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 => NaturallyEmbedded x (FibreBundle x f)

Defined in Data.Manifold.FibreBundle

Methods

embed :: x -> FibreBundle x f

coEmbed :: FibreBundle x f -> x

(CoordDifferential m, f ~ Needle 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 (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

(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]

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

(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

(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)

Methods

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

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

(ParallelTransporting (->) m 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)

(ParallelTransporting (->) m 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)

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)

(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

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

Defined in Data.Manifold.FibreBundle

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')

(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

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

Defined in Data.Manifold.FibreBundle

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)

Defined in Data.Manifold.FibreBundle

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)

Defined in Data.Manifold.FibreBundle

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 (S²_ s) (V2 s')) (FibreBundle (V3 s'') (V3 s'''))

Defined in Data.Manifold.FibreBundle

Methods

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

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

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

Defined in Data.Manifold.FibreBundle

Methods

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

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

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 m v, VectorSpace f) => NaturallyEmbedded (FibreBundle m (ZeroDim s)) (FibreBundle v f)

Defined in Data.Manifold.FibreBundle

Methods

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

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

(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

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-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "FibreBundle" 'PrefixI 'True) (S1 ('MetaSel ('Just "baseSpace") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 b) :*: S1 ('MetaSel ('Just "fibreSpace") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 f)))

data CoordinateIdentifier (FibreBundle b f)

Defined in Math.Manifold.Real.Coordinates

data CoordinateIdentifier (FibreBundle b f)

• = BaseSpaceCoordinate (CoordinateIdentifier b)
• | FibreSpaceCoordinate (b -> CoordinateIdentifier 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_ s) (V2 s'))

Defined in Data.Manifold.FibreBundle

type AxisSpace (FibreBundle (S²_ s) (V2 s')) = ℝP²_ s

type TangentBundle m = FibreBundle m (Needle m)

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

Trivial manifolds

data EmptyMfd v

The empty space can be considered a manifold with any sort of tangent space.

Instances

Empty (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

eliminate :: EmptyMfd v -> x

Eq (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: EmptyMfd v -> EmptyMfd v -> Bool

(/=) :: EmptyMfd v -> EmptyMfd v -> Bool

Ord (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

compare :: EmptyMfd v -> EmptyMfd v -> Ordering

(<) :: EmptyMfd v -> EmptyMfd v -> Bool

(<=) :: EmptyMfd v -> EmptyMfd v -> Bool

(>) :: EmptyMfd v -> EmptyMfd v -> Bool

(>=) :: EmptyMfd v -> EmptyMfd v -> Bool

max :: EmptyMfd v -> EmptyMfd v -> EmptyMfd v

min :: EmptyMfd v -> EmptyMfd v -> EmptyMfd v

(LinearSpace k, OpenManifold k, OpenManifold (Scalar k)) => SemimanifoldWithBoundary (EmptyMfd k)

Defined in Data.Manifold.WithBoundary.Class

Associated Types

type Interior (EmptyMfd k)

type Boundary (EmptyMfd k)

type HalfNeedle (EmptyMfd k)

Methods

(.+^|) :: EmptyMfd k -> Needle (Interior (EmptyMfd k)) -> Either (Boundary (EmptyMfd k), Scalar (Needle (Interior (EmptyMfd k)))) (Interior (EmptyMfd k))

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

fromBoundary :: Boundary (EmptyMfd k) -> EmptyMfd k

(|+^) :: Boundary (EmptyMfd k) -> HalfNeedle (EmptyMfd k) -> EmptyMfd k

separateInterior :: EmptyMfd k -> Either (Boundary (EmptyMfd k)) (Interior (EmptyMfd k))

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

extendToBoundary :: Interior (EmptyMfd k) -> Needle (Interior (EmptyMfd k)) -> Maybe (Boundary (EmptyMfd k))

smfdWBoundWitness :: SmfdWBoundWitness (EmptyMfd k)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (EmptyMfd k))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (EmptyMfd k)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (EmptyMfd k))), Scalar (Needle (Boundary (EmptyMfd k))) ~ Scalar (Needle (Interior (EmptyMfd k)))) => r) -> r

type Boundary (EmptyMfd k)

Defined in Data.Manifold.WithBoundary.Class

type Boundary (EmptyMfd k) = EmptyMfd k

type HalfNeedle (EmptyMfd k)

Defined in Data.Manifold.WithBoundary.Class

type HalfNeedle (EmptyMfd k) = ZeroDim (Scalar k)

type Interior (EmptyMfd k)

