Data.Manifold.FibreBundle

Documentation

pattern TangentBundle :: m -> Needle m -> FibreBundle m (Needle m)

pattern (:@.) :: f -> m -> FibreBundle m f infixr 5

Provided for convenience. Flipped synonym of FibreBundle, restricted to manifolds without boundary (so the type of the whole can be inferred from its interior).

tangentAt :: AdditiveGroup (Needle m) => m -> TangentBundle m

A zero vector in the fibre bundle at the given position. Intended to be used with tangent-modifying lenses such as delta.

data TransportOnNeedleWitness k m f where

Constructors

TransportOnNeedle :: ParallelTransporting (LinearFunction (Scalar (Needle m))) (Needle m) (Needle f) => TransportOnNeedleWitness k m f

data ForgetTransportProperties k m f where

Constructors

ForgetTransportProperties :: ParallelTransporting (->) m f => ForgetTransportProperties k m f

class (PseudoAffine m, Category k, Object k f) => ParallelTransporting k m f where

Minimal complete definition

parallelTransport

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k m f

default transportOnNeedleWitness :: ParallelTransporting (LinearFunction (Scalar (Needle m))) (Needle m) (Needle f) => TransportOnNeedleWitness k m f

forgetTransportProperties :: ForgetTransportProperties k m f

default forgetTransportProperties :: ParallelTransporting (->) m f => ForgetTransportProperties k m f

parallelTransport :: m -> Needle m -> k f f

translateAndInvblyParTransport :: m -> Needle m -> (m, (k f f, k f f))

Instances

(EnhancedCat k (LinearMap ℝ), Object k ℝ²) => ParallelTransporting k S² ℝ²

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k S² ℝ²

forgetTransportProperties :: ForgetTransportProperties k S² ℝ²

parallelTransport :: S² -> Needle S² -> k ℝ² ℝ²

translateAndInvblyParTransport :: S² -> Needle S² -> (S², (k ℝ² ℝ², k ℝ² ℝ²))

(Category k, Object k ℝ) => ParallelTransporting k S¹ ℝ

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k S¹ ℝ

forgetTransportProperties :: ForgetTransportProperties k S¹ ℝ

parallelTransport :: S¹ -> Needle S¹ -> k ℝ ℝ

translateAndInvblyParTransport :: S¹ -> Needle S¹ -> (S¹, (k ℝ ℝ, k ℝ ℝ))

(Category k, Object k ℝ⁴) => ParallelTransporting k ℝ⁴ ℝ⁴

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ⁴ ℝ⁴

forgetTransportProperties :: ForgetTransportProperties k ℝ⁴ ℝ⁴

parallelTransport :: ℝ⁴ -> Needle ℝ⁴ -> k ℝ⁴ ℝ⁴

translateAndInvblyParTransport :: ℝ⁴ -> Needle ℝ⁴ -> (ℝ⁴, (k ℝ⁴ ℝ⁴, k ℝ⁴ ℝ⁴))

(Category k, Object k ℝ³) => ParallelTransporting k ℝ³ ℝ³

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ³ ℝ³

forgetTransportProperties :: ForgetTransportProperties k ℝ³ ℝ³

parallelTransport :: ℝ³ -> Needle ℝ³ -> k ℝ³ ℝ³

translateAndInvblyParTransport :: ℝ³ -> Needle ℝ³ -> (ℝ³, (k ℝ³ ℝ³, k ℝ³ ℝ³))

(Category k, Object k ℝ²) => ParallelTransporting k ℝ² ℝ²

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ² ℝ²

forgetTransportProperties :: ForgetTransportProperties k ℝ² ℝ²

parallelTransport :: ℝ² -> Needle ℝ² -> k ℝ² ℝ²

translateAndInvblyParTransport :: ℝ² -> Needle ℝ² -> (ℝ², (k ℝ² ℝ², k ℝ² ℝ²))

