40299d3"
uuid = "efce3f68-66dc-5838-9240-27a6d6f5f9b6"
version = "0.5.5"

[[deps.XML2_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libiconv_jll", "Pkg", "Zlib_jll"]
git-tree-sha1 = "58443b63fb7e465a8a7210828c91c08b92132dff"
uuid = "02c8fc9c-b97f-50b9-bbe4-9be30ff0a78a"
version = "2.9.14+0"

[[deps.XSLT_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Libgcrypt_jll", "Libgpg_error_jll", "Libiconv_jll", "Pkg", "XML2_jll", "Zlib_jll"]
git-tree-sha1 = "91844873c4085240b95e795f692c4cec4d805f8a"
uuid = "aed1982a-8fda-507f-9586-7b0439959a61"
version = "1.1.34+0"

[[deps.Xorg_libX11_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libxcb_jll", "Xorg_xtrans_jll"]
git-tree-sha1 = "5be649d550f3f4b95308bf0183b82e2582876527"
uuid = "4f6342f7-b3d2-589e-9d20-edeb45f2b2bc"
version = "1.6.9+4"

[[deps.Xorg_libXau_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4e490d5c960c314f33885790ed410ff3a94ce67e"
uuid = "0c0b7dd1-d40b-584c-a123-a41640f87eec"
version = "1.0.9+4"

[[deps.Xorg_libXcursor_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXfixes_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "12e0eb3bc634fa2080c1c37fccf56f7c22989afd"
uuid = "935fb764-8cf2-53bf-bb30-45bb1f8bf724"
version = "1.2.0+4"

[[deps.Xorg_libXdmcp_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fe47bd2247248125c428978740e18a681372dd4"
uuid = "a3789734-cfe1-5b06-b2d0-1dd0d9d62d05"
version = "1.1.3+4"

[[deps.Xorg_libXext_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "b7c0aa8c376b31e4852b360222848637f481f8c3"
uuid = "1082639a-0dae-5f34-9b06-72781eeb8cb3"
version = "1.3.4+4"

[[deps.Xorg_libXfixes_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "0e0dc7431e7a0587559f9294aeec269471c991a4"
uuid = "d091e8ba-531a-589c-9de9-94069b037ed8"
version = "5.0.3+4"

[[deps.Xorg_libXi_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXfixes_jll"]
git-tree-sha1 = "89b52bc2160aadc84d707093930ef0bffa641246"
uuid = "a51aa0fd-4e3c-5386-b890-e753decda492"
version = "1.7.10+4"

[[deps.Xorg_libXinerama_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll"]
git-tree-sha1 = "26be8b1c342929259317d8b9f7b53bf2bb73b123"
uuid = "d1454406-59df-5ea1-beac-c340f2130bc3"
version = "1.1.4+4"

[[deps.Xorg_libXrandr_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libXext_jll", "Xorg_libXrender_jll"]
git-tree-sha1 = "34cea83cb726fb58f325887bf0612c6b3fb17631"
uuid = "ec84b674-ba8e-5d96-8ba1-2a689ba10484"
version = "1.5.2+4"

[[deps.Xorg_libXrender_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Xorg_libX11_jll"]
git-tree-sha1 = "19560f30fd49f4d4efbe7002a1037f8c43d43b96"
uuid = "ea2f1a96-1ddc-540d-b46f-429655e07cfa"
version = "0.9.10+4"

[[deps.Xorg_libpthread_stubs_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "6783737e45d3c59a4a4c4091f5f88cdcf0908cbb"
uuid = "14d82f49-176c-5ed1-bb49-ad3f5cbd8c74"
version = "0.1.0+3"

[[deps.Xorg_libxcb_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "XSLT_jll", "Xorg_libXau_jll", "Xorg_libXdmcp_jll", "Xorg_libpthread_stubs_jll"]
git-tree-sha1 = "daf17f441228e7a3833846cd048892861cff16d6"
uuid = "c7cfdc94-dc32-55de-ac96-5a1b8d977c5b"
version = "1.13.0+3"

[[deps.Xorg_xtrans_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "79c31e7844f6ecf779705fbc12146eb190b7d845"
uuid = "c5fb5394-a638-5e4d-96e5-b29de1b5cf10"
version = "1.4.0+3"

[[deps.Zlib_jll]]
deps = ["Libdl"]
uuid = "83775a58-1f1d-513f-b197-d71354ab007a"
version = "1.2.12+3"

[[deps.ZygoteRules]]
deps = ["MacroTools"]
git-tree-sha1 = "8c1a8e4dfacb1fd631745552c8db35d0deb09ea0"
uuid = "700de1a5-db45-46bc-99cf-38207098b444"
version = "0.2.2"

[[deps.isoband_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "51b5eeb3f98367157a7a12a1fb0aa5328946c03c"
uuid = "9a68df92-36a6-505f-a73e-abb412b6bfb4"
version = "0.2.3+0"