Defined in Data.Manifold.WithBoundary.Class

type Interior (EmptyMfd k) = EmptyMfd k

type Needle (EmptyMfd v)

Defined in Math.LinearMap.Category.Instances

type Needle (EmptyMfd v) = v

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

SemimanifoldWithBoundary ℝay

Defined in Data.Manifold.Cone

Associated Types

type Interior ℝay

type Boundary ℝay

type HalfNeedle ℝay

Methods

(.+^|) :: ℝay -> Needle (Interior ℝay) -> Either (Boundary ℝay, Scalar (Needle (Interior ℝay))) (Interior ℝay)

fromInterior :: Interior ℝay -> ℝay

fromBoundary :: Boundary ℝay -> ℝay

(|+^) :: Boundary ℝay -> HalfNeedle ℝay -> ℝay

separateInterior :: ℝay -> Either (Boundary ℝay) (Interior ℝay)

toInterior :: ℝay -> Maybe (Interior ℝay)

extendToBoundary :: Interior ℝay -> Needle (Interior ℝay) -> Maybe (Boundary ℝay)

smfdWBoundWitness :: SmfdWBoundWitness ℝay

needleIsOpenMfd :: (OpenManifold (Needle (Interior ℝay)) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior ℝay))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary ℝay)), Scalar (Needle (Boundary ℝay)) ~ Scalar (Needle (Interior ℝay))) => r) -> r

HasCoordinates ℝ⁰

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier ℝ⁰

Methods

coordinateAsLens :: CoordinateIdentifier ℝ⁰ -> Lens' ℝ⁰ ℝ

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

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

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)

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

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

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

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

Show (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

showsPrec :: Int -> ZeroDim s -> ShowS

show :: ZeroDim s -> String

showList :: [ZeroDim s] -> ShowS

Binary (ZeroDim a)

Defined in Data.Manifold.Types.Primitive

Methods

put :: ZeroDim a -> Put

get :: Get (ZeroDim a)

putList :: [ZeroDim a] -> Put

Eq (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Methods

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

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

Num k => AdditiveMonoid (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

Methods

zeroHV :: Cℝay (ZeroDim k)

addHVs :: Cℝay (ZeroDim k) -> Cℝay (ZeroDim k) -> Cℝay (ZeroDim k)

AdditiveMonoid (ZeroDim k)

Defined in Data.Monoid.Additive

Methods

zeroHV :: ZeroDim k

addHVs :: ZeroDim k -> ZeroDim k -> ZeroDim k

(Num k, VectorSpace k, Scalar k ~ k) => HalfSpace (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

Associated Types

type FullSubspace (Cℝay (ZeroDim k))

type Ray (Cℝay (ZeroDim k))

type MirrorJoin (Cℝay (ZeroDim k))

Methods

scaleNonNeg :: Ray (Cℝay (ZeroDim k)) -> Cℝay (ZeroDim k) -> Cℝay (ZeroDim k)

fromFullSubspace :: FullSubspace (Cℝay (ZeroDim k)) -> Cℝay (ZeroDim k)

projectToFullSubspace :: Cℝay (ZeroDim k) -> FullSubspace (Cℝay (ZeroDim k))

fullSubspaceIsVectorSpace :: ((VectorSpace (FullSubspace (Cℝay (ZeroDim k))), ScalarSpace (Scalar (FullSubspace (Cℝay (ZeroDim k)))), Scalar (FullSubspace (Cℝay (ZeroDim k))) ~ MirrorJoin (Ray (Cℝay (ZeroDim k)))) => r) -> r

rayIsHalfSpace :: (HalfSpace (Ray (Cℝay (ZeroDim k))) => r) -> r

mirrorJoinIsVectorSpace :: ((VectorSpace (MirrorJoin (Cℝay (ZeroDim k))), Scalar (MirrorJoin (Cℝay (ZeroDim k))) ~ MirrorJoin (Ray (Cℝay (ZeroDim k)))) => r) -> r

fromPositiveHalf :: Cℝay (ZeroDim k) -> MirrorJoin (Cℝay (ZeroDim k))

fromNegativeHalf :: Cℝay (ZeroDim k) -> MirrorJoin (Cℝay (ZeroDim k))

ScalarSpace k => HalfSpace (ZeroDim k)

Defined in Data.Monoid.Additive

Associated Types

type FullSubspace (ZeroDim k)

type Ray (ZeroDim k)

type MirrorJoin (ZeroDim k)

