Cubic Hermite interpolation

.JuliaFormatter.toml

@@ -0,0 +1,11 @@
+short_to_long_function_def = true
+always_for_in = true
+whitespace_ops_in_indices = true
+pipe_to_function_call = true
+import_to_using = true
+always_use_return = true
+whitespace_in_kwargs = false
+remove_extra_newlines = true
+annotate_untyped_fields_with_any = false
+conditional_to_if = false
+ignore = ["tutorials"]

.github/workflows/documenter.yml

@@ -13,7 +13,7 @@ jobs:
       - uses: actions/checkout@v4
       - uses: quarto-dev/quarto-actions/setup@v2
         with:
-          version: "1.3.353"
+          version: "1.5.56"
       - uses: julia-actions/setup-julia@latest
         with:
           version: "1.10"

.gitignore

@@ -1,2 +1,5 @@
 docs/Manifest.toml
 examples/Manifest.toml
+examples/.CondaPkg/
+examples/_freeze/
+docs/.CondaPkg

CondaPkg.toml

@@ -0,0 +1,5 @@
+[deps]
+jupyter = ""
+python = "3.11"
+matplotlib = ">=3.4"
+matplotlib-inline = ""

Project.toml

@@ -5,6 +5,13 @@ version = "0.1.0"
 
 [extras]
 Test = "8dfed614-e22c-5e08-85e1-65c5234f0b40"
+ManifoldDiff = "af67fdf4-a580-4b9f-bbec-742ef357defd"
+Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
+
+[compat]
+ManifoldDiff = "0.3"
+Manifolds = "0.10"
+julia = "1.6"
 
 [targets]
 test = ["Test"]

docs/Project.toml

@@ -1,9 +1,16 @@
 [deps]
+CondaPkg = "992eb4ea-22a4-4c89-a5bb-47a3300528ab"
 Documenter = "e30172f5-a6a5-5a46-863b-614d45cd2de4"
 DocumenterCitations = "daee34ce-89f3-4625-b898-19384cb65244"
 DocumenterInterLinks = "d12716ef-a0f6-4df4-a9f1-a5a34e75c656"
+ManifoldDiff = "af67fdf4-a580-4b9f-bbec-742ef357defd"
+ManifoldExamples = "21be47e3-92bf-4199-8515-27870869dcc6"
+Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
 
 [compat]
+CondaPkg = "0.2"
 Documenter = "1"
 DocumenterCitations = "1"
 DocumenterInterLinks = "0.3"
+Plots = "1"

docs/build/references.bib

@@ -0,0 +1,15 @@
+
+@article{Zimmermann:2020,
+	title = {Hermite {Interpolation} and {Data} {Processing} {Errors} on {Riemannian} {Matrix} {Manifolds}},
+	volume = {42},
+	issn = {1064-8275},
+	url = {https://epubs.siam.org/doi/10.1137/19M1282878},
+	doi = {10.1137/19M1282878},
+	number = {5},
+	journal = {SIAM Journal on Scientific Computing},
+	author = {Zimmermann, Ralf},
+	month = jan,
+	year = {2020},
+	note = {Publisher: Society for Industrial and Applied Mathematics},
+	pages = {A2593--A2619},
+}

docs/make.jl

@@ -2,6 +2,29 @@
 #
 #
 
+if "--help" ∈ ARGS
+    println(
+        """
+docs/make.jl
+
+Render the `Manopt.jl` documentation with optional arguments
+
+Arguments
+* `--help`              - print this help and exit without rendering the documentation
+* `--prettyurls`        – toggle the prettyurls part to true (which is otherwise only true on CI)
+* `--quarto`            – run the Quarto notebooks from the `tutorials/` folder before generating the documentation
+  this has to be run locally at least once for the `tutorials/*.md` files to exist that are included in
+  the documentation (see `--exclude-examples`) for the alternative.
+  If they are generated once they are cached accordingly.
+  Then you can spare time in the rendering by not passing this argument.
+  If quarto is not run, some tutorials are generated as empty files, since they
+  are referenced from within the documentation. These are currently
+  `Optimize.md` and `ImplementOwnManifold.md`.
+""",
+    )
+    exit(0)
+end
+
 #
 # (a) if docs is not the current active environment, switch to it
 # (from https://github.com/JuliaIO/HDF5.jl/pull/1020/) 
@@ -13,24 +36,28 @@ if Base.active_project() != joinpath(@__DIR__, "Project.toml")
     Pkg.instantiate()
 end
 
