docs improvements; split group actions to a separate page

docs/make.jl

@@ -155,6 +155,7 @@ makedocs(;
             "Atlases and charts" => "features/atlases.md",
             "Differentiation" => "features/differentiation.md",
             "Distributions" => "features/distributions.md",
+            "Group actions" => "features/group_actions.md",
             "Integration" => "features/integration.md",
             "Statistics" => "features/statistics.md",
             "Testing" => "features/testing.md",

docs/src/features/group_actions.md

@@ -0,0 +1,63 @@
+# Group actions
+
+Group actions represent actions of a given group on a specified manifold.
+The following operations are available:
+
+* [`action_side`](@ref): whether action acts from the [`LeftSide`](@ref) or [`RightSide`](@ref) (not to be confused with action direction).
+* [`apply`](@ref): performs given action of an element of the group on an object of compatible type.
+* [`apply_diff`](@ref): differential of [`apply`](@ref) with respect to the object it acts upon.
+* [`direction`](@ref): tells whether a given action is [`LeftAction`](@ref), [`RightAction`](@ref).
+* [`inverse_apply`](@ref): performs given action of the inverse of an element of the group on an object of compatible type. By default inverts the element and calls [`apply`](@ref) but it may be have a faster implementation for some actions.
+* [`inverse_apply_diff`](@ref): counterpart of [`apply_diff`](@ref) for [`inverse_apply`](@ref).
+* [`optimal_alignment`](@ref): determine the element of a group that, when it acts upon a point, produces the element closest to another given point in the metric of the G-manifold.
+
+Furthermore, group operation action features the following:
+
+* [`translate`](@ref Main.Manifolds.translate): an operation that performs either ([`LeftAction`](@ref)) on the [`LeftSide`](@ref) or ([`RightAction`](@ref)) on the [`RightSide`](@ref) translation, or actions by inverses of elements ([`RightAction`](@ref) on the [`LeftSide`](@ref) and [`LeftAction`](@ref) on the [`RightSide`](@ref)). This is by default performed by calling [`compose`](@ref) with appropriate order of arguments. This function is separated from `compose` mostly to easily represent its differential, [`translate_diff`](@ref).
+* [`translate_diff`](@ref): differential of [`translate`](@ref Main.Manifolds.translate) with respect to the point being translated.
+* [`adjoint_action`](@ref): adjoint action of a given element of a Lie group on an element of its Lie algebra.
+* [`lie_bracket`](@ref): Lie bracket of two vectors from a Lie algebra corresponding to a given group.
+
+The following group actions are available:
+
+* Group operation action [`GroupOperationAction`](@ref) that describes action of a group on itself.
+* [`RotationAction`](@ref), that is action of [`SpecialOrthogonal`](@ref) group on different manifolds.
+* [`TranslationAction`](@ref), which is the action of [`TranslationGroup`](@ref) group on different manifolds.
+
+```@autodocs
+Modules = [Manifolds]
+Pages = ["groups/group_action.jl"]
+Order = [:type, :function]
+```
+
+## Group operation action
+
+```@autodocs
+Modules = [Manifolds]
+Pages = ["groups/group_operation_action.jl"]
+Order = [:type, :function]
+```
+
+## Rotation action
+
+```@autodocs
+Modules = [Manifolds]
+Pages = ["groups/rotation_action.jl"]
+Order = [:type, :function]
+```
+
+## Translation action
+
+```@autodocs
+Modules = [Manifolds]
+Pages = ["groups/translation_action.jl"]
+Order = [:type, :function]
+```
+
+## Rotation-translation action (special Euclidean)
+
+```@autodocs
+Modules = [Manifolds]
+Pages = ["groups/rotation_translation_action.jl"]
+Order = [:type, :const, :function]
+```

docs/src/manifolds/group.md

@@ -1,4 +1,4 @@
-# [Group manifolds and actions](@id GroupManifoldSection)
+# [Group manifolds](@id GroupManifoldSection)
 
 Lie groups, groups that are Riemannian manifolds with a smooth binary group operation [`AbstractGroupOperation`](@ref), are implemented as [`AbstractDecoratorManifold`](https://juliamanifolds.github.io/ManifoldsBase.jl/stable/decorator.html#ManifoldsBase.AbstractDecoratorManifold) and specifying the group operation using the [`IsGroupManifold`](@ref) or by decorating an existing manifold with a group operation using [`GroupManifold`](@ref).
 
