UTF8 Documentation rework

docs/make.jl

@@ -38,6 +38,7 @@ makedocs(
         "Statistics" => "statistics.md",
         "Distributions" => "distributions.md",
         "Orthonormal bases" => "orthonormal_bases.md",
+        "Notation" => "notation.md",
         "Library" => [
             "Number systems" => "lib/numbers.md",
             "Public" => "lib/public.md",

docs/src/manifolds/euclidean.md

@@ -1,6 +1,6 @@
 # Euclidean space
 
-The Euclidean space $\mathbb R^n$ is a simple model space, since it has curvature constantly zero everywhere; hence, nearly all operations simplify.
+The Euclidean space $ℝ^n$ is a simple model space, since it has curvature constantly zero everywhere; hence, nearly all operations simplify.
 
 ```@autodocs
 Modules = [Manifolds]

docs/src/manifolds/graph.md

@@ -1,11 +1,11 @@
 # Graph manifold
 
 For a given graph $G(V,E)$ implemented using [`LightGraphs.jl`](https://juliagraphs.github.io/LightGraphs.jl/latest/), the [`GraphManifold`](@ref) models a [`PowerManifold`](@ref) either on the nodes or edges of the graph, depending on the [`GraphManifoldType`](@ref).
-i.e., it's either a $\mathcal M^{\lvert V \rvert}$ for the case of a vertex manifold or a $\mathcal M^{\lvert E \rvert}$ for the case of a edge manifold.
+i.e., it's either a $ℳ^{\lvert V \rvert}$ for the case of a vertex manifold or a $ℳ^{\lvert E \rvert}$ for the case of a edge manifold.
 
 ## Example:
 
-To make a graph manifold over $\mathbb{R}^2$ with three vertices and two edges, one can use
+To make a graph manifold over $ℝ^2$ with three vertices and two edges, one can use
 ```@example
 using Manifolds
 using LightGraphs

docs/src/manifolds/product.md

@@ -1,6 +1,6 @@
 # Product manifold
 
-Product manifold $M = M_1 \times M_2 \times \dots M_n$ of manifolds $M_1, M_2, \dots, M_n$.
+Product manifold $M = M_1 ⨉ M_2 ⨉ \dots M_n$ of manifolds $M_1, M_2, \dots, M_n$.
 Points on the product manifold can be constructed using [`ProductRepr`](@ref) with canonical projections $\Pi_i \colon M \to M_i$ for $i \in 1, 2, \dots, n$ provided by [`submanifold_component`](@ref).
 
 ```@autodocs

docs/src/manifolds/rotations.md

@@ -1,8 +1,8 @@
 # Rotations
 
-The manifold $\mathrm{SO}(n)$ of orthogonal matrices with determinant $+1$ in $\mathbb R^{n\times n}$, i.e.
+The manifold $\mathrm{SO}(n)$ of orthogonal matrices with determinant $+1$ in $ℝ^{n⨉ n}$, i.e.
 
-$\mathrm{SO}(n) = \bigl\{R \in \mathbb{R}^{n\times n} \big| RR^{\mathrm{T}} =
+$\mathrm{SO}(n) = \bigl\{R \in ℝ^{n⨉ n} \big| RR^{\mathrm{T}} =
 R^{\mathrm{T}}R = \mathrm{I}_n, \det(R) = 1 \bigr\}$
 
 The Lie group $\mathrm{SO}(n)$ is a subgroup of the orthogonal group $\mathrm{O}(n)$ and also known as the special orthogonal group or the set of rotations group.

docs/src/manifolds/symmetricpositivedefinite.md

@@ -3,7 +3,7 @@
 The symmetric positive definite matrices
 
 ```math
-\mathcal P(n) = \bigl\{ A \in \mathbb R^{n\times n}\ \big|\ A = A^{\mathrm{T}} \text{ and } x^{\mathrm{T}}Ax > 0 \text{ for } 0\neq x \in\mathbb R^n \bigr\}
+\mathcal P(n) = \bigl\{ A \in ℝ^{n⨉ n}\ \big|\ A = A^{\mathrm{T}} \text{ and } x^{\mathrm{T}}Ax > 0 \text{ for } 0\neq x \inℝ^n \bigr\}
 ```
 
