Data.Manifold.Types

Several commonly-used manifolds, represented in some simple way as Haskell data types. All these are in the PseudoAffine class.

Index / ASCII names

type Real0 = ℝ⁰

type Real1 = ℝ

type RealPlus = ℝay

type Real2 = ℝ²

type Real3 = ℝ³

type Sphere0 = S⁰

type Sphere1 = S¹

type Sphere2 = S²

type Projective0 = ℝP⁰

type Projective1 = ℝP¹

type Projective2 = ℝP²

type Disk1 = D¹

type Disk2 = D²

type Cone = CD¹

type OpenCone = Cℝay

data FibreBundle b f

A fibre bundle combines points in the base space b with points in the fibre f. The type FibreBundle b f is thus isomorphic to the tuple space (b,f), but it can have a different topology, the prime example being TangentBundle, where nearby points may have differently-oriented tangent spaces.

Constructors

FibreBundle

Fields

• baseSpace :: !b
• fibreSpace :: !f

Instances

(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

(Show (CoordinateIdentifier b), Show (CoordinateIdentifier f), Eq b, Eq (CoordinateIdentifier f), Arbitrary b, Show b) => Show (CoordinateIdentifier (FibreBundle b f))

Defined in Math.Manifold.Real.Coordinates

Methods

showsPrec :: Int -> CoordinateIdentifier (FibreBundle b f) -> ShowS

show :: CoordinateIdentifier (FibreBundle b f) -> String

showList :: [CoordinateIdentifier (FibreBundle b f)] -> ShowS

(CoordDifferential m, f ~ Needle m, m ~ Interior m, Arbitrary m, Arbitrary (CoordinateIdentifier m), Arbitrary (CoordinateIdentifier f)) => Arbitrary (CoordinateIdentifier (FibreBundle m f))

Defined in Math.Manifold.Real.Coordinates

Methods

arbitrary :: Gen (CoordinateIdentifier (FibreBundle m f))

shrink :: CoordinateIdentifier (FibreBundle m f) -> [CoordinateIdentifier (FibreBundle m f)]

(Show b, Show f) => Show (FibreBundle b f)

Defined in Math.Manifold.Core.PseudoAffine

Methods

showsPrec :: Int -> FibreBundle b f -> ShowS

show :: FibreBundle b f -> String

showList :: [FibreBundle b f] -> ShowS

Generic (FibreBundle b f)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Rep (FibreBundle b f) :: Type -> Type

Methods

from :: FibreBundle b f -> Rep (FibreBundle b f) x

to :: Rep (FibreBundle b f) x -> FibreBundle b f

(Arbitrary m, Arbitrary f) => Arbitrary (FibreBundle m f)

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen (FibreBundle m f)

shrink :: FibreBundle m f -> [FibreBundle m f]

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

Defined in Data.Manifold.FibreBundle

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 (Interior f), Semimanifold f, ParallelTransporting (LinearFunction s) (Needle m) (Needle f), s ~ Scalar (Needle m)) => Semimanifold (FibreBundle m f)

Defined in Data.Manifold.FibreBundle

Associated Types

type Needle (FibreBundle m f) :: Type

type Interior (FibreBundle m f) :: Type

Methods

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

fromInterior :: Interior (FibreBundle m f) -> FibreBundle m f

toInterior :: FibreBundle m f -> Maybe (Interior (FibreBundle m f))

translateP :: Tagged (FibreBundle m f) (Interior (FibreBundle m f) -> Needle (FibreBundle m f) -> Interior (FibreBundle m f))

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

semimanifoldWitness :: SemimanifoldWitness (FibreBundle m f)

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

Defined in Data.Manifold.FibreBundle

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)

(Show m, Show f) => Show (FibreBundle m f)

Defined in Data.Manifold.Types.Primitive

Methods

showsPrec :: Int -> FibreBundle m f -> ShowS

show :: FibreBundle m f -> String

showList :: [FibreBundle m f] -> ShowS

Rotatable (FibreBundle S² ℝ²)

Defined in Data.Manifold.FibreBundle

Associated Types

type AxisSpace (FibreBundle S² ℝ²) :: Type

Methods

rotateAbout :: AxisSpace (FibreBundle S² ℝ²) -> S¹ -> FibreBundle S² ℝ² -> FibreBundle S² ℝ²

(Connected x, Connected y, PseudoAffine (FibreBundle x y)) => Connected (FibreBundle x y)

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: FibreBundle x y -> FibreBundle x y -> Needle (FibreBundle x y)

(HasCoordinates b, HasCoordinates f) => HasCoordinates (FibreBundle b f)

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier (FibreBundle b f) :: Type

Methods

coordinateAsLens :: CoordinateIdentifier (FibreBundle b f) -> Lens' (FibreBundle b f) ℝ

validCoordinateRange :: CoordinateIdentifier (FibreBundle b f) -> FibreBundle b f -> (ℝ, ℝ)

(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

type Rep (FibreBundle b f)

Defined in Math.Manifold.Core.PseudoAffine

type Rep (FibreBundle b f) = D1 (MetaData "FibreBundle" "Math.Manifold.Core.PseudoAffine" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "FibreBundle" PrefixI True) (S1 (MetaSel (Just "baseSpace") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 b) :*: S1 (MetaSel (Just "fibreSpace") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 f)))

type Interior (FibreBundle m f)

Defined in Data.Manifold.FibreBundle

type Interior (FibreBundle m f) = FibreBundle m (Interior f)

type Needle (FibreBundle m f)

Defined in Data.Manifold.FibreBundle

type Needle (FibreBundle m f) = FibreBundle (Needle m) (Needle f)

type AxisSpace (FibreBundle S² ℝ²)

Defined in Data.Manifold.FibreBundle

type AxisSpace (FibreBundle S² ℝ²) = ℝP²

data CoordinateIdentifier (FibreBundle b f)

Defined in Math.Manifold.Real.Coordinates

data CoordinateIdentifier (FibreBundle b f)

• = BaseSpaceCoordinate (CoordinateIdentifier b)
• | FibreSpaceCoordinate (b -> CoordinateIdentifier f)

type TangentBundle m = FibreBundle m (Needle m)

Points on a manifold, combined with vectors in the respective tangent space.

Linear manifolds

data ZeroDim s

Constructors

Origin

Instances

Connected ℝ⁰

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ⁰ -> ℝ⁰ -> Needle ℝ⁰

LtdErrorShow ℝ⁰

Defined in Data.Manifold.Shade

Methods

ltdErrorShowWitness :: LtdErrorShowWitness ℝ⁰

showsPrecShade'_errorLtdC :: Int -> Shade' ℝ⁰ -> ShowS

prettyShowsPrecShade :: Int -> Shade ℝ⁰ -> ShowS

prettyShowsPrecShade' :: Int -> Shade' ℝ⁰ -> ShowS

Refinable ℝ⁰

Defined in Data.Manifold.Shade

Methods

debugView :: Maybe (DebugView ℝ⁰)

subShade' :: Shade' ℝ⁰ -> Shade' ℝ⁰ -> Bool

refineShade' :: Shade' ℝ⁰ -> Shade' ℝ⁰ -> Maybe (Shade' ℝ⁰)

convolveMetric :: Functor p => p ℝ⁰ -> Metric ℝ⁰ -> Metric ℝ⁰ -> Metric ℝ⁰

convolveShade' :: Shade' ℝ⁰ -> Shade' (Needle ℝ⁰) -> Shade' ℝ⁰

HasCoordinates ℝ⁰

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier ℝ⁰ :: Type

Methods

coordinateAsLens :: CoordinateIdentifier ℝ⁰ -> Lens' ℝ⁰ ℝ

validCoordinateRange :: CoordinateIdentifier ℝ⁰ -> ℝ⁰ -> (ℝ, ℝ)

NaturallyEmbedded ℝ⁰ ℝ⁰

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝ⁰ -> ℝ⁰

coEmbed :: ℝ⁰ -> ℝ⁰

Eq (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

(==) :: ZeroDim s -> ZeroDim s -> Bool

(/=) :: ZeroDim s -> ZeroDim s -> Bool

Functor (AbstractSimplex ℝ⁰)

Defined in Data.Simplex.Abstract

Methods

fmap :: (a -> b) -> AbstractSimplex ℝ⁰ a -> AbstractSimplex ℝ⁰ b

(<$) :: a -> AbstractSimplex ℝ⁰ b -> AbstractSimplex ℝ⁰ a

Show (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

showsPrec :: Int -> ZeroDim s -> ShowS

show :: ZeroDim s -> String

showList :: [ZeroDim s] -> ShowS

Applicative (AbstractSimplex ℝ⁰)

Defined in Data.Simplex.Abstract

Methods

pure :: a -> AbstractSimplex ℝ⁰ a

(<*>) :: AbstractSimplex ℝ⁰ (a -> b) -> AbstractSimplex ℝ⁰ a -> AbstractSimplex ℝ⁰ b

liftA2 :: (a -> b -> c) -> AbstractSimplex ℝ⁰ a -> AbstractSimplex ℝ⁰ b -> AbstractSimplex ℝ⁰ c

(*>) :: AbstractSimplex ℝ⁰ a -> AbstractSimplex ℝ⁰ b -> AbstractSimplex ℝ⁰ b

(<*) :: AbstractSimplex ℝ⁰ a -> AbstractSimplex ℝ⁰ b -> AbstractSimplex ℝ⁰ a

Foldable (AbstractSimplex ℝ⁰)

Defined in Data.Simplex.Abstract

Methods

fold :: Monoid m => AbstractSimplex ℝ⁰ m -> m

foldMap :: Monoid m => (a -> m) -> AbstractSimplex ℝ⁰ a -> m

foldr :: (a -> b -> b) -> b -> AbstractSimplex ℝ⁰ a -> b

foldr' :: (a -> b -> b) -> b -> AbstractSimplex ℝ⁰ a -> b

foldl :: (b -> a -> b) -> b -> AbstractSimplex ℝ⁰ a -> b

foldl' :: (b -> a -> b) -> b -> AbstractSimplex ℝ⁰ a -> b

foldr1 :: (a -> a -> a) -> AbstractSimplex ℝ⁰ a -> a

foldl1 :: (a -> a -> a) -> AbstractSimplex ℝ⁰ a -> a

toList :: AbstractSimplex ℝ⁰ a -> [a]

null :: AbstractSimplex ℝ⁰ a -> Bool

length :: AbstractSimplex ℝ⁰ a -> Int

elem :: Eq a => a -> AbstractSimplex ℝ⁰ a -> Bool

maximum :: Ord a => AbstractSimplex ℝ⁰ a -> a

minimum :: Ord a => AbstractSimplex ℝ⁰ a -> a

sum :: Num a => AbstractSimplex ℝ⁰ a -> a

product :: Num a => AbstractSimplex ℝ⁰ a -> a

Traversable (AbstractSimplex ℝ⁰)