-# (b) Did someone say render? Then we render!
+# (b) Did someone say render?
 if "--quarto" ∈ ARGS
     using CondaPkg
     CondaPkg.withenv() do
         @info "Rendering Quarto"
         examples_folder = (@__DIR__) * "/../examples"
         # instantiate the tutorials environment if necessary
         Pkg.activate(examples_folder)
+        # For a breaking release -> also set the tutorials folder to the most recent version
+        Pkg.develop(PackageSpec(; path=(@__DIR__) * "/../"))
         Pkg.resolve()
         Pkg.instantiate()
-        Pkg.build("IJulia") # build IJulia to the right version.
+        Pkg.build("IJulia") # build `IJulia` to the right version.
         Pkg.activate(@__DIR__) # but return to the docs one before
-        run(`quarto render $(examples_folder)`)
+        return run(`quarto render $(examples_folder)`)
     end
 end
 
 # (c) load necessary packages for the docs
 using Documenter: DocMeta, HTML, MathJax3, deploydocs, makedocs
+using DocumenterCitations
+
 using ManifoldExamples
 
 generated_path = joinpath(@__DIR__, "src")
@@ -52,13 +79,19 @@ open(joinpath(generated_path, "contributing.md"), "w") do io
     end
 end
 
+examples_menu = "Examples" => ["Cubic Hermite interpolation" => "examples/hermite.md"]
+
+bib = CitationBibliography(joinpath(@__DIR__, "src", "references.bib"); style=:alpha)
 makedocs(;
     format=HTML(; mathengine=MathJax3(), prettyurls=get(ENV, "CI", nothing) == "true"),
     sitename="ManifoldExamples.jl",
     modules=[ManifoldExamples],
     pages=[
         "Home" => "index.md",
+        examples_menu,
         "Contributing to ManifoldExamples.jl" => "contributing.md",
+        "References" => "references.md",
     ],
+    plugins=[bib],
 )
 deploydocs(; repo="github.com/JuliaManifolds/ManifoldExamples.jl", push_preview=true)

docs/src/references.bib

@@ -0,0 +1,15 @@
+
+@article{Zimmermann:2020,
+	title = {Hermite {Interpolation} and {Data} {Processing} {Errors} on {Riemannian} {Matrix} {Manifolds}},
+	volume = {42},
+	issn = {1064-8275},
+	url = {https://epubs.siam.org/doi/10.1137/19M1282878},
+	doi = {10.1137/19M1282878},
+	number = {5},
+	journal = {SIAM Journal on Scientific Computing},
+	author = {Zimmermann, Ralf},
+	month = jan,
+	year = {2020},
+	note = {Publisher: Society for Industrial and Applied Mathematics},
+	pages = {A2593--A2619},
+}

docs/src/references.md

@@ -0,0 +1,4 @@
+# Literature
+
+```@bibliography
+```

examples/.gitignore

@@ -0,0 +1 @@
+/.quarto/

examples/Project.toml

@@ -1,2 +1,13 @@
 [deps]
+IJulia = "7073ff75-c697-5162-941a-fcdaad2a7d2a"
+ManifoldDiff = "af67fdf4-a580-4b9f-bbec-742ef357defd"
 ManifoldExamples = "21be47e3-92bf-4199-8515-27870869dcc6"
+Manifolds = "1cead3c2-87b3-11e9-0ccd-23c62b72b94e"
+Plots = "91a5bcdd-55d7-5caf-9e0b-520d859cae80"
+PythonPlot = "274fc56d-3b97-40fa-a1cd-1b4a50311bf9"
+RecursiveArrayTools = "731186ca-8d62-57ce-b412-fbd966d074cd"
+
+[compat]
+IJulia = "1"
+Plots = "1"
+PythonPlot = "1"

examples/_quarto.yml

@@ -22,7 +22,7 @@ execute:
 
 format:
   commonmark:
-    variant: -raw_html+tex_math_dollars
+    variant: -raw_html+tex_math_dollars+pipe_tables+footnotes
     wrap: preserve
 