 ```@docs

docs/src/manifolds/torus.md

@@ -1,11 +1,11 @@
 # Torus
-The torus $\mathbb T^d \equiv [-π,π)^d$ is modeled as an [`AbstractPowerManifold`](@ref) of
+The torus $𝕋^d \equiv [-π,π)^d$ is modeled as an [`AbstractPowerManifold`](@ref) of
 the (real-valued) [`Circle`](@ref) and uses [`MultidimentionalArrayPowerRepresentation`](@ref).
-Points on the torus are hence row vectors, $x\in\mathbb R^{d}$.
+Points on the torus are hence row vectors, $x\inℝ^{d}$.
 
 ## Example
 
-The following code can be used to make a three-dimensional torus $\mathbb{T}^3$ and compute a tangent vector.
+The following code can be used to make a three-dimensional torus $𝕋^3$ and compute a tangent vector.
 ```@example
 using Manifolds
 M = Torus(3)

docs/src/manifolds/vector_bundle.md

@@ -1,7 +1,7 @@
 # Vector bundles
 
-Vector bundle $E$ is a manifold that is built on top of another manifold $M$ (base space).
-It is characterized by a continuous function $\Pi \colon E \to M$, such that for each point $x \in M$ the preimage of $x$ by $\Pi$, $\Pi^{-1}(\{x\})$, has a structure of a vector space.
+Vector bundle $E$ is a manifold that is built on top of another manifold $ℳ$ (base space).
+It is characterized by a continuous function $\Pi \colon E \to ℳ$, such that for each point $x \in ℳ$ the preimage of $x$ by $\Pi$, $\Pi^{-1}(\{x\})$, has a structure of a vector space.
 These vector spaces are called fibers.
 Bundle projection can be performed using function [`bundle_projection`](@ref).
 

docs/src/notation.md

@@ -0,0 +1,24 @@
+# Notation overview
+
+Since manifolds include a reasonable amount of elements and functions, the
+following list tries to keep an overview of used notation throughout `Manifolds.jl`.
+The order by terminology. Thy might be used in a plain form within the code or
+when referring to that code. This is for example the case the caligraphic symbols.
+
+Within the documented functions the utf8 symbols are used whenever possible,
+as long as that renders still in TeX within this documentation.
+
+A
+
+| Symbol | Description | Also used | Comment |
+|:--:|:-------------- |:--:|:--- |
+| ``T^*_x ℳ`` | The cotangent space at ``x`` | | |
+| ``ξ`` | A cotangent vector from ``T^*_x ℳ`` | ``ξ_1,ξ_2,…,η,𝛇`` | |
+| ``F`` | A fiber | |
+| ``⟨·,·⟩`` | inner product (in ``T^*_x ℳ``) | ``⟨·,·⟩_x, g_x(·,·)`` |
+| ``ℳ`` | A manifold | ``ℳ_1, ℳ_2,…,𝒩`` | |
+| ``𝒫_{y\gets x}X`` | parallel Transport |
+| ``x`` | A point on ℳ | ``x_1,x_2,…,y,z`` | |
+| ``T_x ℳ`` | The tangent space at ``x`` | | |
+| ``X`` | A tangent vector from ``T_x ℳ`` | ``X_1,X_2,…,Y,Z`` | sometimes with the base point in the index, ``X_x``. |
