Fix several typos in the docs

NEWS.md

@@ -5,6 +5,13 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning](https://semver.org/spec/v2.0.0.html).
 
+## [0.9.15] unreleased
+
+## Fixed
+
+* several typographical errors in the docs
+* unifies to use two backticks ``` `` ``` for math instead of ` $ ` further in the docs
+
 ## [0.9.14] – 2024-01-31
 
 ### Added

docs/make.jl

@@ -70,6 +70,8 @@ makedocs(;
     format=Documenter.HTML(
         prettyurls=false,
         assets=["assets/favicon.ico", "assets/citations.css"],
+        size_threshold_warn=200 * 2^10, # raise slightly from 100 to 200 KiB
+        size_threshold=300 * 2^10,      # raise slightly 200 to to 300 KiB
     ),
     modules=[
         Manifolds,

docs/src/manifolds/rotations.md

@@ -2,8 +2,10 @@
 
 The manifold ``\mathrm{SO}(n)`` of orthogonal matrices with determinant ``+1`` in ``ℝ^{n×n}``, i.e.
 
-``\mathrm{SO}(n) = \bigl\{R ∈ ℝ^{n×n} \big| R R^{\mathrm{T}} =
-R^{\mathrm{T}}R = I_n, \det(R) = 1 \bigr\}$
+```math
+\mathrm{SO}(n) = \bigl\{R ∈ ℝ^{n×n} \big| R R^{\mathrm{T}} =
+R^{\mathrm{T}}R = 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.
 See also [`SpecialOrthogonal`](@ref), which is this manifold equipped with the group operation.
@@ -24,7 +26,7 @@ This convention allows for more efficient operations on tangent vectors.
 Tangent spaces at different points are different vector spaces.
 
 Let ``L_R: \mathrm{SO}(n) → \mathrm{SO}(n)`` where ``R ∈ \mathrm{SO}(n)`` be the left-multiplication by ``R``, that is ``L_R(S) = RS``.
-The tangent space at rotation ``R``, ``T_R \mathrm{SO}(n)``, is related to the tangent space at the identity rotation ``I_n`` by the differential of ``L_R`` at identity, $(\mathrm{d}L_R)_{I_n} : T_{I_n} \mathrm{SO}(n) → T_R \mathrm{SO}(n)``.
+The tangent space at rotation ``R``, ``T_R \mathrm{SO}(n)``, is related to the tangent space at the identity rotation ``I_n`` by the differential of ``L_R`` at identity, ``(\mathrm{d}L_R)_{I_n} : T_{I_n} \mathrm{SO}(n) → T_R \mathrm{SO}(n)``.
 To convert the tangent vector representation at the identity rotation ``X ∈ T_{I_n} \mathrm{SO}(n)`` (i.e., the default) to the matrix representation of the corresponding tangent vector ``Y`` at a rotation ``R`` use the [`embed`](@ref embed(::Manifolds.Rotations, :Any...)) which implements the following multiplication: ``Y = RX ∈ T_R \mathrm{SO}(n)``.
 You can compare the functions [`log`](@ref log(::Manifolds.Rotations, :Any...)) and [`exp`](@ref exp(::Manifolds.Rotations, ::Any...)) to see how it works in practice.
 

docs/src/manifolds/symplecticstiefel.md

@@ -2,23 +2,26 @@
 
 The [`SymplecticStiefel`](@ref) manifold, denoted ``\operatorname{SpSt}(2n, 2k)``,
 represents canonical symplectic bases of ``2k`` dimensonal symplectic subspaces of ``ℝ^{2n×2n}``.
-This means that the columns of each element ``p \in \operatorname{SpSt}(2n, 2k) \subset ℝ^{2n×2k}$
+This means that the columns of each element ``p \in \operatorname{SpSt}(2n, 2k) \subset ℝ^{2n×2k}``
 constitute a canonical symplectic basis of ``\operatorname{span}(p)``.
 The canonical symplectic form is a non-degenerate, bilinear, and skew symmetric map
-``\omega_{2k}\colon 𝔽^{2k}×𝔽^{2k}
-→ 𝔽``, given by
+``\omega_{2k}\colon 𝔽^{2k}×𝔽^{2k} → 𝔽``, given by
 ``\omega_{2k}(x, y) = x^T Q_{2k} y`` for elements ``x, y \in 𝔽^{2k}``, with
+
 ````math
     Q_{2k} =
     \begin{bmatrix}
      0_k  &  I_k \\
     -I_k  &  0_k
     \end{bmatrix}.
 ````
+
 Specifically given an element ``p \in \operatorname{SpSt}(2n, 2k)`` we require that
+
 ````math
     \omega_{2n} (p x, p y) = x^T(p^TQ_{2n}p)y = x^TQ_{2k}y = \omega_{2k}(x, y) \;\forall\; x, y \in 𝔽^{2k},
 ````
+
 leading to the requirement on ``p`` that ``p^TQ_{2n}p = Q_{2k}``.
 In the case that ``k = n``, this manifold reduces to the [`SymplecticMatrices`](@ref) manifold, which is also known as the symplectic group.
 