Defined in Data.Simplex.Abstract

Methods

traverse :: Applicative f => (a -> f b) -> AbstractSimplex ℝ⁰ a -> f (AbstractSimplex ℝ⁰ b)

sequenceA :: Applicative f => AbstractSimplex ℝ⁰ (f a) -> f (AbstractSimplex ℝ⁰ a)

mapM :: Monad m => (a -> m b) -> AbstractSimplex ℝ⁰ a -> m (AbstractSimplex ℝ⁰ b)

sequence :: Monad m => AbstractSimplex ℝ⁰ (m a) -> m (AbstractSimplex ℝ⁰ a)

Semigroup (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

(<>) :: ZeroDim s -> ZeroDim s -> ZeroDim s

sconcat :: NonEmpty (ZeroDim s) -> ZeroDim s

stimes :: Integral b => b -> ZeroDim s -> ZeroDim s

Monoid (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

mempty :: ZeroDim s

mappend :: ZeroDim s -> ZeroDim s -> ZeroDim s

mconcat :: [ZeroDim s] -> ZeroDim s

Binary (ZeroDim a)

Defined in Data.Manifold.Types.Primitive

Methods

put :: ZeroDim a -> Put

get :: Get (ZeroDim a)

putList :: [ZeroDim a] -> Put

AdditiveGroup (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

zeroV :: ZeroDim s

(^+^) :: ZeroDim s -> ZeroDim s -> ZeroDim s

negateV :: ZeroDim s -> ZeroDim s

(^-^) :: ZeroDim s -> ZeroDim s -> ZeroDim s

AdditiveGroup s => InnerSpace (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

(<.>) :: ZeroDim s -> ZeroDim s -> Scalar (ZeroDim s)

VectorSpace (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Associated Types

type Scalar (ZeroDim s) :: Type

Methods

(*^) :: Scalar (ZeroDim s) -> ZeroDim s -> ZeroDim s

HasBasis (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Associated Types

type Basis (ZeroDim s) :: Type

Methods

basisValue :: Basis (ZeroDim s) -> ZeroDim s

decompose :: ZeroDim s -> [(Basis (ZeroDim s), Scalar (ZeroDim s))]

decompose' :: ZeroDim s -> Basis (ZeroDim s) -> Scalar (ZeroDim s)

AffineSpace (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Associated Types

type Diff (ZeroDim s) :: Type

Methods

(.-.) :: ZeroDim s -> ZeroDim s -> Diff (ZeroDim s)

(.+^) :: ZeroDim s -> Diff (ZeroDim s) -> ZeroDim s

RieszDecomposable (ZeroDim ℝ)

Defined in Math.VectorSpace.Docile

Methods

rieszDecomposition :: (FiniteDimensional v, v ~ DualVector v, Scalar v ~ Scalar (ZeroDim ℝ)) => (v +> ZeroDim ℝ) -> [(Basis (ZeroDim ℝ), v)]

TensorDecomposable (ZeroDim ℝ)

Defined in Math.VectorSpace.Docile

Methods

tensorDecomposition :: (ZeroDim ℝ ⊗ w) -> [(Basis (ZeroDim ℝ), w)]

showsPrecBasis :: Functor p => p (ZeroDim ℝ) -> Int -> Basis (ZeroDim ℝ) -> ShowS

(Fractional' s, SemiInner s) => SemiInner (ZeroDim s)

Defined in Math.VectorSpace.Docile

Methods

dualBasisCandidates :: [(Int, ZeroDim s)] -> Forest (Int, DualVector (ZeroDim s))

tensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (ZeroDim s)) => [(Int, ZeroDim s ⊗ w)] -> Forest (Int, DualVector (ZeroDim s ⊗ w))

symTensorDualBasisCandidates :: [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s))] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (DualVector (ZeroDim s)))

symTensorTensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (ZeroDim s)) => [(Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) ⊗ w)] -> Forest (Int, SymmetricTensor (Scalar (ZeroDim s)) (ZeroDim s) +> DualVector w)

Num' s => FiniteDimensional (ZeroDim s)

Defined in Math.VectorSpace.Docile

Associated Types

data SubBasis (ZeroDim s) :: Type

Methods

entireBasis :: SubBasis (ZeroDim s)

enumerateSubBasis :: SubBasis (ZeroDim s) -> [ZeroDim s]

subbasisDimension :: SubBasis (ZeroDim s) -> Int

decomposeLinMap :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s +> w) -> (SubBasis (ZeroDim s), DList w)

decomposeLinMapWithin :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> (ZeroDim s +> w) -> Either (SubBasis (ZeroDim s), DList w) (DList w)

recomposeSB :: SubBasis (ZeroDim s) -> [Scalar (ZeroDim s)] -> (ZeroDim s, [Scalar (ZeroDim s)])

recomposeSBTensor :: (FiniteDimensional w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> SubBasis w -> [Scalar (ZeroDim s)] -> (ZeroDim s ⊗ w, [Scalar (ZeroDim s)])

recomposeLinMap :: (LSpace w, Scalar w ~ Scalar (ZeroDim s)) => SubBasis (ZeroDim s) -> [w] -> (ZeroDim s +> w, [w])

recomposeContraLinMap :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s), Functor f) => (f (Scalar w) -> w) -> f (DualVector (ZeroDim s)) -> ZeroDim s +> w

recomposeContraLinMapTensor :: (FiniteDimensional u, LinearSpace w, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s), Functor f) => (f (Scalar w) -> w) -> f (ZeroDim s +> DualVector u) -> (ZeroDim s ⊗ u) +> w

uncanonicallyFromDual :: DualVector (ZeroDim s) -+> ZeroDim s

uncanonicallyToDual :: ZeroDim s -+> DualVector (ZeroDim s)

Num' s => TensorSpace (ZeroDim s)

Defined in Math.LinearMap.Category.Class

Associated Types

type TensorProduct (ZeroDim s) w :: Type

Methods

scalarSpaceWitness :: ScalarSpaceWitness (ZeroDim s)

linearManifoldWitness :: LinearManifoldWitness (ZeroDim s)

zeroTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => ZeroDim s ⊗ w

toFlatTensor :: ZeroDim s -+> (ZeroDim s ⊗ Scalar (ZeroDim s))

fromFlatTensor :: (ZeroDim s ⊗ Scalar (ZeroDim s)) -+> ZeroDim s

addTensors :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w

subtractTensors :: (TensorSpace (ZeroDim s), TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> ZeroDim s ⊗ w

scaleTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (Scalar (ZeroDim s)) (ZeroDim s ⊗ w) (ZeroDim s ⊗ w)

negateTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -+> (ZeroDim s ⊗ w)

tensorProduct :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s) w (ZeroDim s ⊗ w)

tensorProducts :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => [(ZeroDim s, w)] -> ZeroDim s ⊗ w

transposeTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -+> (w ⊗ ZeroDim s)

fmapTensor :: (TensorSpace w, TensorSpace x, Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear (w -+> x) (ZeroDim s ⊗ w) (ZeroDim s ⊗ x)

fzipTensorWith :: (TensorSpace u, TensorSpace w, TensorSpace x, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear ((w, x) -+> u) (ZeroDim s ⊗ w, ZeroDim s ⊗ x) (ZeroDim s ⊗ u)

coerceFmapTensorProduct :: Functor p => p (ZeroDim s) -> Coercion a b -> Coercion (TensorProduct (ZeroDim s) a) (TensorProduct (ZeroDim s) b)

wellDefinedVector :: ZeroDim s -> Maybe (ZeroDim s)

wellDefinedTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> Maybe (ZeroDim s ⊗ w)

Num' s => LinearSpace (ZeroDim s)

Defined in Math.LinearMap.Category.Class

Associated Types

type DualVector (ZeroDim s) :: Type

Methods

dualSpaceWitness :: DualSpaceWitness (ZeroDim s)

linearId :: ZeroDim s +> ZeroDim s

idTensor :: ZeroDim s ⊗ DualVector (ZeroDim s)

sampleLinearFunction :: (TensorSpace w, Scalar (ZeroDim s) ~ Scalar w) => (ZeroDim s -+> w) -+> (ZeroDim s +> w)

toLinearForm :: DualVector (ZeroDim s) -+> (ZeroDim s +> Scalar (ZeroDim s))

fromLinearForm :: (ZeroDim s +> Scalar (ZeroDim s)) -+> DualVector (ZeroDim s)

coerceDoubleDual :: Coercion (ZeroDim s) (DualVector (DualVector (ZeroDim s)))

trace :: (ZeroDim s +> ZeroDim s) -+> Scalar (ZeroDim s)

contractTensorMap :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s +> (ZeroDim s ⊗ w)) -+> w

contractMapTensor :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ (ZeroDim s +> w)) -+> w

contractTensorFn :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s -+> (ZeroDim s ⊗ w)) -+> w

contractLinearMapAgainst :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s +> w) (w -+> ZeroDim s) (Scalar (ZeroDim s))

applyDualVector :: Bilinear (DualVector (ZeroDim s)) (ZeroDim s) (Scalar (ZeroDim s))

applyLinear :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim s)) => Bilinear (ZeroDim s +> w) (ZeroDim s) w

composeLinear :: (LinearSpace w, TensorSpace x, Scalar w ~ Scalar (ZeroDim s), Scalar x ~ Scalar (ZeroDim s)) => Bilinear (w +> x) (ZeroDim s +> w) (ZeroDim s +> x)

tensorId :: (LinearSpace w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) +> (ZeroDim s ⊗ w)

applyTensorFunctional :: (LinearSpace u, Scalar u ~ Scalar (ZeroDim s)) => Bilinear (DualVector (ZeroDim s ⊗ u)) (ZeroDim s ⊗ u) (Scalar (ZeroDim s))

applyTensorLinMap :: (LinearSpace u, TensorSpace w, Scalar u ~ Scalar (ZeroDim s), Scalar w ~ Scalar (ZeroDim s)) => Bilinear ((ZeroDim s ⊗ u) +> w) (ZeroDim s ⊗ u) w

