### A Pluto.jl notebook ###
# v0.19.0

using Markdown
using InteractiveUtils

# ╔═╡ 9404760f-4015-46b1-8d2b-1bec68024dbb
using Colors, Manopt, Manifolds, PlutoUI

# ╔═╡ 0fabb176-c3e8-11ec-1304-f3d3346be36f
md"""
# Illustration of Jacobi Fields

This tutorial illustrates the usage of Jacobi Fields within
`Manopt.jl`.
For this tutorial you should be familiar with the basic terminology on a
manifold like the exponential and logarithmic map as well as
[shortest geodesic](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.shortest_geodesic-Tuple{Manifold,Any,Any})s.

We first initialize the packages we need
"""

# ╔═╡ cde64039-7fd7-4897-bc31-2e45b7ef18cd
md"and we define some colors from [Paul Tol](https://personal.sron.nl/~pault/)"

# ╔═╡ c8590a06-4825-4e4d-b3fa-7a82a8982063
begin
    black = RGBA{Float64}(colorant"#000000")
    TolVibrantMagenta = RGBA{Float64}(colorant"#EE3377")
    TolVibrantOrange = RGBA{Float64}(colorant"#EE7733")
    TolVibrantCyan = RGBA{Float64}(colorant"#33BBEE")
    TolVibrantTeal = RGBA{Float64}(colorant"#009988")
end;

# ╔═╡ 2028d1ab-3dcb-4f17-a02f-f9ef8c2dfacb
md"And setup our graphics paths"

# ╔═╡ 4e1fc0af-2837-40cc-b6ff-3d8d6d641b6b
begin
    localpath = join(splitpath(@__FILE__)[1:(end - 1)], "/") # files folder
    image_prefix = localpath * "/jacobi_fields"
    @info image_prefix
    render_asy = false # on CI or when you do not have asymptote, this should be false
end;

# ╔═╡ 17803b9a-5b91-4d16-ae52-06a83653dea2
md"""
Assume we have two points on the equator of the [Sphere](https://juliamanifolds.github.io/Manifolds.jl/stable/manifolds/sphere.html) $\mathcal M = \mathbb S^2$
"""

# ╔═╡ bab2db18-0596-4716-a6ce-8bfaf2b8ca24
M = Sphere(2)

# ╔═╡ 9e0e4098-d6c4-49dd-9564-961c73be88d5
p, q = [[1.0, 0.0, 0.0], [0.0, 1.0, 0.0]]

# ╔═╡ ee9a8c21-431e-4448-b758-01e552b1df8c
md"""
their connecting [shortest geodesic](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.shortest_geodesic-Tuple{Manifold,Any,Any}) (sampled at `100` points)
"""

# ╔═╡ 8a0d1039-5cfe-4e4d-9a97-89c3b5705b36
geodesic_curve = shortest_geodesic(M, p, q, [0:0.1:1.0...]);

# ╔═╡ 5e0d7686-7129-41f6-87d7-7298ddb131a5
md"Is just a curve along the equator"

# ╔═╡ b75c32b0-e322-42e5-a8fa-f74893655e18
render_asy && asymptote_export_S2_signals(
    image_prefix * "/jacobi_geodesic.asy";
    curves=[geodesic_curve],
    points=[[p, q]],
    colors=Dict(:curves => [black], :points => [TolVibrantOrange]),
    dot_size=3.5,
    line_width=0.75,
    camera_position=(1.0, 1.0, 0.5),
);

# ╔═╡ 7e3d4b24-ad82-4146-93c4-3bd4011ab151
render_asy && render_asymptote(image_prefix * "/jacobi_geodesic.asy"; render=2);

# ╔═╡ 34622df6-2979-4f05-8be9-224a2ec413f9
PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic.png")

# ╔═╡ f0118459-5343-45dd-aa62-494795019044
md"""
where ``p`` is on the left and ``q`` on the right.
We know solve the following task:

Given a direction $X ∈ T_p\mathcal M$, for example
"""

# ╔═╡ a7b379ac-293e-454e-8ede-6f81bb7caa6e
X = [0.0, 0.4, 0.5]

# ╔═╡ f338d7d0-eef5-4264-af90-33e5bc6ecc62
md"""
we move the start point ``x`` into, how does any point on the geodesic move?