(Category k, Object k ℝ) => ParallelTransporting k ℝ ℝ

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k ℝ ℝ

forgetTransportProperties :: ForgetTransportProperties k ℝ ℝ

parallelTransport :: ℝ -> Needle ℝ -> k ℝ ℝ

translateAndInvblyParTransport :: ℝ -> Needle ℝ -> (ℝ, (k ℝ ℝ, k ℝ ℝ))

(ParallelTransporting k a f, ParallelTransporting k a g, ParallelTransporting (LinearFunction s) (Needle a) (Needle f, Needle g), PseudoAffine f, PseudoAffine g, Morphism k, ObjectPair k f g) => ParallelTransporting k a (f, g)

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k a (f, g)

forgetTransportProperties :: ForgetTransportProperties k a (f, g)

parallelTransport :: a -> Needle a -> k (f, g) (f, g)

translateAndInvblyParTransport :: a -> Needle a -> (a, (k (f, g) (f, g), k (f, g) (f, g)))

(ParallelTransporting k a fa, ParallelTransporting k b fb, PseudoAffine fa, PseudoAffine fb, Scalar (Needle a) ~ s, Scalar (Needle b) ~ s, Scalar (Needle fa) ~ s, Scalar (Needle fb) ~ s, Num' s, Morphism k, ObjectPair k fa fb) => ParallelTransporting k (a, b) (fa, fb)

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness k (a, b) (fa, fb)

forgetTransportProperties :: ForgetTransportProperties k (a, b) (fa, fb)

parallelTransport :: (a, b) -> Needle (a, b) -> k (fa, fb) (fa, fb)

translateAndInvblyParTransport :: (a, b) -> Needle (a, b) -> ((a, b), (k (fa, fb) (fa, fb), k (fa, fb) (fa, fb)))

(PseudoAffine m, s ~ Scalar (Needle m), Num' s) => ParallelTransporting (Discrete :: Type -> Type -> Type) m (ZeroDim s)

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness Discrete m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties Discrete m (ZeroDim s)

parallelTransport :: m -> Needle m -> Discrete (ZeroDim s) (ZeroDim s)

translateAndInvblyParTransport :: m -> Needle m -> (m, (Discrete (ZeroDim s) (ZeroDim s), Discrete (ZeroDim s) (ZeroDim s)))

(PseudoAffine m, s ~ Scalar (Needle m), Num' s) => ParallelTransporting (LinearFunction s) m (ZeroDim s)

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness (LinearFunction s) m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties (LinearFunction s) m (ZeroDim s)

parallelTransport :: m -> Needle m -> LinearFunction s (ZeroDim s) (ZeroDim s)

translateAndInvblyParTransport :: m -> Needle m -> (m, (LinearFunction s (ZeroDim s) (ZeroDim s), LinearFunction s (ZeroDim s) (ZeroDim s)))

(PseudoAffine m, s ~ Scalar (Needle m), Num' s) => ParallelTransporting ((->) :: Type -> Type -> Type) m (ZeroDim s)

Defined in Data.Manifold.FibreBundle

Methods

transportOnNeedleWitness :: TransportOnNeedleWitness (->) m (ZeroDim s)

forgetTransportProperties :: ForgetTransportProperties (->) m (ZeroDim s)

parallelTransport :: m -> Needle m -> ZeroDim s -> ZeroDim s

translateAndInvblyParTransport :: m -> Needle m -> (m, (ZeroDim s -> ZeroDim s, ZeroDim s -> ZeroDim s))

transformEmbeddedTangents :: forall x f v. NaturallyEmbedded (FibreBundle x f) (FibreBundle v v) => (v -> v) -> FibreBundle x f -> FibreBundle x f

ex -> ey, ey -> ez, ez -> ex

Orphan instances

AdditiveGroup f => NaturallyEmbedded x (FibreBundle x f)

Methods

embed :: x -> FibreBundle x f

coEmbed :: FibreBundle x f -> x

(ParallelTransporting (LinearFunction (Scalar f)) m f, AdditiveGroup m, VectorSpace f) => AdditiveGroup (FibreBundle m f)