Semimanifold (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ZeroDim k) :: Type

type Interior (ZeroDim k) :: Type

Methods

(.+~^) :: Interior (ZeroDim k) -> Needle (ZeroDim k) -> ZeroDim k

fromInterior :: Interior (ZeroDim k) -> ZeroDim k

toInterior :: ZeroDim k -> Maybe (Interior (ZeroDim k))

translateP :: Tagged (ZeroDim k) (Interior (ZeroDim k) -> Needle (ZeroDim k) -> Interior (ZeroDim k))

(.-~^) :: Interior (ZeroDim k) -> Needle (ZeroDim k) -> ZeroDim k

semimanifoldWitness :: SemimanifoldWitness (ZeroDim k)

PseudoAffine (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ZeroDim k -> ZeroDim k -> Maybe (Needle (ZeroDim k))

(.-~!) :: ZeroDim k -> ZeroDim k -> Needle (ZeroDim k)

pseudoAffineWitness :: PseudoAffineWitness (ZeroDim k)

Atlas (ZeroDim s)

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex (ZeroDim s) :: Type

Methods

chartReferencePoint :: ChartIndex (ZeroDim s) -> ZeroDim s

interiorChartReferencePoint :: Functor p => p (ZeroDim s) -> ChartIndex (ZeroDim s) -> Interior (ZeroDim s)

lookupAtlas :: ZeroDim s -> ChartIndex (ZeroDim s)

Geodesic (ZeroDim s)

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: ZeroDim s -> ZeroDim s -> Maybe (D¹ -> ZeroDim s)

geodesicWitness :: GeodesicWitness (ZeroDim s)

middleBetween :: ZeroDim s -> ZeroDim s -> Maybe (ZeroDim s)

(PseudoAffine m, m ~ Interior 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, m ~ Interior 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)))

NumPrime s => LocallyCoercible (V0 s) (ZeroDim s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V0 s -> ZeroDim s

coerceNeedle :: Functor p => p (V0 s, ZeroDim s) -> Needle (V0 s) -+> Needle (ZeroDim s)

coerceNeedle' :: Functor p => p (V0 s, ZeroDim s) -> Needle' (V0 s) -+> Needle' (ZeroDim s)

coerceNorm :: Functor p => p (V0 s, ZeroDim s) -> Metric (V0 s) -> Metric (ZeroDim s)

coerceVariance :: Functor p => p (V0 s, ZeroDim s) -> Metric' (V0 s) -> Metric' (ZeroDim s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (ZeroDim s) (V0 s)

interiorLocalCoercion :: Functor p (->) (->) => p (V0 s, ZeroDim s) -> CanonicalDiffeomorphism (Interior (V0 s)) (Interior (ZeroDim s))

NumPrime s => LocallyCoercible (ZeroDim s) (V0 s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ZeroDim s -> V0 s

coerceNeedle :: Functor p => p (ZeroDim s, V0 s) -> Needle (ZeroDim s) -+> Needle (V0 s)

coerceNeedle' :: Functor p => p (ZeroDim s, V0 s) -> Needle' (ZeroDim s) -+> Needle' (V0 s)

coerceNorm :: Functor p => p (ZeroDim s, V0 s) -> Metric (ZeroDim s) -> Metric (V0 s)

coerceVariance :: Functor p => p (ZeroDim s, V0 s) -> Metric' (ZeroDim s) -> Metric' (V0 s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V0 s) (ZeroDim s)

interiorLocalCoercion :: Functor p (->) (->) => p (ZeroDim s, V0 s) -> CanonicalDiffeomorphism (Interior (ZeroDim s)) (Interior (V0 s))

NumPrime s => LocallyCoercible (ZeroDim s) (ZeroDim s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ZeroDim s -> ZeroDim s

coerceNeedle :: Functor p => p (ZeroDim s, ZeroDim s) -> Needle (ZeroDim s) -+> Needle (ZeroDim s)

coerceNeedle' :: Functor p => p (ZeroDim s, ZeroDim s) -> Needle' (ZeroDim s) -+> Needle' (ZeroDim s)

coerceNorm :: Functor p => p (ZeroDim s, ZeroDim s) -> Metric (ZeroDim s) -> Metric (ZeroDim s)

coerceVariance :: Functor p => p (ZeroDim s, ZeroDim s) -> Metric' (ZeroDim s) -> Metric' (ZeroDim s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (ZeroDim s) (ZeroDim s)

interiorLocalCoercion :: Functor p (->) (->) => p (ZeroDim s, ZeroDim s) -> CanonicalDiffeomorphism (Interior (ZeroDim s)) (Interior (ZeroDim s))

(PseudoAffine m, m ~ Interior 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))

(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

data AbstractSimplex ℝ⁰ x

Defined in Data.Simplex.Abstract

data AbstractSimplex ℝ⁰ x = ℝ⁰Simplex !x

data CoordinateIdentifier ℝ⁰

Defined in Math.Manifold.Real.Coordinates

data CoordinateIdentifier ℝ⁰

type Scalar (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Scalar (ZeroDim s) = s

type Basis (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Basis (ZeroDim s) = Void

type Diff (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Diff (ZeroDim s) = ZeroDim s

data SubBasis (ZeroDim s)

Defined in Math.VectorSpace.Docile

data SubBasis (ZeroDim s) = ZeroBasis

type DualVector (ZeroDim s)

Defined in Math.LinearMap.Category.Class

type DualVector (ZeroDim s) = ZeroDim s

type Interior (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

type Interior (ZeroDim k) = ZeroDim k

type Needle (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ZeroDim k) = ZeroDim k

type ChartIndex (ZeroDim s)

Defined in Data.Manifold.Atlas

type ChartIndex (ZeroDim s) = ()

type TensorProduct (ZeroDim s) v

Defined in Math.LinearMap.Category.Class

type TensorProduct (ZeroDim s) v = ZeroDim s

type ℝ = Double

type ℝ⁰ = ZeroDim ℝ

type ℝ¹ = V1 ℝ

type ℝ² = V2 ℝ

type ℝ³ = V3 ℝ

type ℝ⁴ = V4 ℝ

Hyperspheres

General form: Stiefel manifolds

newtype Stiefel1 v

Constructors

Stiefel1

Fields

• getStiefel1N :: DualVector v

Instances

Show (DualVector v) => Show (Stiefel1 v)

Defined in Data.Manifold.Types.Stiefel

Methods

showsPrec :: Int -> Stiefel1 v -> ShowS

show :: Stiefel1 v -> String

showList :: [Stiefel1 v] -> ShowS

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => Semimanifold (Stiefel1 v)

Defined in Data.Manifold.Types

Associated Types

type Needle (Stiefel1 v) :: Type

type Interior (Stiefel1 v) :: Type

Methods

(.+~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

fromInterior :: Interior (Stiefel1 v) -> Stiefel1 v

toInterior :: Stiefel1 v -> Maybe (Interior (Stiefel1 v))

translateP :: Tagged (Stiefel1 v) (Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Interior (Stiefel1 v))

(.-~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

semimanifoldWitness :: SemimanifoldWitness (Stiefel1 v)

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => PseudoAffine (Stiefel1 v)

Defined in Data.Manifold.Types

Methods

(.-~.) :: Stiefel1 v -> Stiefel1 v -> Maybe (Needle (Stiefel1 v))

(.-~!) :: Stiefel1 v -> Stiefel1 v -> Needle (Stiefel1 v)

pseudoAffineWitness :: PseudoAffineWitness (Stiefel1 v)

(Geodesic v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), LinearSpace v, Scalar v ~ ℝ, Geodesic (DualVector v), InnerSpace (DualVector v)) => Geodesic (Stiefel1 v)

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (D¹ -> Stiefel1 v)

geodesicWitness :: GeodesicWitness (Stiefel1 v)

middleBetween :: Stiefel1 v -> Stiefel1 v -> Maybe (Stiefel1 v)

type Interior (Stiefel1 v)

Defined in Data.Manifold.Types

type Interior (Stiefel1 v) = Stiefel1 v

type Needle (Stiefel1 v)

Defined in Data.Manifold.Types

type Needle (Stiefel1 v)

stiefel1Project

Arguments

:: LinearSpace v

=> DualVector v

Must be nonzero.

-> Stiefel1 v

stiefel1Embed :: (HilbertSpace v, RealFloat (Scalar v)) => Stiefel1 v -> v

Specific examples

class (PseudoAffine v, InnerSpace v, NaturallyEmbedded (UnitSphere v) (DualVector v)) => HasUnitSphere v where

Minimal complete definition

Nothing

Associated Types

type UnitSphere v :: *

Methods

stiefel :: UnitSphere v -> Stiefel1 v

unstiefel :: Stiefel1 v -> UnitSphere v

Instances

HasUnitSphere ℝ

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ :: Type

Methods

stiefel :: UnitSphere ℝ -> Stiefel1 ℝ

unstiefel :: Stiefel1 ℝ -> UnitSphere ℝ

HasUnitSphere ℝ³

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ³ :: Type

Methods

stiefel :: UnitSphere ℝ³ -> Stiefel1 ℝ³

unstiefel :: Stiefel1 ℝ³ -> UnitSphere ℝ³

HasUnitSphere ℝ²

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ² :: Type

Methods

stiefel :: UnitSphere ℝ² -> Stiefel1 ℝ²

unstiefel :: Stiefel1 ℝ² -> UnitSphere ℝ²

data S⁰

The zero-dimensional sphere is actually just two points. Implementation might therefore change to ℝ⁰ + ℝ⁰: the disjoint sum of two single-point spaces.