Methods

scaleNonNeg :: Ray (ZeroDim k) -> ZeroDim k -> ZeroDim k

fromFullSubspace :: FullSubspace (ZeroDim k) -> ZeroDim k

projectToFullSubspace :: ZeroDim k -> FullSubspace (ZeroDim k)

fullSubspaceIsVectorSpace :: ((VectorSpace (FullSubspace (ZeroDim k)), ScalarSpace (Scalar (FullSubspace (ZeroDim k))), Scalar (FullSubspace (ZeroDim k)) ~ MirrorJoin (Ray (ZeroDim k))) => r) -> r

rayIsHalfSpace :: (HalfSpace (Ray (ZeroDim k)) => r) -> r

mirrorJoinIsVectorSpace :: ((VectorSpace (MirrorJoin (ZeroDim k)), Scalar (MirrorJoin (ZeroDim k)) ~ MirrorJoin (Ray (ZeroDim k))) => r) -> r

fromPositiveHalf :: ZeroDim k -> MirrorJoin (ZeroDim k)

fromNegativeHalf :: ZeroDim k -> MirrorJoin (ZeroDim k)

Num' s => LinearSpace (ZeroDim s)

Defined in Math.LinearMap.Category.Class

Associated Types

type DualVector (ZeroDim s)

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 :: VSCCoercion (Scalar (ZeroDim s)) (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

useTupleLinearSpaceComponents :: ZeroDim s ~ (x, y) => ((LinearSpace x, LinearSpace y, Scalar x ~ Scalar y) => φ) -> φ

Num' s => TensorSpace (ZeroDim s)

Defined in Math.LinearMap.Category.Class

Associated Types

type TensorProduct (ZeroDim s) w

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)

tensorUnsafeFromArrayWithOffset :: forall w α (n :: Nat) (m :: Nat). (Dimensional n (ZeroDim s), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (ZeroDim s), Vector α (Scalar (ZeroDim s))) => Int -> α (Scalar (ZeroDim s)) -> ZeroDim s ⊗ w

tensorUnsafeWriteArrayWithOffset :: forall w (α :: Type -> Type) σ (n :: Nat) (m :: Nat). (Dimensional n (ZeroDim s), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (ZeroDim s), Vector α (Scalar (ZeroDim s))) => Mutable α σ (Scalar (ZeroDim s)) -> Int -> (ZeroDim s ⊗ w) -> ST σ ()

coerceFmapTensorProduct :: (Functor p, TensorSpace a, Scalar a ~ Scalar (ZeroDim s), TensorSpace b, Scalar b ~ Scalar (ZeroDim s)) => p (ZeroDim s) -> VSCCoercion (Scalar (ZeroDim s)) 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 => FiniteDimensional (ZeroDim s)

Defined in Math.VectorSpace.Docile

Associated Types

data SubBasis (ZeroDim s)

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)

tensorEquality :: (TensorSpace w, Eq w, Scalar w ~ Scalar (ZeroDim s)) => (ZeroDim s ⊗ w) -> (ZeroDim s ⊗ w) -> Bool

dualFinitenessWitness :: DualFinitenessWitness (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)]

(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)

TensorDecomposable (ZeroDim ℝ)

Defined in Math.VectorSpace.Docile

Methods

tensorDecomposition :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim ℝ)) => (ZeroDim ℝ ⊗ w) -> [(Basis (ZeroDim ℝ), w)]

tensorDecompose' :: (TensorSpace w, Scalar w ~ Scalar (ZeroDim ℝ)) => (ZeroDim ℝ ⊗ w) -> Basis (ZeroDim ℝ) -> w

showsPrecBasis :: Int -> Basis (ZeroDim ℝ) -> ShowS

NumPrime s => Atlas (ZeroDim s)

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex (ZeroDim s)

Methods

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

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

(Num' s, OpenManifold s) => Geodesic (ZeroDim s)

Defined in Data.Manifold.Riemannian

Methods

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

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

(Num' k, ProjectableBoundary k, OpenManifold k) => ProjectableBoundary (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

Methods

projectToBoundary :: ZeroDim k -> Boundary (ZeroDim k) -> Maybe (Needle (Boundary (ZeroDim k)), Scalar (Needle (Interior (ZeroDim k))))

marginFromBoundary :: Boundary (ZeroDim k) -> Scalar (Needle (Interior (ZeroDim k))) -> ZeroDim k

needleBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Needle (Interior (ZeroDim k))) => r) -> r

scalarBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Scalar (Needle (Interior (ZeroDim k)))) => r) -> r