+| ``B`` | The Vector bundle | |

docs/src/orthonormal_bases.md

@@ -1,8 +1,8 @@
 # Orthonormal bases
 
 The following functions and types provide support for orthonormal bases of the tangent space of different manifolds.
-An orthonormal basis of the tangent space $T_x M$ of (real) dimension $N$ has a real-coefficient basis $e_1, e_2, \dots, e_N$ if $\mathrm{Re}(g_x(e_i, e_j)) = \delta_{ij}$ for each $i,j \in \{1, 2, \dots, N\}$ where $g_x$ is the Riemannian metric at point $x$.
-A vector $v$ from the tangent space $T_x M$ can be expressed as a sum $v = v^i e_i$ where coefficients $v^i$ are calculated as $v^i = \mathrm{Re}(g_x(v, e_i))$.
+An orthonormal basis of the tangent space $T_x ℳ$ of (real) dimension $N$ has a real-coefficient basis $e_1, e_2, \dots, e_N$ if $\mathrm{Re}(g_x(e_i, e_j)) = \delta_{ij}$ for each $i,j \in \{1, 2, \dots, N\}$ where $g_x$ is the Riemannian metric at point $x$.
+A vector $v$ from the tangent space $T_x ℳ$ can be expressed as a sum $v = v^i e_i$ where coefficients $v^i$ are calculated as $v^i = \mathrm{Re}(g_x(v, e_i))$.
 
 The main types are:
 * [`ArbitraryOrthonormalBasis`](@ref), which is designed to work when no special properties of the tangent space basis are required.

src/groups/group.jl

@@ -237,7 +237,7 @@ end
     inv(G::AbstractGroupManifold, x)
 
 Inverse $x^{-1} ∈ G$ of an element $x ∈ G$, such that
-$x \circ x^{-1} = x^{-1} \circ x = e ∈ G$.
+$x ◦ x^{-1} = x^{-1} ◦ x = e ∈ G$.
 """
 inv(M::Manifold, x) = inv(M, x, is_decorator_manifold(M))
 inv(M::Manifold, x, ::Val{true}) = inv(M.manifold, x)
@@ -253,7 +253,7 @@ end
     inv!(G::AbstractGroupManifold, y, x)
 
 Inverse $x^{-1} ∈ G$ of an element $x ∈ G$, such that
-$x \circ x^{-1} = x^{-1} \circ x = e ∈ G$.
+$x ◦ x^{-1} = x^{-1} ◦ x = e ∈ G$.
 The result is saved to `y`.
 """
 inv!(M::Manifold, y, x) = inv!(M, y, x, is_decorator_manifold(M))
@@ -265,7 +265,7 @@ end
 @doc doc"""
     identity(G::AbstractGroupManifold, x)
 
-Identity element $e ∈ G$, such that for any element $x ∈ G$, $x \circ e = e \circ x = x$.
+Identity element $e ∈ G$, such that for any element $x ∈ G$, $x ◦ e = e ◦ x = x$.
 The returned element is of a similar type to `x`.
 """
 identity(M::Manifold, x) = identity(M, x, is_decorator_manifold(M))
@@ -315,7 +315,7 @@ isapprox(::GT, ::E, ::E; kwargs...) where {GT<:GroupManifold,E<:Identity{GT}} =
 @doc doc"""
     compose(G::AbstractGroupManifold, x, y)
 
-Compose elements $x,y ∈ G$ using the group operation $x \circ y$.
+Compose elements $x,y ∈ G$ using the group operation $x ◦ y$.
 """
 compose(M::Manifold, x, y) = compose(M, x, y, is_decorator_manifold(M))