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

Eq S⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S⁰ -> S⁰ -> Bool

(/=) :: S⁰ -> S⁰ -> Bool

Show S⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S⁰ -> ShowS

show :: S⁰ -> String

showList :: [S⁰] -> ShowS

Generic S⁰

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S⁰ :: Type -> Type

Methods

from :: S⁰ -> Rep S⁰ x

to :: Rep S⁰ x -> S⁰

Function S⁰

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S⁰ -> b) -> S⁰ :-> b

Arbitrary S⁰

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen S⁰

shrink :: S⁰ -> [S⁰]

CoArbitrary S⁰

Defined in Data.Manifold.Types.Primitive

Methods

coarbitrary :: S⁰ -> Gen b -> Gen b

Binary S⁰

Defined in Data.Manifold.Types.Primitive

Methods

put :: S⁰ -> Put

get :: Get S⁰

putList :: [S⁰] -> Put

Semimanifold S⁰

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle S⁰ :: Type

type Interior S⁰ :: Type

Methods

(.+~^) :: Interior S⁰ -> Needle S⁰ -> S⁰

fromInterior :: Interior S⁰ -> S⁰

toInterior :: S⁰ -> Maybe (Interior S⁰)

translateP :: Tagged S⁰ (Interior S⁰ -> Needle S⁰ -> Interior S⁰)

(.-~^) :: Interior S⁰ -> Needle S⁰ -> S⁰

semimanifoldWitness :: SemimanifoldWitness S⁰

PseudoAffine S⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: S⁰ -> S⁰ -> Maybe (Needle S⁰)

(.-~!) :: S⁰ -> S⁰ -> Needle S⁰

pseudoAffineWitness :: PseudoAffineWitness S⁰

Show S⁰

Defined in Data.Manifold.Types.Primitive

Methods

showsPrec :: Int -> S⁰ -> ShowS

show :: S⁰ -> String

showList :: [S⁰] -> ShowS

Atlas S⁰

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S⁰ :: Type

Methods

chartReferencePoint :: ChartIndex S⁰ -> S⁰

interiorChartReferencePoint :: Functor p => p S⁰ -> ChartIndex S⁰ -> Interior S⁰

lookupAtlas :: S⁰ -> ChartIndex S⁰

Geodesic S⁰

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: S⁰ -> S⁰ -> Maybe (D¹ -> S⁰)

geodesicWitness :: GeodesicWitness S⁰

middleBetween :: S⁰ -> S⁰ -> Maybe S⁰

NaturallyEmbedded S⁰ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S⁰ -> ℝ

coEmbed :: ℝ -> S⁰

type Rep S⁰

Defined in Math.Manifold.Core.Types.Internal

type Rep S⁰ = D1 (MetaData "S\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "PositiveHalfSphere" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "NegativeHalfSphere" PrefixI False) (U1 :: Type -> Type))

type Interior S⁰

Defined in Math.Manifold.Core.PseudoAffine

type Interior S⁰ = S⁰

type Needle S⁰

Defined in Math.Manifold.Core.PseudoAffine

type Needle S⁰ = ZeroDim ℝ

type ChartIndex S⁰

Defined in Data.Manifold.Atlas

type ChartIndex S⁰ = S⁰

newtype S¹

The unit circle.

Constructors

S¹Polar

Fields

• φParamS¹ :: Double

  Must be in range [-π, π[.

Instances

Eq S¹

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S¹ -> S¹ -> Bool

(/=) :: S¹ -> S¹ -> Bool

Show S¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S¹ -> ShowS

show :: S¹ -> String

showList :: [S¹] -> ShowS

Generic S¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S¹ :: Type -> Type

Methods

from :: S¹ -> Rep S¹ x

to :: Rep S¹ x -> S¹

Function S¹

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S¹ -> b) -> S¹ :-> b

Arbitrary S¹

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen S¹

shrink :: S¹ -> [S¹]

CoArbitrary S¹

Defined in Data.Manifold.Types.Primitive

Methods

coarbitrary :: S¹ -> Gen b -> Gen b

Binary S¹

Defined in Data.Manifold.Types.Primitive

Methods

put :: S¹ -> Put

get :: Get S¹

putList :: [S¹] -> Put

Semimanifold S¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle S¹ :: Type

type Interior S¹ :: Type

Methods

(.+~^) :: Interior S¹ -> Needle S¹ -> S¹

fromInterior :: Interior S¹ -> S¹

toInterior :: S¹ -> Maybe (Interior S¹)

translateP :: Tagged S¹ (Interior S¹ -> Needle S¹ -> Interior S¹)

(.-~^) :: Interior S¹ -> Needle S¹ -> S¹

semimanifoldWitness :: SemimanifoldWitness S¹

PseudoAffine S¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: S¹ -> S¹ -> Maybe (Needle S¹)

(.-~!) :: S¹ -> S¹ -> Needle S¹

pseudoAffineWitness :: PseudoAffineWitness S¹

Show S¹

Defined in Data.Manifold.Types.Primitive

Methods

showsPrec :: Int -> S¹ -> ShowS

show :: S¹ -> String

showList :: [S¹] -> ShowS

Rotatable S¹

Defined in Math.Rotations.Class

Associated Types

type AxisSpace S¹ :: Type

Methods

rotateAbout :: AxisSpace S¹ -> S¹ -> S¹ -> S¹

Connected S¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S¹ -> S¹ -> Needle S¹

Atlas S¹

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S¹ :: Type

Methods

chartReferencePoint :: ChartIndex S¹ -> S¹

interiorChartReferencePoint :: Functor p => p S¹ -> ChartIndex S¹ -> Interior S¹

lookupAtlas :: S¹ -> ChartIndex S¹

Geodesic S¹

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: S¹ -> S¹ -> Maybe (D¹ -> S¹)

geodesicWitness :: GeodesicWitness S¹

middleBetween :: S¹ -> S¹ -> Maybe S¹

HasAzimuth S¹

Defined in Math.Manifold.Real.Coordinates

Methods

azimuth :: Coordinate S¹

CoordDifferential S¹

Defined in Math.Manifold.Real.Coordinates

Methods

delta :: CoordinateIdentifier S¹ -> Coordinate (TangentBundle S¹)

HasCoordinates S¹

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier S¹ :: Type

Methods

coordinateAsLens :: CoordinateIdentifier S¹ -> Lens' S¹ ℝ

validCoordinateRange :: CoordinateIdentifier S¹ -> S¹ -> (ℝ, ℝ)

NaturallyEmbedded S¹ ℝ²

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S¹ -> ℝ²

coEmbed :: ℝ² -> S¹

(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))

Eq (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

(==) :: CoordinateIdentifier S¹ -> CoordinateIdentifier S¹ -> Bool

(/=) :: CoordinateIdentifier S¹ -> CoordinateIdentifier S¹ -> Bool

Show (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

showsPrec :: Int -> CoordinateIdentifier S¹ -> ShowS

show :: CoordinateIdentifier S¹ -> String

showList :: [CoordinateIdentifier S¹] -> ShowS

Arbitrary (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

arbitrary :: Gen (CoordinateIdentifier S¹)

shrink :: CoordinateIdentifier S¹ -> [CoordinateIdentifier S¹]

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

type Rep S¹

Defined in Math.Manifold.Core.Types.Internal

type Rep S¹ = D1 (MetaData "S\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "S\185Polar" PrefixI True) (S1 (MetaSel (Just "\966ParamS\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Interior S¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior S¹ = S¹

type Needle S¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle S¹ = ℝ

type AxisSpace S¹

Defined in Math.Rotations.Class

type AxisSpace S¹ = ℝP⁰

type ChartIndex S¹

Defined in Data.Manifold.Atlas

type ChartIndex S¹ = S⁰

data CoordinateIdentifier S¹

Defined in Math.Manifold.Real.Coordinates

data CoordinateIdentifier S¹ = S¹Azimuth

pattern S¹ :: Double -> S¹

data S²

The ordinary unit sphere.

Constructors

S²Polar

Fields

• ϑParamS² :: !Double

  Range [0, π[.

• φParamS² :: !Double

  Range [-π, π[.

Instances

Eq S²

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S² -> S² -> Bool

(/=) :: S² -> S² -> Bool

Show S²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S² -> ShowS

show :: S² -> String

showList :: [S²] -> ShowS

Generic S²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S² :: Type -> Type

Methods

from :: S² -> Rep S² x

to :: Rep S² x -> S²

Function S²

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S² -> b) -> S² :-> b

Arbitrary S²

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen S²

shrink :: S² -> [S²]

CoArbitrary S²

Defined in Data.Manifold.Types.Primitive

Methods

coarbitrary :: S² -> Gen b -> Gen b

Binary S²

Defined in Data.Manifold.Types.Primitive

Methods

put :: S² -> Put

get :: Get S²

putList :: [S²] -> Put

Semimanifold S²

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle S² :: Type

type Interior S² :: Type

Methods

(.+~^) :: Interior S² -> Needle S² -> S²

fromInterior :: Interior S² -> S²

toInterior :: S² -> Maybe (Interior S²)

translateP :: Tagged S² (Interior S² -> Needle S² -> Interior S²)

(.-~^) :: Interior S² -> Needle S² -> S²

semimanifoldWitness :: SemimanifoldWitness S²

PseudoAffine S²

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: S² -> S² -> Maybe (Needle S²)

(.-~!) :: S² -> S² -> Needle S²

pseudoAffineWitness :: PseudoAffineWitness S²

Show S²

Defined in Data.Manifold.Types.Primitive

Methods

showsPrec :: Int -> S² -> ShowS

show :: S² -> String

showList :: [S²] -> ShowS

Rotatable S²

Defined in Math.Rotations.Class

Associated Types

type AxisSpace S² :: Type

Methods

rotateAbout :: AxisSpace S² -> S¹ -> S² -> S²

Connected S²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S² -> S² -> Needle S²

Atlas S²

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S² :: Type

Methods

chartReferencePoint :: ChartIndex S² -> S²

interiorChartReferencePoint :: Functor p => p S² -> ChartIndex S² -> Interior S²

lookupAtlas :: S² -> ChartIndex S²

HasZenithDistance S²

Defined in Math.Manifold.Real.Coordinates

Methods

zenithAngle :: Coordinate S²

HasAzimuth S²

Defined in Math.Manifold.Real.Coordinates

Methods

azimuth :: Coordinate S²

CoordDifferential S²

Defined in Math.Manifold.Real.Coordinates

Methods

delta :: CoordinateIdentifier S² -> Coordinate (TangentBundle S²)

HasCoordinates S²

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier S² :: Type

Methods

coordinateAsLens :: CoordinateIdentifier S² -> Lens' S² ℝ

validCoordinateRange :: CoordinateIdentifier S² -> S² -> (ℝ, ℝ)

NaturallyEmbedded S² ℝ³

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S² -> ℝ³

coEmbed :: ℝ³ -> S²

(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))

Eq (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

(==) :: CoordinateIdentifier S² -> CoordinateIdentifier S² -> Bool

(/=) :: CoordinateIdentifier S² -> CoordinateIdentifier S² -> Bool

Show (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

showsPrec :: Int -> CoordinateIdentifier S² -> ShowS

show :: CoordinateIdentifier S² -> String

showList :: [CoordinateIdentifier S²] -> ShowS

Arbitrary (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

arbitrary :: Gen (CoordinateIdentifier S²)

shrink :: CoordinateIdentifier S² -> [CoordinateIdentifier S²]

