# Ease of notation

The following terms introduce a nicer notation for some operations, for example using the ∈ operator, $p ∈ \mathcal M$, to determine whether $p$ is a point on the Manifold $\mathcal M$.

Base.inFunction
Base.in(p, M::Manifold; kwargs...)
p ∈ M

Check, whether a point p is a valid point (i.e. in) a Manifold M. This method employs is_manifold_point deaticating the error throwing option.

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Base.in(p, TpM::TangentSpaceAtPoint; kwargs...)
X ∈ TangentSpaceAtPoint(M,p)

Check whether X is a tangent vector from (in) the tangent space $T_p\mathcal M$, i.e. the TangentSpaceAtPoint at p on the Manifold M. This method uses is_tangent_vector deactivating the error throw option.

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# Public documentation

The following functions are of interest for extending and using the ProductManifold.

Manifolds.ShapeSpecificationType
ShapeSpecification(reshapers, manifolds::Manifold...)

A structure for specifying array size and offset information for linear storage of points and tangent vectors on the product manifold of manifolds.

The first argument, reshapers, indicates how a view representing a point in the ProductArray will be reshaped. It can either be an object of type AbstractReshaper that will be applied to all views or a tuple of such objects that will be applied to subsequent manifolds.

Two main reshaping methods are provided by types StaticReshaper that is faster for manifolds represented by small arrays (up to about 100 elements) and ArrayReshaper that is faster for larger arrays.

For example, consider the shape specification for the product of a sphere and group of rotations:

julia> M1 = Sphere(2)
Sphere{2}()

julia> M2 = Manifolds.Rotations(2)
Manifolds.Rotations{2}()

julia> reshaper = Manifolds.StaticReshaper()
Manifolds.StaticReshaper()

julia> shape = Manifolds.ShapeSpecification(reshaper, M1, M2)
Manifolds.ShapeSpecification{(1:3, 4:7),Tuple{Tuple{3},Tuple{2,2}},
Tuple{Manifolds.StaticReshaper,Manifolds.StaticReshaper}}(
(Manifolds.StaticReshaper(), Manifolds.StaticReshaper()))

TRanges contains ranges in the linear storage that correspond to a specific manifold. Sphere(2) needs three numbers and is first, so it is allocated the first three elements of the linear storage (1:3). Rotations(2) needs four numbers and is second, so the next four numbers are allocated to it (4:7). TSizes describe how the linear storage must be reshaped to correctly represent points. In this case, Sphere(2) expects a three-element vector, so the corresponding size is Tuple{3}. On the other hand, Rotations(2) expects two-by-two matrices, so its size specification is Tuple{2,2}.

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Manifolds.submanifold_componentFunction
submanifold_component(M::Manifold, p, i::Integer)
submanifold_component(M::Manifold, p, ::Val(i)) where {i}
submanifold_component(p, i::Integer)
submanifold_component(p, ::Val(i)) where {i}

Project the product array p on M to its ith component. A new array is returned.

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Manifolds.submanifold_componentsFunction
submanifold_components(M::Manifold, p)
submanifold_components(p)

Get the projected components of p on the submanifolds of M. The components are returned in a Tuple.

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Manifolds.ProductArrayType
ProductArray(shape::ShapeSpecification, data)

An array-based representation for points and tangent vectors on the product manifold. data contains underlying representation of points arranged according to TRanges and TSizes from shape. Internal views for each specific sub-point are created and stored in parts.

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Manifolds.ProductReprType
ProductRepr(parts)

A more general but slower representation of points and tangent vectors on a product manifold.

Example:

A product point on a product manifold Sphere(2) × Euclidean(2) might be created as

ProductRepr([1.0, 0.0, 0.0], [2.0, 3.0])

where [1.0, 0.0, 0.0] is the part corresponding to the sphere factor and [2.0, 3.0] is the part corresponding to the euclidean manifold.

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Manifolds.prod_pointFunction
prod_point(M::ShapeSpecification, pts...)

Construct a product point from product manifold M based on point pts represented by ProductArray.

Example

To construct a point on the product manifold $S^2 × ℝ^2$ from points on the sphere and in the euclidean space represented by, respectively, [1.0, 0.0, 0.0] and [-3.0, 2.0] you need to construct shape specification first. It describes how linear storage of ProductArray corresponds to array representations expected by Sphere(2) and Euclidean(2).

M1 = Sphere(2)
M2 = Euclidean(2)
reshaper = Manifolds.StaticReshaper()
Mshape = Manifolds.ShapeSpecification(reshaper, M1, M2)

Next, the desired point on the product manifold can be obtained by calling Manifolds.prod_point(Mshape, [1.0, 0.0, 0.0], [-3.0, 2.0]).

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Manifolds.make_reshapeFunction
make_reshape(reshaper::AbstractReshaper, ::Type{Size}, data) where Size

Reshape array data to size Size using method provided by reshaper.

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## Specific exception types

For some manifolds it is useful to keep an extra index, at which point on the manifold, the error occurred as well as to collect all errors that occurred on a manifold. This page contains the manifold-specific error messages this package introduces.