Or mathematically: Compute ``D_p g(t; p,q)`` for some fixed ``t∈[0,1]``
and a given direction ``X_p``.
Of course two cases are quite easy: For ``t=0`` we are in ``x`` and how ``x`` “moves”
is already known, so ``D_x g(0;p,q) = X``. On the other side, for ``t=1``,
``g(1; p,q) = q`` which is fixed, so ``D_p g(1; p,q)`` is the zero tangent vector
(in ``T_q\mathcal M``).

For all other cases we employ a [`jacobi_field`](https://manoptjl.org/stable/functions/Jacobi_fields.html), which is a (tangent)
vector field along the [shortest geodesic](https://juliamanifolds.github.io/Manifolds.jl/stable/interface.html#ManifoldsBase.shortest_geodesic-Tuple{Manifold,Any,Any}) given as follows: The _geodesic variation_
``\Gamma_{g,X}(s,t)`` is defined for some ``\varepsilon > 0`` as
````math
\Gamma_{g,X}(s,t):=\exp{\gamma_{p,X}(s)}[t\log_{g(s;p,X)}p],\qquad s∈(-\varepsilon,\varepsilon),\ t∈[0,1].
````
Intuitively we make a small step ``s`` into direction ``ξ`` using the geodesic
``g(⋅; p,X)`` and from ``r=g(s; p,X)`` we follow (in ``t``) the geodesic
``g(⋅; r,q)``. The corresponding Jacobi field ``J_{g,X}``
along ``g(⋅; p,q)`` is given by

````math
J_{g,X}(t):=\frac{D}{\partial s}\Gamma_{g,X}(s,t)\Bigl\rvert_{s=0}
````

which is an ODE and we know the boundary conditions ``J_{g,X}(0)=X`` and
``J_{g,X}(t) = 0``. In symmetric spaces we can compute the solution, since the
system of ODEs decouples, see for example [^doCarmo1992],
Chapter 4.2. Within `Manopt.jl` this is implemented as
`jacobi_field(M,p,q,t,X[,β])`, where the optional parameter (function)
``β`` specifies, which Jacobi field we want to evaluate and the one used here is
the default.

We can hence evaluate that on the points on the geodesic at
"""

# ╔═╡ b457c7c8-7f5c-4b92-a064-0f36b188f7a1
T = [0:0.1:1.0...];

# ╔═╡ c2ba61e5-7f74-4cb7-b68d-9057ab896859
md"namely"

# ╔═╡ 3e044a29-dff6-4174-9adf-f1228672b91d
r = shortest_geodesic(M, p, q, T);

# ╔═╡ c17abc34-6e5d-41d9-b428-b984bc8f7cd9
md"The tangent vector now moves as a differential along the geodesic as"

# ╔═╡ d7308356-100f-4766-9afe-588960b925dd
W = jacobi_field.(Ref(M), Ref(p), Ref(q), T, Ref(X));

# ╔═╡ fab5b16b-e05e-4a8a-b001-236422d9e245
md"""Which can also be called using `differential_geodesic_startpoint`.
We can add to the image above by creating extended tangent vectors
the include their base points"""

# ╔═╡ 0713167c-80c6-4907-b1e3-e104d598a804
V = [Tuple([a, b]) for (a, b) in zip(r, W)];

# ╔═╡ e26bb3fb-5f7e-470a-a2a1-7be37eb1bcb9
md"to add the vectors as one further set to the Asymptote export."

# ╔═╡ aa621823-57e3-4f53-8b65-1afbb80943bc
render_asy && asymptote_export_S2_signals(
    image_prefix * "/jacobi_geodesic_diff_start.asy";
    curves=[geodesic_curve],
    points=[[p, q], r],
    tangent_vectors=[V],
    colors=Dict(
        :curves => [black],
        :points => [TolVibrantOrange, TolVibrantCyan],
        :tvectors => [TolVibrantCyan],
    ),
    dot_sizes=[3.5, 2.0],
    line_width=0.75,
    camera_position=(1.0, 1.0, 0.5),
);