docs/src/manifolds/tucker.md

@@ -7,3 +7,8 @@ Order = [:type, :function]
 ```
 
 ## Literature
+
+```@bibliography
+Pages = ["tucker.md"]
+Canonical=false
+```

src/groups/special_euclidean.jl

@@ -1,16 +1,16 @@
 @doc raw"""
     SpecialEuclidean(n)
 
-Special Euclidean group $\mathrm{SE}(n)$, the group of rigid motions.
+Special Euclidean group ``\mathrm{SE}(n)``, the group of rigid motions.
 
-``\mathrm{SE}(n)`` is the semidirect product of the [`TranslationGroup`](@ref) on $ℝ^n$ and
+``\mathrm{SE}(n)`` is the semidirect product of the [`TranslationGroup`](@ref) on ``ℝ^n`` and
 [`SpecialOrthogonal`](@ref)`(n)`
 
 ````math
 \mathrm{SE}(n) ≐ \mathrm{T}(n) ⋊_θ \mathrm{SO}(n),
 ````
 
-where ``θ`` is the canonical action of ``\mathrm{SO}(n)`` on $\mathrm{T}(n)$ by vector rotation.
+where ``θ`` is the canonical action of ``\mathrm{SO}(n)`` on ``\mathrm{T}(n)`` by vector rotation.
 
 This constructor is equivalent to calling
 
@@ -20,8 +20,8 @@ SOn = SpecialOrthogonal(n)
 SemidirectProductGroup(Tn, SOn, RotationAction(Tn, SOn))
 ```
 
-Points on $\mathrm{SE}(n)$ may be represented as points on the underlying product manifold
-$\mathrm{T}(n) × \mathrm{SO}(n)$. For group-specific functions, they may also be
+Points on ``\mathrm{SE}(n)`` may be represented as points on the underlying product manifold
+``\mathrm{T}(n) × \mathrm{SO}(n)``. For group-specific functions, they may also be
 represented as affine matrices with size `(n + 1, n + 1)` (see [`affine_matrix`](@ref)), for
 which the group operation is [`MultiplicationOperation`](@ref).
 """
@@ -167,9 +167,9 @@ end
 @doc raw"""
     affine_matrix(G::SpecialEuclidean, p) -> AbstractMatrix
 
-Represent the point $p ∈ \mathrm{SE}(n)$ as an affine matrix.
-For $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n), R ∈ \mathrm{SO}(n)$, the
-affine representation is the $n + 1 × n + 1$ matrix
+Represent the point ``p ∈ \mathrm{SE}(n)`` as an affine matrix.
+For ``p = (t, R) ∈ \mathrm{SE}(n)``, where ``t ∈ \mathrm{T}(n), R ∈ \mathrm{SO}(n)``, the
+affine representation is the ``n + 1 × n + 1`` matrix
 
 ````math
 \begin{pmatrix}
@@ -178,7 +178,7 @@ R & t \\
 \end{pmatrix}.
 ````
 
-This function embeds $\mathrm{SE}(n)$ in the general linear group $\mathrm{GL}(n+1)$.
+This function embeds ``\mathrm{SE}(n)`` in the general linear group ``\mathrm{GL}(n+1)``.
 It is an isometric embedding and group homomorphism [RicoMartinez:1988](@cite).
 