Rotatable (FibreBundle S² ℝ²)

Defined in Data.Manifold.FibreBundle

Associated Types

type AxisSpace (FibreBundle S² ℝ²) :: Type

Methods

rotateAbout :: AxisSpace (FibreBundle S² ℝ²) -> S¹ -> FibreBundle S² ℝ² -> FibreBundle S² ℝ²

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

type Rep S²

Defined in Math.Manifold.Core.Types.Internal

type Rep S² = D1 (MetaData "S\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "S\178Polar" PrefixI True) (S1 (MetaSel (Just "\977ParamS\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966ParamS\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

type Interior S²

Defined in Data.Manifold.PseudoAffine

type Interior S² = S²

type Needle S²

Defined in Data.Manifold.PseudoAffine

type Needle S² = ℝ²

type AxisSpace S²

Defined in Math.Rotations.Class

type AxisSpace S² = ℝP²

type ChartIndex S²

Defined in Data.Manifold.Atlas

type ChartIndex S² = S⁰

data CoordinateIdentifier S²

Defined in Math.Manifold.Real.Coordinates

data CoordinateIdentifier S²

• = S²ZenithAngle
• | S²Azimuth

type AxisSpace (FibreBundle S² ℝ²)

Defined in Data.Manifold.FibreBundle

type AxisSpace (FibreBundle S² ℝ²) = ℝP²

pattern S² :: Double -> Double -> S²

Projective spaces

data ℝP⁰

Constructors

ℝPZero

Instances

Eq ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: ℝP⁰ -> ℝP⁰ -> Bool

(/=) :: ℝP⁰ -> ℝP⁰ -> Bool

Show ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP⁰ -> ShowS

show :: ℝP⁰ -> String

showList :: [ℝP⁰] -> ShowS

Generic ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP⁰ :: Type -> Type

Methods

from :: ℝP⁰ -> Rep ℝP⁰ x

to :: Rep ℝP⁰ x -> ℝP⁰

Arbitrary ℝP⁰

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen ℝP⁰

shrink :: ℝP⁰ -> [ℝP⁰]

Binary ℝP⁰

Defined in Data.Manifold.Types.Primitive

Methods

put :: ℝP⁰ -> Put

get :: Get ℝP⁰

putList :: [ℝP⁰] -> Put

Semimanifold ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle ℝP⁰ :: Type

type Interior ℝP⁰ :: Type

Methods

(.+~^) :: Interior ℝP⁰ -> Needle ℝP⁰ -> ℝP⁰

fromInterior :: Interior ℝP⁰ -> ℝP⁰

toInterior :: ℝP⁰ -> Maybe (Interior ℝP⁰)

translateP :: Tagged ℝP⁰ (Interior ℝP⁰ -> Needle ℝP⁰ -> Interior ℝP⁰)

(.-~^) :: Interior ℝP⁰ -> Needle ℝP⁰ -> ℝP⁰

semimanifoldWitness :: SemimanifoldWitness ℝP⁰

PseudoAffine ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP⁰ -> ℝP⁰ -> Maybe (Needle ℝP⁰)

(.-~!) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

pseudoAffineWitness :: PseudoAffineWitness ℝP⁰

Connected ℝP⁰

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

type Rep ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP⁰ = D1 (MetaData "\8477P\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "\8477PZero" PrefixI False) (U1 :: Type -> Type))

type Interior ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

type Interior ℝP⁰ = ℝP⁰

type Needle ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

type Needle ℝP⁰ = ZeroDim ℝ

newtype ℝP¹

Constructors

HemisphereℝP¹Polar

Fields

• φParamℝP¹ :: Double

  Range [-π/2,π/2[.

Instances

Show ℝP¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP¹ -> ShowS

show :: ℝP¹ -> String

showList :: [ℝP¹] -> ShowS

Generic ℝP¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP¹ :: Type -> Type

Methods

from :: ℝP¹ -> Rep ℝP¹ x

to :: Rep ℝP¹ x -> ℝP¹

Arbitrary ℝP¹

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen ℝP¹

shrink :: ℝP¹ -> [ℝP¹]

Binary ℝP¹

Defined in Data.Manifold.Types.Primitive

Methods

put :: ℝP¹ -> Put

get :: Get ℝP¹

putList :: [ℝP¹] -> Put

Semimanifold ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle ℝP¹ :: Type

type Interior ℝP¹ :: Type

Methods

(.+~^) :: Interior ℝP¹ -> Needle ℝP¹ -> ℝP¹

fromInterior :: Interior ℝP¹ -> ℝP¹

toInterior :: ℝP¹ -> Maybe (Interior ℝP¹)

translateP :: Tagged ℝP¹ (Interior ℝP¹ -> Needle ℝP¹ -> Interior ℝP¹)

(.-~^) :: Interior ℝP¹ -> Needle ℝP¹ -> ℝP¹

semimanifoldWitness :: SemimanifoldWitness ℝP¹

PseudoAffine ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP¹ -> ℝP¹ -> Maybe (Needle ℝP¹)

(.-~!) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

pseudoAffineWitness :: PseudoAffineWitness ℝP¹

Connected ℝP¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

type Rep ℝP¹

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP¹ = D1 (MetaData "\8477P\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "Hemisphere\8477P\185Polar" PrefixI True) (S1 (MetaSel (Just "\966Param\8477P\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Interior ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior ℝP¹ = ℝP¹

type Needle ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle ℝP¹ = ℝ

pattern ℝP¹ :: Double -> ℝP¹

data ℝP²

The two-dimensional real projective space, implemented as a disk with opposing points on the rim glued together. Image this disk as the northern hemisphere of a unit sphere; ℝP² is the space of all straight lines passing through the origin of ℝ³, and each of these lines is represented by the point at which it passes through the hemisphere.

Constructors

HemisphereℝP²Polar

Fields

• ϑParamℝP² :: !Double

  Range [0, π/2].

• φParamℝP² :: !Double

  Range [-π, π[.

Instances

Show ℝP²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP² -> ShowS

show :: ℝP² -> String

showList :: [ℝP²] -> ShowS

Generic ℝP²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP² :: Type -> Type

Methods

from :: ℝP² -> Rep ℝP² x

to :: Rep ℝP² x -> ℝP²

Arbitrary ℝP²

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen ℝP²

shrink :: ℝP² -> [ℝP²]

Binary ℝP²

Defined in Data.Manifold.Types.Primitive

Methods

put :: ℝP² -> Put

get :: Get ℝP²

putList :: [ℝP²] -> Put

Semimanifold ℝP²

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle ℝP² :: Type

type Interior ℝP² :: Type

Methods

(.+~^) :: Interior ℝP² -> Needle ℝP² -> ℝP²

fromInterior :: Interior ℝP² -> ℝP²

toInterior :: ℝP² -> Maybe (Interior ℝP²)

translateP :: Tagged ℝP² (Interior ℝP² -> Needle ℝP² -> Interior ℝP²)

(.-~^) :: Interior ℝP² -> Needle ℝP² -> ℝP²

semimanifoldWitness :: SemimanifoldWitness ℝP²

PseudoAffine ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: ℝP² -> ℝP² -> Maybe (Needle ℝP²)

(.-~!) :: ℝP² -> ℝP² -> Needle ℝP²

pseudoAffineWitness :: PseudoAffineWitness ℝP²

Connected ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP² -> ℝP² -> Needle ℝP²

NaturallyEmbedded ℝP² ℝ³

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝP² -> ℝ³

coEmbed :: ℝ³ -> ℝP²

type Rep ℝP²

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP² = D1 (MetaData "\8477P\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "Hemisphere\8477P\178Polar" PrefixI True) (S1 (MetaSel (Just "\977Param\8477P\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966Param\8477P\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

type Interior ℝP²

Defined in Data.Manifold.PseudoAffine

type Interior ℝP² = ℝP²

type Needle ℝP²

Defined in Data.Manifold.PseudoAffine

type Needle ℝP² = ℝ²

pattern ℝP² :: Double -> Double -> ℝP²

Intervals/disks/cones

newtype D¹

The “one-dimensional disk” – really just the line segment between the two points -1 and 1 of S⁰, i.e. this is simply a closed interval.

Constructors

D¹

Fields

• xParamD¹ :: Double

  Range [-1, 1].

Instances

Show D¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D¹ -> ShowS

show :: D¹ -> String

showList :: [D¹] -> ShowS

Generic D¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep D¹ :: Type -> Type

Methods

from :: D¹ -> Rep D¹ x

to :: Rep D¹ x -> D¹

Arbitrary D¹

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen D¹

shrink :: D¹ -> [D¹]

Binary D¹

Defined in Data.Manifold.Types.Primitive

Methods

put :: D¹ -> Put

get :: Get D¹

putList :: [D¹] -> Put

Semimanifold D¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle D¹ :: Type

type Interior D¹ :: Type

Methods

(.+~^) :: Interior D¹ -> Needle D¹ -> D¹

fromInterior :: Interior D¹ -> D¹

toInterior :: D¹ -> Maybe (Interior D¹)

translateP :: Tagged D¹ (Interior D¹ -> Needle D¹ -> Interior D¹)

(.-~^) :: Interior D¹ -> Needle D¹ -> D¹

semimanifoldWitness :: SemimanifoldWitness D¹

PseudoAffine D¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: D¹ -> D¹ -> Maybe (Needle D¹)

(.-~!) :: D¹ -> D¹ -> Needle D¹

pseudoAffineWitness :: PseudoAffineWitness D¹

IntervalLike D¹

Defined in Data.Manifold.Riemannian

Methods

toClosedInterval :: D¹ -> D¹

NaturallyEmbedded D¹ ℝ

Defined in Data.Manifold.Types.Primitive

Methods

embed :: D¹ -> ℝ

coEmbed :: ℝ -> D¹

type Rep D¹

Defined in Math.Manifold.Core.Types.Internal

type Rep D¹ = D1 (MetaData "D\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "D\185" PrefixI True) (S1 (MetaSel (Just "xParamD\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Interior D¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior D¹ = ℝ

type Needle D¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle D¹ = ℝ

data D²

The standard, closed unit disk. Homeomorphic to the cone over S¹, but not in the the obvious, “flat” way. (In is not homeomorphic, despite the almost identical ADT definition, to the projective space ℝP²!)

Constructors

D²Polar

Fields

• rParamD² :: !Double

  Range [0, 1].

• φParamD² :: !Double

  Range [-π, π[.