 compose(M::Manifold, x, y, ::Val{true}) = compose(M.manifold, x, y)
@@ -343,8 +343,8 @@ For group elements $x,y ∈ G$, translate $y$ by $x$ with the specified conventi
 left $L_x$ or right $R_x$, defined as
 ```math
 \begin{aligned}
-L_x &: y ↦ x \circ y\\
-R_x &: y ↦ y \circ x.
+L_x &: y ↦ x ◦ y\\
+R_x &: y ↦ y ◦ x.
 \end{aligned}
 ```
 """
@@ -369,8 +369,8 @@ For group elements $x,y ∈ G$, translate $y$ by $x$ with the specified conventi
 left $L_x$ or right $R_x$, defined as
 ```math
 \begin{aligned}
-L_x &: y ↦ x \circ y\\
-R_x &: y ↦ y \circ x.
+L_x &: y ↦ x ◦ y\\
+R_x &: y ↦ y ◦ x.
 \end{aligned}
 ```
 Result of the operation is saved in `z`.
@@ -396,8 +396,8 @@ For group elements $x,y ∈ G$, inverse translate $y$ by $x$ with the specified
 either left $L_x^{-1}$ or right $R_x^{-1}$, defined as
 ```math
 \begin{aligned}
-L_x^{-1} &: y ↦ x^{-1} \circ y\\
-R_x^{-1} &: y ↦ y \circ x^{-1}.
+L_x^{-1} &: y ↦ x^{-1} ◦ y\\
+R_x^{-1} &: y ↦ y ◦ x^{-1}.
 \end{aligned}
 ```
 """
@@ -422,8 +422,8 @@ For group elements $x,y ∈ G$, inverse translate $y$ by $x$ with the specified
 either left $L_x^{-1}$ or right $R_x^{-1}$, defined as
 ```math
 \begin{aligned}
-L_x^{-1} &: y ↦ x^{-1} \circ y\\
-R_x^{-1} &: y ↦ y \circ x^{-1}.
+L_x^{-1} &: y ↦ x^{-1} ◦ y\\
+R_x^{-1} &: y ↦ y ◦ x^{-1}.
 \end{aligned}
 ```
 Result is saved in `z`.
@@ -450,8 +450,8 @@ differential of the translation by $x$ on $v$, written as $(\mathrm{d}τ_x)_y (v
 specified left or right convention. The differential transports vectors:
 ```math
 \begin{aligned}
-(\mathrm{d}L_x)_y (v) &: T_y G → T_{x \circ y} G\\
-(\mathrm{d}R_x)_y (v) &: T_y G → T_{y \circ x} G\\
+(\mathrm{d}L_x)_y (v) &: T_y G → T_{x ◦ y} G\\
+(\mathrm{d}R_x)_y (v) &: T_y G → T_{y ◦ x} G\\
 \end{aligned}
 ```
 """
@@ -494,8 +494,8 @@ $((\mathrm{d}τ_x)_y)^{-1} (v) = (\mathrm{d}τ_{x^{-1}})_y (v)$, with the specif
 right convention. The differential transports vectors:
 ```math
 \begin{aligned}
-((\mathrm{d}L_x)_y)^{-1} (v) &: T_y G → T_{x^{-1} \circ y} G\\
-((\mathrm{d}R_x)_y)^{-1} (v) &: T_y G → T_{y \circ x^{-1}} G\\
+((\mathrm{d}L_x)_y)^{-1} (v) &: T_y G → T_{x^{-1} ◦ y} G\\
+((\mathrm{d}R_x)_y)^{-1} (v) &: T_y G → T_{y ◦ x^{-1}} G\\
 \end{aligned}
 ```
 """

src/groups/group_action.jl

@@ -33,7 +33,7 @@ direction(::AbstractGroupAction{AD}) where {AD} = AD()
 
 Apply action `a` to the point `x`. The action is specified by `A`.
 Unless otherwise specified, right actions are defined in terms of the left action. For
-point $x ∈ M$ and action element $a$, the right action is
+point $x ∈ ℳ$ and action element $a$, the right action is
 