-jupyter: "julia-1.10"
+jupyter: julia-1.10

examples/hermite.qmd

@@ -0,0 +1,115 @@
+---
+title: "Cubic Hermite interpolation on manifolds"
+author: "Mateusz Baran"
+date: 05/09/2024
+---
+
+## Introduction
+
+This example shows how to perform cubic Hermite interpolation on Riemannian manifolds. The idea is described in [Zimmermann:2020](@cite).
+
+First, let's import necessary libraries.
+
+```{julia}
+using Manifolds, ManifoldDiff, Plots
+using ManifoldDiff: differential_log_argument
+pythonplot()
+
+```
+
+The main interpolation function directly follows Eq (2.5) from [Zimmermann:2020](@cite).
+
+```{julia}
+
+function manifold_hermite_interpolation(
+    M::AbstractManifold,
+    p,
+    q,
+    Xp,
+    Xq,
+    t::Real,
+)
+    Y_qp = log(M, q, p)
+    Xp_to_q = differential_log_argument(M, q, p, Xp)
+    a0t = 2*t^3 - 3*t^2 + 1 # simplified (A.1)
+    b0t = t^3 - 2*t^2 + t # simplified (A.3)
+    b1t = t^3 - t^2 # simplified (A.4)
+    return exp(M, q, a0t .* Y_qp .+ b0t .* Xp_to_q .+ b1t .* Xq)
+end
+```
+
+We can now plot the interpolating line between two points on a sphere ($p$ and $q$) with tangent vectors $X_p$ and $X_q$. The curve $c\colon [0, 1] \to \mathbb{S}^2$ defined by `manifold_hermite_interpolation` now has the following interpolation properties:
+
+1. $c(0) = p$
+2. $c(1) = q$
+3. $\dot{c}(0) = X_p$
+4. $\dot{c}(1) = X_q$
+5. In the Euclidean case, the `manifold_hermite_interpolation` conicides with cubic Hermite interpolation.
+
+```{julia}
+
+M = Sphere(2)
+p = [0.8266841314682074, 0.3288540904434144, 0.45656142410117206]
+Xp = [0.15493539779687937, 0.5824002702016382, -0.7000314284584177]
+
+q = [0.0, 1.0, 0.0]
+Xq = [-2, 0.0, 5]
+
+scene = plot(M, [p, q]; wireframe_color=colorant"#CCCCCC", markersize=10, camera=(140.0, 10.0))
+plot!(scene, M, [p, q], [Xp, Xq]; wireframe = false, linewidth=1.5)
+
+interp_line = [manifold_hermite_interpolation(M, p, q, Xp, Xq, t) for t in 0.0:0.01:1.0]
+plot!(scene, M, interp_line; wireframe = false, linewidth=1.5)
+
+```
+
+Now, let's add interpolating curve with reversed start and end, and reflected tangent vectors.
+Note that for, contrary to the Eulidean case, the order of points does matter for cubic interpolation on a sphere.
+
+```{julia}
+
+scene = plot(M, [p, q]; wireframe_color=colorant"#CCCCCC", markersize=10, camera=(140.0, 10.0))
+plot!(scene, M, [p, q], [Xp, Xq]; wireframe = false, linewidth=1.5)
+
+plot!(scene, M, interp_line; wireframe = false, linewidth=1.5)
+interp_line2 = [manifold_hermite_interpolation(M, q, p, -Xq, -Xp, t) for t in 0.0:0.01:1.0]
+plot!(scene, M, interp_line2; wireframe = false, linewidth=1.5)
+
+```
+
+One possible way of making an interpolation method that is independent of the order of points in computing the average between these two interplations (see `manifold_hermite_interpolation_symmetric` below).
+
+
+```{julia}
+
+function manifold_hermite_interpolation_symmetric(
+    M::AbstractManifold,
+    p,
+    q,
+    Xp,
+    Xq,
+    t::Real,
+)
+    r_pq = manifold_hermite_interpolation(M, p, q, Xp, Xq, t)
+    r_qp = manifold_hermite_interpolation(M, q, p, -Xq, -Xp, 1-t)
+    return mid_point(M, r_pq, r_qp)
+end
+scene = plot(M, [p, q]; wireframe_color=colorant"#CCCCCC", markersize=10, camera=(140.0, 10.0))
+plot!(scene, M, [p, q], [Xp, Xq]; wireframe = false, linewidth=1.5)
+
+plot!(scene, M, interp_line; wireframe = false, linewidth=1.5)
+plot!(scene, M, interp_line2; wireframe = false, linewidth=1.5)
+
+interp_line3 = [manifold_hermite_interpolation_symmetric(M, q, p, -Xq, -Xp, t) for t in 0.0:0.01:1.0]
+plot!(scene, M, interp_line3; wireframe = false, linewidth=1.5)
+
+```
+
+## Literature
+
+````{=commonmark}
+```@bibliography
+Pages = ["hermite.md"]
+Canonical=false
+```
+````

test/runtests.jl

@@ -1,5 +1,3 @@
 using ManifoldExamples, Test
 
-@testset "Manifold Examples" begin
-
-end
+@testset "Manifold Examples" begin end