Instances

Show D²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D² -> ShowS

show :: D² -> String

showList :: [D²] -> ShowS

Generic D²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep D² :: Type -> Type

Methods

from :: D² -> Rep D² x

to :: Rep D² x -> D²

Arbitrary D²

Defined in Data.Manifold.Types.Primitive

Methods

arbitrary :: Gen D²

shrink :: D² -> [D²]

Binary D²

Defined in Data.Manifold.Types.Primitive

Methods

put :: D² -> Put

get :: Get D²

putList :: [D²] -> Put

type Rep D²

Defined in Math.Manifold.Core.Types.Internal

type Rep D² = D1 (MetaData "D\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "D\178Polar" PrefixI True) (S1 (MetaSel (Just "rParamD\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966ParamD\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

pattern D² :: Double -> Double -> D²

type ℝay = Cℝay ℝ⁰

Better known as ℝ⁺ (which is not a legal Haskell name), the ray of positive numbers (including zero, i.e. closed on one end).

data CD¹ x

A (closed) cone over a space x is the product of x with the closed interval D¹ of “heights”, except on its “tip”: here, x is smashed to a single point.

This construct becomes (homeomorphic-to-) an actual geometric cone (and to D²) in the special case x = S¹.

Constructors

CD¹

Fields

• hParamCD¹ :: !Double

  Range [0, 1]

• pParamCD¹ :: !x

  Irrelevant at h = 0.

Instances

Show x => Show (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> CD¹ x -> ShowS

show :: CD¹ x -> String

showList :: [CD¹ x] -> ShowS

Generic (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (CD¹ x) :: Type -> Type

Methods

from :: CD¹ x -> Rep (CD¹ x) x0

to :: Rep (CD¹ x) x0 -> CD¹ x

Binary y => Binary (CD¹ y)

Defined in Data.Manifold.Types.Primitive

Methods

put :: CD¹ y -> Put

get :: Get (CD¹ y)

putList :: [CD¹ y] -> Put

ConeSemimfd m => Semimanifold (CD¹ m)

Defined in Data.Manifold.Cone

Associated Types

type Needle (CD¹ m) :: Type

type Interior (CD¹ m) :: Type

Methods

(.+~^) :: Interior (CD¹ m) -> Needle (CD¹ m) -> CD¹ m

fromInterior :: Interior (CD¹ m) -> CD¹ m

toInterior :: CD¹ m -> Maybe (Interior (CD¹ m))

translateP :: Tagged (CD¹ m) (Interior (CD¹ m) -> Needle (CD¹ m) -> Interior (CD¹ m))

(.-~^) :: Interior (CD¹ m) -> Needle (CD¹ m) -> CD¹ m

semimanifoldWitness :: SemimanifoldWitness (CD¹ m)

type Rep (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

type Rep (CD¹ x) = D1 (MetaData "CD\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "CD\185" PrefixI True) (S1 (MetaSel (Just "hParamCD\185") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "pParamCD\185") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 x)))

type Interior (CD¹ m)

Defined in Data.Manifold.Cone

type Interior (CD¹ m)

type Needle (CD¹ m)

Defined in Data.Manifold.Cone

type Needle (CD¹ m)

data Cℝay x

An open cone is homeomorphic to a closed cone without the “lid”, i.e. without the “last copy” of x, at the far end of the height interval. Since that means the height does not include its supremum, it is actually more natural to express it as the entire real ray, hence the name.

Constructors

Cℝay

Fields

• hParamCℝay :: !Double

  Range [0, ∞[

• pParamCℝay :: !x

  Irrelevant at h = 0.

Instances

Show x => Show (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> Cℝay x -> ShowS

show :: Cℝay x -> String

showList :: [Cℝay x] -> ShowS

Generic (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (Cℝay x) :: Type -> Type

Methods

from :: Cℝay x -> Rep (Cℝay x) x0

to :: Rep (Cℝay x) x0 -> Cℝay x

Binary y => Binary (Cℝay y)

Defined in Data.Manifold.Types.Primitive

Methods

put :: Cℝay y -> Put

get :: Get (Cℝay y)

putList :: [Cℝay y] -> Put

ConeSemimfd m => Semimanifold (Cℝay m)

Defined in Data.Manifold.Cone

Associated Types

type Needle (Cℝay m) :: Type

type Interior (Cℝay m) :: Type

Methods

(.+~^) :: Interior (Cℝay m) -> Needle (Cℝay m) -> Cℝay m

fromInterior :: Interior (Cℝay m) -> Cℝay m

toInterior :: Cℝay m -> Maybe (Interior (Cℝay m))

translateP :: Tagged (Cℝay m) (Interior (Cℝay m) -> Needle (Cℝay m) -> Interior (Cℝay m))

(.-~^) :: Interior (Cℝay m) -> Needle (Cℝay m) -> Cℝay m

semimanifoldWitness :: SemimanifoldWitness (Cℝay m)

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

type Rep (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

type Rep (Cℝay x) = D1 (MetaData "C\8477ay" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "C\8477ay" PrefixI True) (S1 (MetaSel (Just "hParamC\8477ay") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "pParamC\8477ay") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 x)))

type Interior (Cℝay m)

Defined in Data.Manifold.Cone

type Interior (Cℝay m)

type Needle (Cℝay m)

Defined in Data.Manifold.Cone

type Needle (Cℝay m)

Affine subspaces

Lines

data Line x

Constructors

Line

Fields

• lineHandle :: x
• lineDirection :: Stiefel1 (Needle' x)

lineAsPlaneIntersection :: forall x. (WithField ℝ Manifold x, FiniteDimensional (Needle' x)) => Line x -> [Cutplane x]

Hyperplanes

data Cutplane x

Oriented hyperplanes, naïvely generalised to PseudoAffine manifolds: Cutplane p w represents the set of all points q such that (q.-~.p) ^<.> w ≡ 0.

In vector spaces this is indeed a hyperplane; for general manifolds it should behave locally as a plane, globally as an (n−1)-dimensional submanifold.

Constructors

Cutplane

Fields

• sawHandle :: x
• cutNormal :: Stiefel1 (Needle x)

Instances

(Show x, Show (Needle' x)) => Show (Cutplane x)

Defined in Data.Manifold.Types

Methods

showsPrec :: Int -> Cutplane x -> ShowS

show :: Cutplane x -> String

showList :: [Cutplane x] -> ShowS

normalPlane

Arguments

:: x

Some point lying in the desired plane.

-> Needle' x

Co-vector perpendicular to the plane. Must be nonzero.

-> Cutplane x

fathomCutDistance

Arguments

:: (WithField ℝ PseudoAffine x, LinearSpace (Needle x))

=> Cutplane x

Hyperplane to measure the distance from.

-> Metric' x

Metric to use for measuring that distance. This can only be accurate if the metric is valid both around the cut-plane's sawHandle, and around the points you measure. (Strictly speaking, we would need parallel transport to ensure this).

-> x

Point to measure the distance to.

-> Maybe ℝ

A signed number, giving the distance from plane to point with indication on which side the point lies. Nothing if the point isn't reachable from the plane.

sideOfCut :: (WithField ℝ PseudoAffine x, LinearSpace (Needle x)) => Cutplane x -> x -> Maybe S⁰

cutPosBetween :: WithField ℝ Manifold x => Cutplane x -> (x, x) -> Maybe D¹

Linear mappings

data LinearMap s v w

The tensor product between one space's dual space and another space is the space spanned by vector–dual-vector pairs, in bra-ket notation written as

m = ∑ |w⟩⟨v|

Any linear mapping can be written as such a (possibly infinite) sum. The TensorProduct data structure only stores the linear independent parts though; for simple finite-dimensional spaces this means e.g. LinearMap ℝ ℝ³ ℝ³ effectively boils down to an ordinary matrix type, namely an array of column-vectors |w⟩.

(The ⟨v| dual-vectors are then simply assumed to come from the canonical basis.)

For bigger spaces, the tensor product may be implemented in a more efficient sparse structure; this can be defined in the TensorSpace instance.

Instances

(Show (SubBasis (DualVector u)), Show (SubBasis v)) => Show (SubBasis (LinearMap s u v))

Defined in Math.VectorSpace.Docile

Methods

showsPrec :: Int -> SubBasis (LinearMap s u v) -> ShowS

show :: SubBasis (LinearMap s u v) -> String

showList :: [SubBasis (LinearMap s u v)] -> ShowS

Num' s => Morphism (LinearMap s)

Defined in Math.LinearMap.Category.Class

Methods

first :: (ObjectPair (LinearMap s) b d, ObjectPair (LinearMap s) c d) => LinearMap s b c -> LinearMap s (b, d) (c, d)

second :: (ObjectPair (LinearMap s) d b, ObjectPair (LinearMap s) d c) => LinearMap s b c -> LinearMap s (d, b) (d, c)

(***) :: (ObjectPair (LinearMap s) b b', ObjectPair (LinearMap s) c c') => LinearMap s b c -> LinearMap s b' c' -> LinearMap s (b, b') (c, c')

Num' s => PreArrow (LinearMap s)

Defined in Math.LinearMap.Category.Class

Methods

(&&&) :: (Object (LinearMap s) b, ObjectPair (LinearMap s) c c') => LinearMap s b c -> LinearMap s b c' -> LinearMap s b (c, c')

terminal :: Object (LinearMap s) b => LinearMap s b (UnitObject (LinearMap s))

fst :: ObjectPair (LinearMap s) x y => LinearMap s (x, y) x

snd :: ObjectPair (LinearMap s) x y => LinearMap s (x, y) y

Category (LinearMap s)

Defined in Math.LinearMap.Category.Class

Associated Types

type Object (LinearMap s) o :: Constraint

Methods

id :: Object (LinearMap s) a => LinearMap s a a

(.) :: (Object (LinearMap s) a, Object (LinearMap s) b, Object (LinearMap s) c) => LinearMap s b c -> LinearMap s a b -> LinearMap s a c

Num' s => Cartesian (LinearMap s)

