Math.Manifold.Embedding.Simple.Class

Some manifolds are “naturally” embedded within some bigger space. For instance, the topological spheres are readily identified with the geometric unit spheres in real vector spaces.

An embedding is a pretty strong relationship, but often all that's needed is being able to map single points from the manifold to the enclosing space. This module offers a class which does just that.

Documentation

class NaturallyEmbedded m v where

Methods

embed :: m -> v

coEmbed :: v -> m

Instances

NaturallyEmbedded S⁰ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S⁰ -> ℝ

coEmbed :: ℝ -> S⁰

NaturallyEmbedded S¹ ℝ²

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S¹ -> ℝ²

coEmbed :: ℝ² -> S¹

NaturallyEmbedded S² ℝ³

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S² -> ℝ³

coEmbed :: ℝ³ -> S²

NaturallyEmbedded ℝP² ℝ³

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝP² -> ℝ³

coEmbed :: ℝ³ -> ℝP²

NaturallyEmbedded D¹ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: D¹ -> ℝ

coEmbed :: ℝ -> D¹

NaturallyEmbedded ℝ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ -> ℝ

coEmbed :: ℝ -> ℝ

NaturallyEmbedded ℝ⁰ ℝ⁰

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ⁰ -> ℝ⁰

coEmbed :: ℝ⁰ -> ℝ⁰

NaturallyEmbedded ℝ⁴ ℝ⁴

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ⁴ -> ℝ⁴

coEmbed :: ℝ⁴ -> ℝ⁴

NaturallyEmbedded ℝ³ ℝ³

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ³ -> ℝ³

coEmbed :: ℝ³ -> ℝ³

NaturallyEmbedded ℝ² ℝ²

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ² -> ℝ²

coEmbed :: ℝ² -> ℝ²

(VectorSpace y, VectorSpace z) => NaturallyEmbedded x ((x, y), z)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: x -> ((x, y), z)

coEmbed :: ((x, y), z) -> x

VectorSpace y => NaturallyEmbedded x (x, y)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: x -> (x, y)

coEmbed :: (x, y) -> x

(AdditiveGroup f, x ~ Interior x) => NaturallyEmbedded x (FibreBundle x f)

Defined in Data.Manifold.FibreBundle

Methods

embed :: x -> FibreBundle x f

coEmbed :: FibreBundle x f -> x

NaturallyEmbedded x p => NaturallyEmbedded (Cℝay x) (p, ℝ)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: Cℝay x -> (p, ℝ)

coEmbed :: (p, ℝ) -> Cℝay x

(AdditiveGroup y, AdditiveGroup g) => NaturallyEmbedded (FibreBundle x f) (FibreBundle (x, y) (f, g))

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle x f -> FibreBundle (x, y) (f, g)

coEmbed :: FibreBundle (x, y) (f, g) -> FibreBundle x f

(NaturallyEmbedded m v, VectorSpace f) => NaturallyEmbedded (FibreBundle m ℝ⁰) (FibreBundle v f)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle m ℝ⁰ -> FibreBundle v f

coEmbed :: FibreBundle v f -> FibreBundle m ℝ⁰

NaturallyEmbedded (FibreBundle S¹ ℝ) (FibreBundle ℝ² ℝ²)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle S¹ ℝ -> FibreBundle ℝ² ℝ²

coEmbed :: FibreBundle ℝ² ℝ² -> FibreBundle S¹ ℝ

NaturallyEmbedded (FibreBundle S² ℝ²) (FibreBundle ℝ³ ℝ³)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle S² ℝ² -> FibreBundle ℝ³ ℝ³

coEmbed :: FibreBundle ℝ³ ℝ³ -> FibreBundle S² ℝ²

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ v) (FibreBundle ℝ w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle ℝ v -> FibreBundle ℝ w

coEmbed :: FibreBundle ℝ w -> FibreBundle ℝ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ⁴ v) (FibreBundle ℝ⁴ w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle ℝ⁴ v -> FibreBundle ℝ⁴ w

coEmbed :: FibreBundle ℝ⁴ w -> FibreBundle ℝ⁴ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ³ v) (FibreBundle ℝ³ w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle ℝ³ v -> FibreBundle ℝ³ w

coEmbed :: FibreBundle ℝ³ w -> FibreBundle ℝ³ v

NaturallyEmbedded v w => NaturallyEmbedded (FibreBundle ℝ² v) (FibreBundle ℝ² w)

Defined in Data.Manifold.FibreBundle

Methods

embed :: FibreBundle ℝ² v -> FibreBundle ℝ² w

coEmbed :: FibreBundle ℝ² w -> FibreBundle ℝ² v