# ╔═╡ d02cff4d-8e88-4979-b536-6f367af8ab32
render_asy && render_asymptote(image_prefix * "/jacobi_geodesic_diff_start.asy"; render=2);

# ╔═╡ a17f81a9-066f-435c-b5ba-ae4809ac835b
PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic_diff_start.png")

# ╔═╡ b2ac06aa-4b9f-4869-8b4c-c6f30661b8fe
md"""
The interpretation is as follows: If an infinitesimal change of the start point in direction ``X`` would happen, the infinitesimal change of a point along the line would change as indicated.

Note that each new vector is a tangent vector to its point (up to a small numerical tolerance), so the blue vectors are not just “shifted and scaled versions” of ``X``.
"""

# ╔═╡ fa6e15b8-2201-4095-8ebd-ddcd749fe11d
all([is_vector(M, a[1], a[2]; atol=1e-15) for a in V])

# ╔═╡ 99526567-14aa-4526-9a8d-48e2a3ba15a6
md"If we further move the end point, too, we can derive that Differential in direction"

# ╔═╡ 84090442-ea5f-4a1e-9c66-b0e828463853
begin
    Y = [0.2, 0.0, -0.5]
    W2 = differential_geodesic_endpoint.(Ref(M), Ref(p), Ref(q), T, Ref(Y))
    V2 = [Tuple([a, b]) for (a, b) in zip(r, W2)]
end;

# ╔═╡ bf3205fa-5348-42f9-8c1d-16e3fbcbc06c
md"and we can combine both vector fields we obtained"

# ╔═╡ c319b655-b3e5-4190-bf3c-d5af55da42ba
V3 = [Tuple([a, b]) for (a, b) in zip(r, W2 + W)];

# ╔═╡ 63b0d8ae-9a1b-46ff-978c-753f1cbdefe9
render_asy && asymptote_export_S2_signals(
    image_prefix * "/jacobi_geodesic_complete.asy";
    curves=[geodesic_curve],
    points=[[p, q], r],
    tangent_vectors=[V, V2, V3],
    colors=Dict(
        :curves => [black],
        :points => [TolVibrantOrange, TolVibrantCyan],
        :tvectors => [TolVibrantCyan, TolVibrantMagenta, TolVibrantTeal],
    ),
    dot_sizes=[3.5, 2.0],
    line_width=0.75,
    camera_position=(1.0, 1.0, 0.0),
);

# ╔═╡ 18278235-a049-4bd5-91dd-bddd31bcb8fb
render_asy && render_asymptote(image_prefix * "/jacobi_geodesic_complete.asy"; render=2);

# ╔═╡ cd1fa853-16ea-4975-b91b-5e6c1943523d
PlutoUI.LocalResource(image_prefix * "/jacobi_geodesic_complete.png")

# ╔═╡ ee803867-bc00-43b3-85c5-c4471370b6a8
md"""
Here the first vector field is still in blue, the second is in magenta, and their combined effect is in teal. Sure as a differential this does not make much sense, maybe as an infinitesimal movement of both start and end point cmobined.
"""

# ╔═╡ 83d85bac-85c7-462b-bf1c-097e27a3ca9a
md"""
## Literature

[^doCarmo1992]:
    > do Carmo, Manfredo __Riemannian Geometry__,  Birkhäuser Boston, 1992, ISBN: 0-8176-3490-8.
"""

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"""

# ╔═╡ Cell order:
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# ╠═63b0d8ae-9a1b-46ff-978c-753f1cbdefe9
# ╠═18278235-a049-4bd5-91dd-bddd31bcb8fb
# ╠═cd1fa853-16ea-4975-b91b-5e6c1943523d
# ╟─ee803867-bc00-43b3-85c5-c4471370b6a8
# ╟─83d85bac-85c7-462b-bf1c-097e27a3ca9a
# ╟─00000000-0000-0000-0000-000000000001
# ╟─00000000-0000-0000-0000-000000000002