Defined in Math.LinearMap.Category.Class

Associated Types

type PairObjects (LinearMap s) a b :: Constraint

type UnitObject (LinearMap s) :: Type

Methods

swap :: (ObjectPair (LinearMap s) a b, ObjectPair (LinearMap s) b a) => LinearMap s (a, b) (b, a)

attachUnit :: (unit ~ UnitObject (LinearMap s), ObjectPair (LinearMap s) a unit) => LinearMap s a (a, unit)

detachUnit :: (unit ~ UnitObject (LinearMap s), ObjectPair (LinearMap s) a unit) => LinearMap s (a, unit) a

regroup :: (ObjectPair (LinearMap s) a b, ObjectPair (LinearMap s) b c, ObjectPair (LinearMap s) a (b, c), ObjectPair (LinearMap s) (a, b) c) => LinearMap s (a, (b, c)) ((a, b), c)

regroup' :: (ObjectPair (LinearMap s) a b, ObjectPair (LinearMap s) b c, ObjectPair (LinearMap s) a (b, c), ObjectPair (LinearMap s) (a, b) c) => LinearMap s ((a, b), c) (a, (b, c))

Num' s => EnhancedCat (LinearMap s) (LinearFunction s)

Defined in Math.LinearMap.Category.Class

Methods

arr :: (Object (LinearFunction s) b, Object (LinearFunction s) c, Object (LinearMap s) b, Object (LinearMap s) c) => LinearFunction s b c -> LinearMap s b c

Num' s => EnhancedCat (LinearFunction s) (LinearMap s)

Defined in Math.LinearMap.Category.Class

Methods

arr :: (Object (LinearMap s) b, Object (LinearMap s) c, Object (LinearFunction s) b, Object (LinearFunction s) c) => LinearMap s b c -> LinearFunction s b c

EnhancedCat (Affine s) (LinearMap s)

Defined in Data.Function.Affine

Methods

arr :: (Object (LinearMap s) b, Object (LinearMap s) c, Object (Affine s) b, Object (Affine s) c) => LinearMap s b c -> Affine s b c

Num' s => EnhancedCat ((->) :: Type -> Type -> Type) (LinearMap s)

Defined in Math.LinearMap.Category.Class

Methods

arr :: (Object (LinearMap s) b, Object (LinearMap s) c, Object (->) b, Object (->) c) => LinearMap s b c -> b -> c

(Num' s, LinearSpace v, Scalar v ~ s) => Monoidal (LinearMap s v) (LinearFunction s) (LinearFunction s)

Defined in Math.LinearMap.Category.Class

Methods

pureUnit :: LinearFunction s (UnitObject (LinearFunction s)) (LinearMap s v (UnitObject (LinearFunction s)))

fzipWith :: (ObjectPair (LinearFunction s) a b, Object (LinearFunction s) c, ObjectPair (LinearFunction s) (LinearMap s v a) (LinearMap s v b), Object (LinearFunction s) (LinearMap s v c)) => LinearFunction s (a, b) c -> LinearFunction s (LinearMap s v a, LinearMap s v b) (LinearMap s v c)

(LinearSpace v, Num' s, Scalar v ~ s) => Functor (LinearMap s v) (LinearFunction s) (LinearFunction s)

Defined in Math.LinearMap.Category.Class

Methods

fmap :: (Object (LinearFunction s) a, Object (LinearFunction s) (LinearMap s v a), Object (LinearFunction s) b, Object (LinearFunction s) (LinearMap s v b)) => LinearFunction s a b -> LinearFunction s (LinearMap s v a) (LinearMap s v b)

(LinearSpace v, Scalar v ~ s) => Functor (LinearMap s v) (Coercion :: Type -> Type -> Type) (Coercion :: Type -> Type -> Type)

Defined in Math.LinearMap.Category.Class

Methods

fmap :: (Object Coercion a, Object Coercion (LinearMap s v a), Object Coercion b, Object Coercion (LinearMap s v b)) => Coercion a b -> Coercion (LinearMap s v a) (LinearMap s v b)

(LinearSpace v, TensorSpace w, Scalar v ~ s, Scalar w ~ s) => AdditiveGroup (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

Methods

zeroV :: LinearMap s v w

(^+^) :: LinearMap s v w -> LinearMap s v w -> LinearMap s v w

negateV :: LinearMap s v w -> LinearMap s v w

(^-^) :: LinearMap s v w -> LinearMap s v w -> LinearMap s v w

(LinearSpace v, TensorSpace w, Scalar v ~ s, Scalar w ~ s) => VectorSpace (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

Associated Types

type Scalar (LinearMap s v w) :: Type

Methods

(*^) :: Scalar (LinearMap s v w) -> LinearMap s v w -> LinearMap s v w

(LinearSpace u, TensorSpace v, s ~ Scalar u, s ~ Scalar v) => AffineSpace (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

Associated Types

type Diff (LinearMap s u v) :: Type

Methods

(.-.) :: LinearMap s u v -> LinearMap s u v -> Diff (LinearMap s u v)

(.+^) :: LinearMap s u v -> Diff (LinearMap s u v) -> LinearMap s u v

(LinearSpace u, SemiInner (DualVector u), SemiInner v, Scalar u ~ s, Scalar v ~ s) => SemiInner (LinearMap s u v)

Defined in Math.VectorSpace.Docile

Methods

dualBasisCandidates :: [(Int, LinearMap s u v)] -> Forest (Int, DualVector (LinearMap s u v))

tensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (LinearMap s u v)) => [(Int, LinearMap s u v ⊗ w)] -> Forest (Int, DualVector (LinearMap s u v ⊗ w))

symTensorDualBasisCandidates :: [(Int, SymmetricTensor (Scalar (LinearMap s u v)) (LinearMap s u v))] -> Forest (Int, SymmetricTensor (Scalar (LinearMap s u v)) (DualVector (LinearMap s u v)))

symTensorTensorDualBasisCandidates :: (SemiInner w, Scalar w ~ Scalar (LinearMap s u v)) => [(Int, SymmetricTensor (Scalar (LinearMap s u v)) (LinearMap s u v) ⊗ w)] -> Forest (Int, SymmetricTensor (Scalar (LinearMap s u v)) (LinearMap s u v) +> DualVector w)

(LSpace u, FiniteDimensional (DualVector u), FiniteDimensional v, Scalar u ~ s, Scalar v ~ s, Scalar (DualVector v) ~ s, Fractional' (Scalar v)) => FiniteDimensional (LinearMap s u v)

Defined in Math.VectorSpace.Docile

Associated Types

data SubBasis (LinearMap s u v) :: Type

Methods

entireBasis :: SubBasis (LinearMap s u v)

enumerateSubBasis :: SubBasis (LinearMap s u v) -> [LinearMap s u v]

subbasisDimension :: SubBasis (LinearMap s u v) -> Int

decomposeLinMap :: (LSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v +> w) -> (SubBasis (LinearMap s u v), DList w)

decomposeLinMapWithin :: (LSpace w, Scalar w ~ Scalar (LinearMap s u v)) => SubBasis (LinearMap s u v) -> (LinearMap s u v +> w) -> Either (SubBasis (LinearMap s u v), DList w) (DList w)

recomposeSB :: SubBasis (LinearMap s u v) -> [Scalar (LinearMap s u v)] -> (LinearMap s u v, [Scalar (LinearMap s u v)])

recomposeSBTensor :: (FiniteDimensional w, Scalar w ~ Scalar (LinearMap s u v)) => SubBasis (LinearMap s u v) -> SubBasis w -> [Scalar (LinearMap s u v)] -> (LinearMap s u v ⊗ w, [Scalar (LinearMap s u v)])

recomposeLinMap :: (LSpace w, Scalar w ~ Scalar (LinearMap s u v)) => SubBasis (LinearMap s u v) -> [w] -> (LinearMap s u v +> w, [w])

recomposeContraLinMap :: (LinearSpace w, Scalar w ~ Scalar (LinearMap s u v), Functor f) => (f (Scalar w) -> w) -> f (DualVector (LinearMap s u v)) -> LinearMap s u v +> w

recomposeContraLinMapTensor :: (FiniteDimensional u0, LinearSpace w, Scalar u0 ~ Scalar (LinearMap s u v), Scalar w ~ Scalar (LinearMap s u v), Functor f) => (f (Scalar w) -> w) -> f (LinearMap s u v +> DualVector u0) -> (LinearMap s u v ⊗ u0) +> w

uncanonicallyFromDual :: DualVector (LinearMap s u v) -+> LinearMap s u v

uncanonicallyToDual :: LinearMap s u v -+> DualVector (LinearMap s u v)

(LinearSpace u, TensorSpace v, Scalar u ~ s, Scalar v ~ s) => TensorSpace (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

Associated Types

type TensorProduct (LinearMap s u v) w :: Type

Methods

scalarSpaceWitness :: ScalarSpaceWitness (LinearMap s u v)

linearManifoldWitness :: LinearManifoldWitness (LinearMap s u v)

zeroTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => LinearMap s u v ⊗ w

toFlatTensor :: LinearMap s u v -+> (LinearMap s u v ⊗ Scalar (LinearMap s u v))

fromFlatTensor :: (LinearMap s u v ⊗ Scalar (LinearMap s u v)) -+> LinearMap s u v

addTensors :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) -> (LinearMap s u v ⊗ w) -> LinearMap s u v ⊗ w

subtractTensors :: (TensorSpace (LinearMap s u v), TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) -> (LinearMap s u v ⊗ w) -> LinearMap s u v ⊗ w

scaleTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => Bilinear (Scalar (LinearMap s u v)) (LinearMap s u v ⊗ w) (LinearMap s u v ⊗ w)

negateTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) -+> (LinearMap s u v ⊗ w)

tensorProduct :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => Bilinear (LinearMap s u v) w (LinearMap s u v ⊗ w)

tensorProducts :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => [(LinearMap s u v, w)] -> LinearMap s u v ⊗ w

transposeTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) -+> (w ⊗ LinearMap s u v)

fmapTensor :: (TensorSpace w, TensorSpace x, Scalar w ~ Scalar (LinearMap s u v), Scalar x ~ Scalar (LinearMap s u v)) => Bilinear (w -+> x) (LinearMap s u v ⊗ w) (LinearMap s u v ⊗ x)

fzipTensorWith :: (TensorSpace u0, TensorSpace w, TensorSpace x, Scalar u0 ~ Scalar (LinearMap s u v), Scalar w ~ Scalar (LinearMap s u v), Scalar x ~ Scalar (LinearMap s u v)) => Bilinear ((w, x) -+> u0) (LinearMap s u v ⊗ w, LinearMap s u v ⊗ x) (LinearMap s u v ⊗ u0)

coerceFmapTensorProduct :: Functor p => p (LinearMap s u v) -> Coercion a b -> Coercion (TensorProduct (LinearMap s u v) a) (TensorProduct (LinearMap s u v) b)

wellDefinedVector :: LinearMap s u v -> Maybe (LinearMap s u v)

wellDefinedTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) -> Maybe (LinearMap s u v ⊗ w)