 ````math
 x ⋅ a ≐ a^{-1} ⋅ x.
@@ -82,15 +82,15 @@ end
 @doc doc"""
     apply_diff(A::AbstractGroupAction, a, x, v)
 
-For group point $x ∈ M$ and tangent vector $v ∈ T_x M$, compute the action of the
+For group point $x ∈ ℳ$ and tangent vector $v ∈ T_x ℳ$, compute the action of the
 differential of the action of $a ∈ G$ on $v$, specified by rule `A`. Written as
 $(\mathrm{d}τ_a)_x (v)$, with the specified left or right convention, the differential
 transports vectors
 
 ````math
 \begin{aligned}
-(\mathrm{d}L_a)_x (v) &: T_x M → T_{a ⋅ x} M\\
-(\mathrm{d}R_a)_x (v) &: T_x M → T_{x ⋅ a} M
+(\mathrm{d}L_a)_x (v) &: T_x ℳ → T_{a ⋅ x} ℳ\\
+(\mathrm{d}R_a)_x (v) &: T_x ℳ → T_{x ⋅ a} ℳ
 \end{aligned}
 ````
 """
@@ -105,15 +105,15 @@ end
 @doc doc"""
     inverse_apply_diff(A::AbstractGroupAction, a, x, v)
 
-For group point $x ∈ M$ and tangent vector $v ∈ T_x M$, compute the action of the
+For group point $x ∈ ℳ$ and tangent vector $v ∈ T_x ℳ$, compute the action of the
 differential of the inverse action of $a ∈ G$ on $v$, specified by rule `A`. Written as
 $(\mathrm{d}τ_a)_x^{-1} (v)$, with the specified left or right convention, the
 differential transports vectors
 
 ````math
 \begin{aligned}
-(\mathrm{d}L_a)_x^{-1} (v) &: T_x M → T_{a^{-1} ⋅ x} M\\
-(\mathrm{d}R_a)_x^{-1} (v) &: T_x M → T_{x ⋅ a^{-1}} M
+(\mathrm{d}L_a)_x^{-1} (v) &: T_x ℳ → T_{a^{-1} ⋅ x} ℳ\\
+(\mathrm{d}R_a)_x^{-1} (v) &: T_x ℳ → T_{x ⋅ a^{-1}} ℳ
 \end{aligned}
 ````
 """
@@ -139,7 +139,7 @@ the element closest to `x2` in the metric of the G-manifold:
 ```math
 \arg\min_{g ∈ G} d_M(g ⋅ x_1, x_2)
 ```
-where $G$ is the group that acts on the G-manifold $M$.
+where $G$ is the group that acts on the G-manifold $ℳ$.
 """
 function optimal_alignment(A::AbstractGroupAction, x1, x2)
     error("optimal_alignment not implemented for $(typeof(A)) and points $(typeof(x1)) and $(typeof(x2)).")

src/groups/semidirect_product_group.jl

@@ -24,7 +24,7 @@ $G = N ⋊_θ H$, where $θ: H × N \to N$ is an automorphism action
 of $H$ on $N$. The group $G$ has the composition rule
 
 ````math
-g \circ g' = (n, h) \circ (n', h') = (n \circ θ_h(n'), h \circ h')
+g ◦ g' = (n, h) ◦ (n', h') = (n ◦ θ_h(n'), h ◦ h')
 ````
 
 and the inverse

src/groups/special_euclidean.jl

@@ -21,7 +21,7 @@ SemidirectProductGroup(Tn, SOn, RotationAction(Tn, SOn))
 ```
 
 Points on $\mathrm{SE}(n)$ may be represented as points on the underlying product manifold
-$\mathrm{T}(n) \times \mathrm{SO}(n)$ or as affine matrices with size `(n + 1, n + 1)`.
+$\mathrm{T}(n) ⨉ \mathrm{SO}(n)$ or as affine matrices with size `(n + 1, n + 1)`.
 """
 const SpecialEuclidean{N} = SemidirectProductGroup{
     TranslationGroup{Tuple{N},ℝ},

src/manifolds/CholeskySpace.jl

@@ -9,7 +9,7 @@ are for example summarized in Table 1 of [^Lin2019].
 
     CholeskySpace(n)
 
-Generate the manifold of $n\times n$ lower triangular matrices with positive diagonal.
+Generate the manifold of $n⨉ n$ lower triangular matrices with positive diagonal.
 
 [^Lin2019]:
     > Lin, Zenhua: "Riemannian Geometry of Symmetric Positive Definite Matrices via
@@ -83,7 +83,7 @@ matrices `x`, `y` that are lower triangular with positive diagonal. The formula
 reads
 
 ````math
-d_{\mathcal M}(x,y) = \sqrt{\sum_{i>j} (x_{ij}-y_{ij})^2 +
+d_{ℳ}(x,y) = \sqrt{\sum_{i>j} (x_{ij}-y_{ij})^2 +
 \sum_{j=1}^m (\log x_{jj} - \log y_{jj})^2
 }
 ````
@@ -191,11 +191,11 @@ Parallely transport the tangent vector `v` at `x` along the geodesic to `y`
 on to the [`CholeskySpace`](@ref) manifold `M`. The formula reads
 