[[deps.libaom_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "3a2ea60308f0996d26f1e5354e10c24e9ef905d4"
uuid = "a4ae2306-e953-59d6-aa16-d00cac43593b"
version = "3.4.0+0"

[[deps.libass_jll]]
deps = ["Artifacts", "Bzip2_jll", "FreeType2_jll", "FriBidi_jll", "HarfBuzz_jll", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "5982a94fcba20f02f42ace44b9894ee2b140fe47"
uuid = "0ac62f75-1d6f-5e53-bd7c-93b484bb37c0"
version = "0.15.1+0"

[[deps.libblastrampoline_jll]]
deps = ["Artifacts", "Libdl", "OpenBLAS_jll"]
uuid = "8e850b90-86db-534c-a0d3-1478176c7d93"
version = "5.1.1+0"

[[deps.libfdk_aac_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "daacc84a041563f965be61859a36e17c4e4fcd55"
uuid = "f638f0a6-7fb0-5443-88ba-1cc74229b280"
version = "2.0.2+0"

[[deps.libpng_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg", "Zlib_jll"]
git-tree-sha1 = "94d180a6d2b5e55e447e2d27a29ed04fe79eb30c"
uuid = "b53b4c65-9356-5827-b1ea-8c7a1a84506f"
version = "1.6.38+0"

[[deps.libsixel_jll]]
deps = ["Artifacts", "JLLWrappers", "JpegTurbo_jll", "Libdl", "Pkg", "libpng_jll"]
git-tree-sha1 = "d4f63314c8aa1e48cd22aa0c17ed76cd1ae48c3c"
uuid = "075b6546-f08a-558a-be8f-8157d0f608a5"
version = "1.10.3+0"

[[deps.libvorbis_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Ogg_jll", "Pkg"]
git-tree-sha1 = "b910cb81ef3fe6e78bf6acee440bda86fd6ae00c"
uuid = "f27f6e37-5d2b-51aa-960f-b287f2bc3b7a"
version = "1.3.7+1"

[[deps.nghttp2_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "8e850ede-7688-5339-a07c-302acd2aaf8d"
version = "1.48.0+0"

[[deps.p7zip_jll]]
deps = ["Artifacts", "Libdl"]
uuid = "3f19e933-33d8-53b3-aaab-bd5110c3b7a0"
version = "17.4.0+0"

[[deps.x264_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "4fea590b89e6ec504593146bf8b988b2c00922b2"
uuid = "1270edf5-f2f9-52d2-97e9-ab00b5d0237a"
version = "2021.5.5+0"

[[deps.x265_jll]]
deps = ["Artifacts", "JLLWrappers", "Libdl", "Pkg"]
git-tree-sha1 = "ee567a171cce03570d77ad3a43e90218e38937a9"
uuid = "dfaa095f-4041-5dcd-9319-2fabd8486b76"
version = "3.5.0+0"
"""

# ╔═╡ Cell order:
# ╠═04e7ac42-4b1b-11ed-1cb3-e76ac86d4203
# ╟─8159d5bc-2c3f-4657-80c4-661e9162b79d
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# ╠═27038d25-32b3-433d-ad47-f7ae330fef1c
# ╠═e0c54d00-c698-4ea6-a710-37fc284ed83e
# ╟─f0d60996-ae46-4bfa-8408-b6e28cba1f6b
# ╟─96fcb06e-2517-4c5b-b350-375d1ecb3258
# ╠═87272f4b-f898-4edb-8842-0464ffc4bf01
# ╟─0fe4dd5d-9838-4236-8ef9-bfe9afb76ac7
# ╠═9efc27d4-caa0-4e9c-85e6-e9dbdbafbb9c
# ╠═40487cc6-707f-4ca3-b735-6d5f50b07f74
# ╠═f2276b50-49f4-478f-9a96-0e374a37fe2f
# ╟─c18609fe-b510-447f-9d2a-d1e6eb1da3c2
# ╠═2532be0e-6d48-4775-b78f-e48f04e5c3a3
# ╟─9ee3fbaf-a579-480e-ae33-9d6eb8b07fbc
# ╟─34cd1003-3790-4db1-bcf8-2e98cf9b98b6
# ╟─6db745f9-afa4-4b5a-8871-b58585165c9e
# ╟─fea185a1-bfae-4bd8-9b2a-f889e54f3f70
# ╠═c1660206-d21a-4812-9dd3-bda91b633c0b
# ╠═a30fa94f-5669-4265-a541-03d16dbd5745
# ╟─a941fd19-faf5-49d0-8f68-ae2fbe45130d
# ╟─922461b0-55a0-447b-b59d-cfff7b448858
# ╠═2abef887-32a8-40ba-89e1-b6adbfe9174f
# ╠═eea2fbc7-1ba8-49f8-916c-743984abe15d
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