(LinearSpace u, LinearSpace v, Scalar u ~ s, Scalar v ~ s) => LinearSpace (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

Associated Types

type DualVector (LinearMap s u v) :: Type

Methods

dualSpaceWitness :: DualSpaceWitness (LinearMap s u v)

linearId :: LinearMap s u v +> LinearMap s u v

idTensor :: LinearMap s u v ⊗ DualVector (LinearMap s u v)

sampleLinearFunction :: (TensorSpace w, Scalar (LinearMap s u v) ~ Scalar w) => (LinearMap s u v -+> w) -+> (LinearMap s u v +> w)

toLinearForm :: DualVector (LinearMap s u v) -+> (LinearMap s u v +> Scalar (LinearMap s u v))

fromLinearForm :: (LinearMap s u v +> Scalar (LinearMap s u v)) -+> DualVector (LinearMap s u v)

coerceDoubleDual :: Coercion (LinearMap s u v) (DualVector (DualVector (LinearMap s u v)))

trace :: (LinearMap s u v +> LinearMap s u v) -+> Scalar (LinearMap s u v)

contractTensorMap :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v +> (LinearMap s u v ⊗ w)) -+> w

contractMapTensor :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ (LinearMap s u v +> w)) -+> w

contractTensorFn :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v -+> (LinearMap s u v ⊗ w)) -+> w

contractLinearMapAgainst :: (LinearSpace w, Scalar w ~ Scalar (LinearMap s u v)) => Bilinear (LinearMap s u v +> w) (w -+> LinearMap s u v) (Scalar (LinearMap s u v))

applyDualVector :: Bilinear (DualVector (LinearMap s u v)) (LinearMap s u v) (Scalar (LinearMap s u v))

applyLinear :: (TensorSpace w, Scalar w ~ Scalar (LinearMap s u v)) => Bilinear (LinearMap s u v +> w) (LinearMap s u v) w

composeLinear :: (LinearSpace w, TensorSpace x, Scalar w ~ Scalar (LinearMap s u v), Scalar x ~ Scalar (LinearMap s u v)) => Bilinear (w +> x) (LinearMap s u v +> w) (LinearMap s u v +> x)

tensorId :: (LinearSpace w, Scalar w ~ Scalar (LinearMap s u v)) => (LinearMap s u v ⊗ w) +> (LinearMap s u v ⊗ w)

applyTensorFunctional :: (LinearSpace u0, Scalar u0 ~ Scalar (LinearMap s u v)) => Bilinear (DualVector (LinearMap s u v ⊗ u0)) (LinearMap s u v ⊗ u0) (Scalar (LinearMap s u v))

applyTensorLinMap :: (LinearSpace u0, TensorSpace w, Scalar u0 ~ Scalar (LinearMap s u v), Scalar w ~ Scalar (LinearMap s u v)) => Bilinear ((LinearMap s u v ⊗ u0) +> w) (LinearMap s u v ⊗ u0) w

(LinearSpace v, TensorSpace w, Scalar v ~ s, Scalar w ~ s) => Semimanifold (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

Associated Types

type Needle (LinearMap s v w) :: Type

type Interior (LinearMap s v w) :: Type

Methods

(.+~^) :: Interior (LinearMap s v w) -> Needle (LinearMap s v w) -> LinearMap s v w

fromInterior :: Interior (LinearMap s v w) -> LinearMap s v w

toInterior :: LinearMap s v w -> Maybe (Interior (LinearMap s v w))

translateP :: Tagged (LinearMap s v w) (Interior (LinearMap s v w) -> Needle (LinearMap s v w) -> Interior (LinearMap s v w))

(.-~^) :: Interior (LinearMap s v w) -> Needle (LinearMap s v w) -> LinearMap s v w

semimanifoldWitness :: SemimanifoldWitness (LinearMap s v w)

(LinearSpace v, TensorSpace w, Scalar v ~ s, Scalar w ~ s) => PseudoAffine (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

Methods

(.-~.) :: LinearMap s v w -> LinearMap s v w -> Maybe (Needle (LinearMap s v w))

(.-~!) :: LinearMap s v w -> LinearMap s v w -> Needle (LinearMap s v w)

pseudoAffineWitness :: PseudoAffineWitness (LinearMap s v w)

(LinearSpace v, Scalar v ~ s, TensorSpace w, Scalar w ~ s) => Atlas (LinearMap s v w)

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex (LinearMap s v w) :: Type

Methods

chartReferencePoint :: ChartIndex (LinearMap s v w) -> LinearMap s v w

interiorChartReferencePoint :: Functor p => p (LinearMap s v w) -> ChartIndex (LinearMap s v w) -> Interior (LinearMap s v w)

lookupAtlas :: LinearMap s v w -> ChartIndex (LinearMap s v w)

(LinearSpace v, Scalar v ~ ℝ, TensorSpace w, Scalar w ~ ℝ) => Geodesic (LinearMap ℝ v w)

Defined in Data.Manifold.Riemannian

Methods

geodesicBetween :: LinearMap ℝ v w -> LinearMap ℝ v w -> Maybe (D¹ -> LinearMap ℝ v w)

geodesicWitness :: GeodesicWitness (LinearMap ℝ v w)

middleBetween :: LinearMap ℝ v w -> LinearMap ℝ v w -> Maybe (LinearMap ℝ v w)

(HilbertSpace v, SemiInner v, FiniteDimensional v, LtdErrorShow v, Scalar v ~ ℝ) => LtdErrorShow (LinearMap ℝ v (ℝ, ℝ))

Defined in Data.Manifold.Shade

Methods

ltdErrorShowWitness :: LtdErrorShowWitness (LinearMap ℝ v (ℝ, ℝ))

showsPrecShade'_errorLtdC :: Int -> Shade' (LinearMap ℝ v (ℝ, ℝ)) -> ShowS

prettyShowsPrecShade :: Int -> Shade (LinearMap ℝ v (ℝ, ℝ)) -> ShowS

prettyShowsPrecShade' :: Int -> Shade' (LinearMap ℝ v (ℝ, ℝ)) -> ShowS

(HilbertSpace v, SemiInner v, FiniteDimensional v, LtdErrorShow v, Scalar v ~ ℝ) => LtdErrorShow (LinearMap ℝ v ℝ)

Defined in Data.Manifold.Shade

Methods

ltdErrorShowWitness :: LtdErrorShowWitness (LinearMap ℝ v ℝ)

showsPrecShade'_errorLtdC :: Int -> Shade' (LinearMap ℝ v ℝ) -> ShowS

prettyShowsPrecShade :: Int -> Shade (LinearMap ℝ v ℝ) -> ShowS

prettyShowsPrecShade' :: Int -> Shade' (LinearMap ℝ v ℝ) -> ShowS

(SimpleSpace a, SimpleSpace b, Refinable a, Refinable b, Scalar a ~ ℝ, Scalar b ~ ℝ, Scalar (DualVector a) ~ ℝ, Scalar (DualVector b) ~ ℝ, Scalar (DualVector (DualVector a)) ~ ℝ, Scalar (DualVector (DualVector b)) ~ ℝ) => Refinable (LinearMap ℝ a b)

Defined in Data.Manifold.Shade

Methods

debugView :: Maybe (DebugView (LinearMap ℝ a b))

subShade' :: Shade' (LinearMap ℝ a b) -> Shade' (LinearMap ℝ a b) -> Bool

refineShade' :: Shade' (LinearMap ℝ a b) -> Shade' (LinearMap ℝ a b) -> Maybe (Shade' (LinearMap ℝ a b))

convolveMetric :: Functor p => p (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b) -> Metric (LinearMap ℝ a b)

convolveShade' :: Shade' (LinearMap ℝ a b) -> Shade' (Needle (LinearMap ℝ a b)) -> Shade' (LinearMap ℝ a b)

type UnitObject (LinearMap s)

Defined in Math.LinearMap.Category.Class

type UnitObject (LinearMap s) = ZeroDim s

type Object (LinearMap s) v

Defined in Math.LinearMap.Category.Class

type Object (LinearMap s) v = (LinearSpace v, Scalar v ~ s)

type PairObjects (LinearMap s) a b

Defined in Math.LinearMap.Category.Class

type PairObjects (LinearMap s) a b = ()

type Scalar (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

type Scalar (LinearMap s v w) = s

type Diff (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

type Diff (LinearMap s u v) = LinearMap s u v

data SubBasis (LinearMap s u v)

Defined in Math.VectorSpace.Docile

data SubBasis (LinearMap s u v) = LinMapBasis !(SubBasis (DualVector u)) !(SubBasis v)

type DualVector (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

type DualVector (LinearMap s u v) = Tensor s u (DualVector v)

type Interior (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

type Interior (LinearMap s v w) = LinearMap s v w

type Needle (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

type Needle (LinearMap s v w) = LinearMap s v w

type ChartIndex (LinearMap s v w)

Defined in Data.Manifold.Atlas

type ChartIndex (LinearMap s v w) = ()

type TensorProduct (LinearMap s u v) w

Defined in Math.LinearMap.Category.Class

type TensorProduct (LinearMap s u v) w = TensorProduct (DualVector u) (Tensor s v w)

type LocalLinear x y = LinearMap (Scalar (Needle x)) (Needle x) (Needle y)

Misc

type StiefelScalar s = (RealFloat s, Unbox s)

Orphan instances

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => Semimanifold (Stiefel1 v)

Associated Types

type Needle (Stiefel1 v) :: Type

type Interior (Stiefel1 v) :: Type

Methods

(.+~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

fromInterior :: Interior (Stiefel1 v) -> Stiefel1 v

toInterior :: Stiefel1 v -> Maybe (Interior (Stiefel1 v))

translateP :: Tagged (Stiefel1 v) (Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Interior (Stiefel1 v))

(.-~^) :: Interior (Stiefel1 v) -> Needle (Stiefel1 v) -> Stiefel1 v

semimanifoldWitness :: SemimanifoldWitness (Stiefel1 v)

(LinearSpace v, FiniteFreeSpace v, FiniteFreeSpace (DualVector v), StiefelScalar (Scalar v)) => PseudoAffine (Stiefel1 v)

Methods

(.-~.) :: Stiefel1 v -> Stiefel1 v -> Maybe (Needle (Stiefel1 v))

(.-~!) :: Stiefel1 v -> Stiefel1 v -> Needle (Stiefel1 v)

pseudoAffineWitness :: PseudoAffineWitness (Stiefel1 v)