 See also [`screw_matrix`](@ref) for matrix representations of the Lie algebra.
@@ -278,9 +278,9 @@ end
 @doc raw"""
     screw_matrix(G::SpecialEuclidean, X) -> AbstractMatrix
 
-Represent the Lie algebra element $X ∈ 𝔰𝔢(n) = T_e \mathrm{SE}(n)$ as a screw matrix.
-For $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $Ω ∈ 𝔰𝔬(n) = T_e \mathrm{SO}(n)$, the screw representation is
-the $n + 1 × n + 1$ matrix
+Represent the Lie algebra element ``X ∈ 𝔰𝔢(n) = T_e \mathrm{SE}(n)`` as a screw matrix.
+For ``X = (b, Ω) ∈ 𝔰𝔢(n)``, where ``Ω ∈ 𝔰𝔬(n) = T_e \mathrm{SO}(n)``, the screw representation is
+the ``n + 1 × n + 1`` matrix
 
 ````math
 \begin{pmatrix}
@@ -289,7 +289,7 @@ the $n + 1 × n + 1$ matrix
 \end{pmatrix}.
 ````
 
-This function embeds $𝔰𝔢(n)$ in the general linear Lie algebra $𝔤𝔩(n+1)$ but it's not
+This function embeds ``𝔰𝔢(n)`` in the general linear Lie algebra ``𝔤𝔩(n+1)`` but it's not
 a homomorphic embedding (see [`SpecialEuclideanInGeneralLinear`](@ref) for a homomorphic one).
 
 See also [`affine_matrix`](@ref) for matrix representations of the Lie group.
@@ -351,13 +351,13 @@ end
 @doc raw"""
     exp_lie(G::SpecialEuclidean{n}, X)
 
-Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(n)$, where $b ∈ 𝔱(n)$ and $Ω ∈ 𝔰𝔬(n)$:
+Compute the group exponential of ``X = (b, Ω) ∈ 𝔰𝔢(n)``, where ``b ∈ 𝔱(n)`` and ``Ω ∈ 𝔰𝔬(n)``:
 
 ````math
 \exp X = (t, R),
 ````
 
-where $t ∈ \mathrm{T}(n)$ and $R = \exp Ω$ is the group exponential on $\mathrm{SO}(n)$.
+where ``t ∈ \mathrm{T}(n)`` and ``R = \exp Ω`` is the group exponential on ``\mathrm{SO}(n)``.
 
 In the [`screw_matrix`](@ref) representation, the group exponential is the matrix
 exponential (see [`exp_lie`](@ref)).
@@ -367,39 +367,39 @@ exp_lie(::SpecialEuclidean, ::Any)
 @doc raw"""
     exp_lie(G::SpecialEuclidean{TypeParameter{Tuple{2}}}, X)
 
-Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(2)$, where $b ∈ 𝔱(2)$ and $Ω ∈ 𝔰𝔬(2)$:
+Compute the group exponential of ``X = (b, Ω) ∈ 𝔰𝔢(2)``, where ``b ∈ 𝔱(2)`` and ``Ω ∈ 𝔰𝔬(2)``:
 
 ````math
 \exp X = (t, R) = (U(θ) b, \exp Ω),
 ````
 
-where $t ∈ \mathrm{T}(2)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(2)$,
+where ``t ∈ \mathrm{T}(2)``, ``R = \exp Ω`` is the group exponential on ``\mathrm{SO}(2)``,
 
 ````math
 U(θ) = \frac{\sin θ}{θ} I_2 + \frac{1 - \cos θ}{θ^2} Ω,
 ````
 
-and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$
+and ``θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e``
 (see [`norm`](@ref norm(M::Rotations, p, X))) is the angle of the rotation.
 """
 exp_lie(::SpecialEuclidean{TypeParameter{Tuple{2}}}, ::Any)
 
 @doc raw"""
     exp_lie(G::SpecialEuclidean{TypeParameter{Tuple{3}}}, X)
 
-Compute the group exponential of $X = (b, Ω) ∈ 𝔰𝔢(3)$, where $b ∈ 𝔱(3)$ and $Ω ∈ 𝔰𝔬(3)$:
+Compute the group exponential of ``X = (b, Ω) ∈ 𝔰𝔢(3)``, where ``b ∈ 𝔱(3)`` and ``Ω ∈ 𝔰𝔬(3)``:
 
 ````math
 \exp X = (t, R) = (U(θ) b, \exp Ω),
 ````
 
-where $t ∈ \mathrm{T}(3)$, $R = \exp Ω$ is the group exponential on $\mathrm{SO}(3)$,
+where ``t ∈ \mathrm{T}(3)``, ``R = \exp Ω`` is the group exponential on ``\mathrm{SO}(3)``,
 
 ````math
 U(θ) = I_3 + \frac{1 - \cos θ}{θ^2} Ω + \frac{θ - \sin θ}{θ^3} Ω^2,
 ````
 
-and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$
+and ``θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e``
 (see [`norm`](@ref norm(M::Rotations, p, X))) is the angle of the rotation.
 """
 exp_lie(::SpecialEuclidean{TypeParameter{Tuple{3}}}, ::Any)
@@ -471,14 +471,14 @@ end
 @doc raw"""
     log_lie(G::SpecialEuclidean, p)
 
-Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(n)$, where $t ∈ \mathrm{T}(n)$
-and $R ∈ \mathrm{SO}(n)$:
+Compute the group logarithm of ``p = (t, R) ∈ \mathrm{SE}(n)``, where ``t ∈ \mathrm{T}(n)``
+and ``R ∈ \mathrm{SO}(n)``:
 
 ````math
 \log p = (b, Ω),
 ````
 
-where $b ∈ 𝔱(n)$ and $Ω = \log R ∈ 𝔰𝔬(n)$ is the group logarithm on $\mathrm{SO}(n)$.
+where ``b ∈ 𝔱(n)`` and ``Ω = \log R ∈ 𝔰𝔬(n)`` is the group logarithm on ``\mathrm{SO}(n)``.
 
 In the [`affine_matrix`](@ref) representation, the group logarithm is the matrix logarithm
 (see [`log_lie`](@ref)):
@@ -488,41 +488,41 @@ log_lie(::SpecialEuclidean, ::Any)
 @doc raw"""
     log_lie(G::SpecialEuclidean{TypeParameter{Tuple{2}}}, p)
 
-Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(2)$, where $t ∈ \mathrm{T}(2)$
-and $R ∈ \mathrm{SO}(2)$:
+Compute the group logarithm of ``p = (t, R) ∈ \mathrm{SE}(2)``, where ``t ∈ \mathrm{T}(2)``
+and ``R ∈ \mathrm{SO}(2)``:
 
 ````math
 \log p = (b, Ω) = (U(θ)^{-1} t, \log R),
 ````
 
-where $b ∈ 𝔱(2)$, $Ω = \log R ∈ 𝔰𝔬(2)$ is the group logarithm on $\mathrm{SO}(2)$,
+where ``b ∈ 𝔱(2)``, ``Ω = \log R ∈ 𝔰𝔬(2)`` is the group logarithm on ``\mathrm{SO}(2)``,
 
 ````math
 U(θ) = \frac{\sin θ}{θ} I_2 + \frac{1 - \cos θ}{θ^2} Ω,
 ````
 
-and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$
+and ``θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e``
 (see [`norm`](@ref norm(M::Rotations, p, X))) is the angle of the rotation.
 """
 log_lie(::SpecialEuclidean{TypeParameter{Tuple{2}}}, ::Any)
 
 @doc raw"""
     log_lie(G::SpecialEuclidean{TypeParameter{Tuple{3}}}, p)
 
-Compute the group logarithm of $p = (t, R) ∈ \mathrm{SE}(3)$, where $t ∈ \mathrm{T}(3)$
-and $R ∈ \mathrm{SO}(3)$:
+Compute the group logarithm of ``p = (t, R) ∈ \mathrm{SE}(3)``, where ``t ∈ \mathrm{T}(3)``
+and ``R ∈ \mathrm{SO}(3)``:
 
 ````math
 \log p = (b, Ω) = (U(θ)^{-1} t, \log R),
 ````
 
-where $b ∈ 𝔱(3)$, $Ω = \log R ∈ 𝔰𝔬(3)$ is the group logarithm on $\mathrm{SO}(3)$,
+where ``b ∈ 𝔱(3)``, ``Ω = \log R ∈ 𝔰𝔬(3)`` is the group logarithm on ``\mathrm{SO}(3)``,
 
 ````math
 U(θ) = I_3 + \frac{1 - \cos θ}{θ^2} Ω + \frac{θ - \sin θ}{θ^3} Ω^2,
 ````
 
-and $θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e$
+and ``θ = \frac{1}{\sqrt{2}} \lVert Ω \rVert_e``
 (see [`norm`](@ref norm(M::Rotations, p, X))) is the angle of the rotation.
 """
 log_lie(::SpecialEuclidean{TypeParameter{Tuple{3}}}, ::Any)
@@ -647,8 +647,8 @@ end
 @doc raw"""
     SpecialEuclideanInGeneralLinear
 
-An explicit isometric and homomorphic embedding of $\mathrm{SE}(n)$ in $\mathrm{GL}(n+1)$
-and $𝔰𝔢(n)$ in $𝔤𝔩(n+1)$.
+An explicit isometric and homomorphic embedding of ``\mathrm{SE}(n)`` in ``\mathrm{GL}(n+1)``
+and ``𝔰𝔢(n)`` in ``𝔤𝔩(n+1)``.
 Note that this is *not* a transparently isometric embedding.
 
 # Constructor