@@ -180,70 +180,6 @@ Pages = ["groups/translation_group.jl"]
 Order = [:constant, :type, :function]
 ```
 
-## Group actions
-
-Group actions represent actions of a given group on a specified manifold.
-The following operations are available:
-
-* [`action_side`](@ref): whether action acts from the [`LeftSide`](@ref) or [`RightSide`](@ref) (not to be confused with action direction).
-* [`apply`](@ref): performs given action of an element of the group on an object of compatible type.
-* [`apply_diff`](@ref): differential of [`apply`](@ref) with respect to the object it acts upon.
-* [`direction`](@ref): tells whether a given action is [`LeftAction`](@ref), [`RightAction`](@ref).
-* [`inverse_apply`](@ref): performs given action of the inverse of an element of the group on an object of compatible type. By default inverts the element and calls [`apply`](@ref) but it may be have a faster implementation for some actions.
-* [`inverse_apply_diff`](@ref): counterpart of [`apply_diff`](@ref) for [`inverse_apply`](@ref).
-* [`optimal_alignment`](@ref): determine the element of a group that, when it acts upon a point, produces the element closest to another given point in the metric of the G-manifold.
-
-Furthermore, group operation action features the following:
-
-* [`translate`](@ref Main.Manifolds.translate): an operation that performs either ([`LeftAction`](@ref)) on the [`LeftSide`](@ref) or ([`RightAction`](@ref)) on the [`RightSide`](@ref) translation, or actions by inverses of elements ([`RightAction`](@ref) on the [`LeftSide`](@ref) and [`LeftAction`](@ref) on the [`RightSide`](@ref)). This is by default performed by calling [`compose`](@ref) with appropriate order of arguments. This function is separated from `compose` mostly to easily represent its differential, [`translate_diff`](@ref).
-* [`translate_diff`](@ref): differential of [`translate`](@ref Main.Manifolds.translate) with respect to the point being translated.
-* [`adjoint_action`](@ref): adjoint action of a given element of a Lie group on an element of its Lie algebra.
-* [`lie_bracket`](@ref): Lie bracket of two vectors from a Lie algebra corresponding to a given group.
-
-The following group actions are available:
-
-* Group operation action [`GroupOperationAction`](@ref) that describes action of a group on itself.
-* [`RotationAction`](@ref), that is action of [`SpecialOrthogonal`](@ref) group on different manifolds.
-* [`TranslationAction`](@ref), which is the action of [`TranslationGroup`](@ref) group on different manifolds.
-
-```@autodocs
-Modules = [Manifolds]
-Pages = ["groups/group_action.jl"]
-Order = [:type, :function]
-```
-
-### Group operation action
-
-```@autodocs
-Modules = [Manifolds]
-Pages = ["groups/group_operation_action.jl"]
-Order = [:type, :function]
-```
-
-### Rotation action
-
-```@autodocs
-Modules = [Manifolds]
-Pages = ["groups/rotation_action.jl"]
-Order = [:type, :function]
-```
-
-### Translation action
-
-```@autodocs
-Modules = [Manifolds]
-Pages = ["groups/translation_action.jl"]
-Order = [:type, :function]
-```
-
-### Rotation-translation action (special Euclidean)
-
-```@autodocs
-Modules = [Manifolds]
-Pages = ["groups/rotation_translation_action.jl"]
-Order = [:type, :function]
-```
-
 ## Metrics on groups
 
 Lie groups by default typically forward all metric-related operations like exponential or logarithmic map to the underlying manifold, for example [`SpecialOrthogonal`](@ref) uses methods for [`Rotations`](@ref) (which is, incidentally, bi-invariant), or [`SpecialEuclidean`](@ref) uses product metric of the translation and rotation parts (which is not invariant under group operation).

src/Manifolds.jl

@@ -932,6 +932,7 @@ export AbstractGroupAction,
     RightSide,
     RotationAction,
     RotationTranslationAction,
+    RotationTranslationActionOnVector,
     SemidirectProductGroup,
     SpecialEuclidean,
     SpecialLinear,

src/groups/group_operation_action.jl

@@ -6,7 +6,7 @@ An element `p` of the group can act upon another another element by either:
 * left action from the left side: ``L_p: q ↦ p \circ q``,
 * right action from the left side: ``L'_p: q ↦ p^{-1} \circ q``,
 * right action from the right side: ``R_p: q ↦ q \circ p``,
-* left action from the right side: ``R'_p &: q ↦ q \circ p^{-1}``.
+* left action from the right side: ``R'_p: q ↦ q \circ p^{-1}``.
 
 # Constructor
 

src/groups/rotation_action.jl

@@ -274,7 +274,7 @@ of computation are described in Section 2.2.1 of [SrivastavaKlassen:2016](@cite)
 The formula reads
 ```math
 O^{*} = \begin{cases}
-UV^T & \text{if } \operatorname{det}(p q^{\mathrm{T}})\\
+UV^T & \text{if } \operatorname{det}(p q^{\mathrm{T}}) \geq 0\\
 U K V^{\mathrm{T}} & \text{otherwise}
 \end{cases}
 ```

src/groups/rotation_translation_action.jl

@@ -32,6 +32,13 @@ function Base.show(io::IO, A::RotationTranslationAction)
     return print(io, "RotationTranslationAction($(A.manifold), $(A.SEn), $(direction(A)))")
 end
 
+"""
+    RotationTranslationActionOnVector{TAD,𝔽,TE,TSE}
+
+Alias for [`RotationTranslationAction`](@ref) where the manifold `M` is [`Euclidean`](@ref)
+or [`TranslationGroup`](@ref) with size of type `TE`, and [`SpecialEuclidean`](@ref)
+group has size type `TSE`.
+"""
 const RotationTranslationActionOnVector{TAD,𝔽,TE,TSE} = RotationTranslationAction{
     TAD,
     <:Union{Euclidean{TE,𝔽},TranslationGroup{TE,𝔽}},