 ````math
-\mathcal P_{y\gets x}(v) = \lfloor v \rfloor
+\mathcal 𝒫_{y\gets x}(v) = \lfloor v \rfloor
 + \operatorname{diag}(y)\operatorname{diag}(x)^{-1}\operatorname{diag}(v),
 ````
 
-where $\lfloor\cdot\rfloor$ denotes the strictly lower triangular matrix,
+where $\lfloor·\rfloor$ denotes the strictly lower triangular matrix,
 and $\operatorname{diag}$ extracts the diagonal matrix.
 """
 vector_transport_to(::CholeskySpace, ::Any, ::Any, ::Any, ::ParallelTransport)

src/manifolds/Circle.jl

@@ -1,8 +1,8 @@
 @doc doc"""
     Circle{F} <: Manifold
 
-The circle $\mathbb S^1$ as a manifold ere manifold represented by
-real-valued data in $[-\pi,\pi)$ or complex-valued data $z\in \mathbb C$ of absolute value
+The circle $𝕊^1$ as a manifold ere manifold represented by
+real-valued data in $[-\pi,\pi)$ or complex-valued data $z\in ℂ$ of absolute value
 $\lvert z\rvert = 1$.
 # Constructor
 
@@ -21,7 +21,7 @@ Circle(f::AbstractNumbers = ℝ) = Circle{f}()
 
 Check whether `x` is a point on the [`Circle`](@ref) `M`.
 For the real-valued case, `x` is an angle and hence it checks that $x \in [-\pi,\pi)$.
-for the complex-valued case its a unit number, $x \in \mathbb C$ with $\lvert x \rvert = 1$.
+for the complex-valued case its a unit number, $x \in ℂ$ with $\lvert x \rvert = 1$.
 """
 check_manifold_point(::Circle, ::Any...)
 
@@ -100,7 +100,7 @@ Compute the exponential map on the [`Circle`](@ref).
 ````math
 \exp_xv = (x+v)_{2\pi},
 ````
-where $(\cdot)$ is the (symmetric) remainder with respect to division by $2\pi$,
+where $(·)$ is the (symmetric) remainder with respect to division by $2\pi$,
 i.e. in $[-\pi,\pi)$.
 
 For the complex-valued case the formula is the same as for the [`Sphere`](@ref)
@@ -203,7 +203,7 @@ Compute the logarithmic map on the [`Circle`](@ref) `M`.
 ````math
 \exp_xv = (y,x)_{2\pi},
 ````
-where $(\cdot)$ is the (symmetric) remainder with respect to division by $2\pi$,
+where $(·)$ is the (symmetric) remainder with respect to division by $2\pi$,
 i.e. in $[-\pi,\pi)$.
 
 For the complex-valued case the formula is the same as for the [`Sphere`](@ref)
@@ -244,15 +244,15 @@ end
     manifold_dimension(M::Circle)
 
 Return the dimension of the [`Circle`](@ref) `M`,
-i.e. $\operatorname{dim}(\mathbb S^1) = 1$.
+i.e. $\operatorname{dim}(𝕊^1) = 1$.
 """
 manifold_dimension(::Circle) = 1
 
 @doc doc"""
     mean(M::Circle, x::AbstractVector[, w::AbstractWeights])
 
 Compute the Riemannian [`mean`](@ref mean(M::Manifold, args...)) of `x` on the
-[`Circle`](@ref) $\mathbb S^1$ by the wrapped mean, i.e. the remainder of the
+[`Circle`](@ref) $𝕊^1$ by the wrapped mean, i.e. the remainder of the
 mean modulo 2π.
 """
 mean(::Circle, ::Any)
@@ -320,7 +320,7 @@ For the real-valued case this results in the identity.
 For the complex-valud case, the formula is the same as for the [`Sphere`](@ref)`(1)` in the
 complex plane.
 ````math
-P_{y\gets x}(v) = v - \frac{\langle \log_xy,v\rangle_x}{d^2_{\mathbb C}(x,y)}
+𝒫_{y\gets x}(v) = v - \frac{⟨\log_xy,v⟩_x}{d^2_{ℂ}(x,y)}
 \bigl(\log_xy + \log_yx \bigr),
 ````
 where [`log`](@ref) denotes the logarithmic map on `M`.