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

Minimal complete definition

embed, coEmbed

Methods

embed :: m -> v

coEmbed :: v -> m

Instances

NaturallyEmbedded S⁰ ℝ

Methods

embed :: S⁰ -> ℝ

coEmbed :: ℝ -> S⁰

NaturallyEmbedded S¹ ℝ²

Methods

embed :: S¹ -> ℝ²

coEmbed :: ℝ² -> S¹

NaturallyEmbedded S² ℝ³

Methods

embed :: S² -> ℝ³

coEmbed :: ℝ³ -> S²

NaturallyEmbedded ℝP² ℝ³

Methods

embed :: ℝP² -> ℝ³

coEmbed :: ℝ³ -> ℝP²

NaturallyEmbedded D¹ ℝ

Methods

embed :: D¹ -> ℝ

coEmbed :: ℝ -> D¹

NaturallyEmbedded ℝ ℝ

Methods

embed :: ℝ -> ℝ

coEmbed :: ℝ -> ℝ

NaturallyEmbedded ℝ⁰ ℝ⁰

Methods

embed :: ℝ⁰ -> ℝ⁰

coEmbed :: ℝ⁰ -> ℝ⁰

NaturallyEmbedded ℝ⁴ ℝ⁴

Methods

embed :: ℝ⁴ -> ℝ⁴

coEmbed :: ℝ⁴ -> ℝ⁴

NaturallyEmbedded ℝ³ ℝ³

Methods

embed :: ℝ³ -> ℝ³

coEmbed :: ℝ³ -> ℝ³

NaturallyEmbedded ℝ² ℝ²

Methods

embed :: ℝ² -> ℝ²

coEmbed :: ℝ² -> ℝ²

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

Methods

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

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

VectorSpace y => NaturallyEmbedded x (x, y)

Methods

embed :: x -> (x, y)

coEmbed :: (x, y) -> x

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

Methods

embed :: Cℝay x -> (p, ℝ)

coEmbed :: (p, ℝ) -> Cℝay x