Methods

zeroV :: FibreBundle m f

(^+^) :: FibreBundle m f -> FibreBundle m f -> FibreBundle m f

negateV :: FibreBundle m f -> FibreBundle m f

(^-^) :: FibreBundle m f -> FibreBundle m f -> FibreBundle m f

(ParallelTransporting ((->) :: Type -> Type -> Type) m f, PseudoAffine f, ParallelTransporting (LinearFunction s) (Needle m) (Needle f), s ~ Scalar (Needle m)) => PseudoAffine (FibreBundle m f)

Methods

(.-~.) :: FibreBundle m f -> FibreBundle m f -> Maybe (Needle (FibreBundle m f))

(.-~!) :: FibreBundle m f -> FibreBundle m f -> Needle (FibreBundle m f)

pseudoAffineWitness :: PseudoAffineWitness (FibreBundle m f)

(ParallelTransporting ((->) :: Type -> Type -> Type) m f, Semimanifold f, ParallelTransporting (LinearFunction s) (Needle m) (Needle f), s ~ Scalar (Needle m)) => Semimanifold (FibreBundle m f)

Associated Types

type Needle (FibreBundle m f)

Methods

(.+~^) :: FibreBundle m f -> Needle (FibreBundle m f) -> FibreBundle m f

(.-~^) :: FibreBundle m f -> Needle (FibreBundle m f) -> FibreBundle m f

semimanifoldWitness :: SemimanifoldWitness (FibreBundle m f)

(s ~ ℝ, s' ~ ℝ) => Rotatable (FibreBundle (S²_ s) (V2 s'))

Associated Types

type AxisSpace (FibreBundle (S²_ s) (V2 s'))

Methods

rotateAbout :: AxisSpace (FibreBundle (S²_ s) (V2 s')) -> S¹ -> FibreBundle (S²_ s) (V2 s') -> FibreBundle (S²_ s) (V2 s')

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

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 (ZeroDim s)) (FibreBundle v f)

Methods

embed :: FibreBundle m (ZeroDim s) -> FibreBundle v f

coEmbed :: FibreBundle v f -> FibreBundle m (ZeroDim s)

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V2 s) v) (FibreBundle (V2 s') w)

Methods

embed :: FibreBundle (V2 s) v -> FibreBundle (V2 s') w

coEmbed :: FibreBundle (V2 s') w -> FibreBundle (V2 s) v

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V3 s) v) (FibreBundle (V3 s') w)

Methods

embed :: FibreBundle (V3 s) v -> FibreBundle (V3 s') w

coEmbed :: FibreBundle (V3 s') w -> FibreBundle (V3 s) v

(NaturallyEmbedded v w, s' ~ s) => NaturallyEmbedded (FibreBundle (V4 s) v) (FibreBundle (V4 s') w)

Methods

embed :: FibreBundle (V4 s) v -> FibreBundle (V4 s') w

coEmbed :: FibreBundle (V4 s') w -> FibreBundle (V4 s) v

(RealFloat s, InnerSpace s, s ~ s', s ~ s'', s ~ s''') => NaturallyEmbedded (FibreBundle (S¹_ s) s') (FibreBundle (V2 s'') (V2 s'''))

Methods

embed :: FibreBundle (S¹_ s) s' -> FibreBundle (V2 s'') (V2 s''')

coEmbed :: FibreBundle (V2 s'') (V2 s''') -> FibreBundle (S¹_ s) s'

(RealFloat' s, InnerSpace s, s ~ s', s ~ s'', s ~ s''') => NaturallyEmbedded (FibreBundle (S²_ s) (V2 s')) (FibreBundle (V3 s'') (V3 s'''))

Methods

embed :: FibreBundle (S²_ s) (V2 s') -> FibreBundle (V3 s'') (V3 s''')

coEmbed :: FibreBundle (V3 s'') (V3 s''') -> FibreBundle (S²_ s) (V2 s')

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

Methods

embed :: FibreBundle ℝ v -> FibreBundle ℝ w

coEmbed :: FibreBundle ℝ w -> FibreBundle ℝ v