(Num' k, OpenManifold k) => PseudoAffineWithBoundary (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

Methods

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

(.-|) :: ZeroDim k -> Boundary (ZeroDim k) -> Maybe (HalfNeedle (ZeroDim k))

(!-|) :: ZeroDim k -> Boundary (ZeroDim k) -> HalfNeedle (ZeroDim k)

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

SemimanifoldWithBoundary (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

Associated Types

type Interior (CD¹ ℝ⁰)

type Boundary (CD¹ ℝ⁰)

type HalfNeedle (CD¹ ℝ⁰)

Methods

(.+^|) :: CD¹ ℝ⁰ -> Needle (Interior (CD¹ ℝ⁰)) -> Either (Boundary (CD¹ ℝ⁰), Scalar (Needle (Interior (CD¹ ℝ⁰)))) (Interior (CD¹ ℝ⁰))

fromInterior :: Interior (CD¹ ℝ⁰) -> CD¹ ℝ⁰

fromBoundary :: Boundary (CD¹ ℝ⁰) -> CD¹ ℝ⁰

(|+^) :: Boundary (CD¹ ℝ⁰) -> HalfNeedle (CD¹ ℝ⁰) -> CD¹ ℝ⁰

separateInterior :: CD¹ ℝ⁰ -> Either (Boundary (CD¹ ℝ⁰)) (Interior (CD¹ ℝ⁰))

toInterior :: CD¹ ℝ⁰ -> Maybe (Interior (CD¹ ℝ⁰))

extendToBoundary :: Interior (CD¹ ℝ⁰) -> Needle (Interior (CD¹ ℝ⁰)) -> Maybe (Boundary (CD¹ ℝ⁰))

smfdWBoundWitness :: SmfdWBoundWitness (CD¹ ℝ⁰)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (CD¹ ℝ⁰))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (CD¹ ℝ⁰)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (CD¹ ℝ⁰))), Scalar (Needle (Boundary (CD¹ ℝ⁰))) ~ Scalar (Needle (Interior (CD¹ ℝ⁰)))) => r) -> r

(Num' k, OpenManifold k) => SemimanifoldWithBoundary (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

Associated Types

type Interior (ZeroDim k)

type Boundary (ZeroDim k)

type HalfNeedle (ZeroDim k)

Methods

(.+^|) :: ZeroDim k -> Needle (Interior (ZeroDim k)) -> Either (Boundary (ZeroDim k), Scalar (Needle (Interior (ZeroDim k)))) (Interior (ZeroDim k))

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

fromBoundary :: Boundary (ZeroDim k) -> ZeroDim k

(|+^) :: Boundary (ZeroDim k) -> HalfNeedle (ZeroDim k) -> ZeroDim k

separateInterior :: ZeroDim k -> Either (Boundary (ZeroDim k)) (Interior (ZeroDim k))

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

extendToBoundary :: Interior (ZeroDim k) -> Needle (Interior (ZeroDim k)) -> Maybe (Boundary (ZeroDim k))

smfdWBoundWitness :: SmfdWBoundWitness (ZeroDim k)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (ZeroDim k))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (ZeroDim k)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (ZeroDim k))), Scalar (Needle (Boundary (ZeroDim k))) ~ Scalar (Needle (Interior (ZeroDim k)))) => r) -> r

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)

Semimanifold (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ZeroDim k)

Methods

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

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

semimanifoldWitness :: SemimanifoldWitness (ZeroDim k)

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

AffineSpace (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Associated Types

type Diff (ZeroDim s)

Methods

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

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

HasBasis (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

Associated Types

type Basis (ZeroDim s)

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)

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)

Methods

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

(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)))

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)

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)

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)

(Num s, s ~ s') => NaturallyEmbedded (ZeroDim s) (ZeroDim s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ZeroDim s -> ZeroDim s'

coEmbed :: ZeroDim s' -> ZeroDim s

(PseudoAffine m, s ~ Scalar (Needle m), Num' s) => ParallelTransporting (->) 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 (ZeroDim s)) (FibreBundle v f)

Defined in Data.Manifold.FibreBundle

Methods

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

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

type Boundary ℝay

Defined in Data.Manifold.Cone

type Boundary ℝay = ℝ⁰

type HalfNeedle ℝay

Defined in Data.Manifold.Cone

type HalfNeedle ℝay = ℝay

type Interior ℝay

Defined in Data.Manifold.Cone

type Interior ℝay = ℝ

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 FullSubspace (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type FullSubspace (Cℝay (ZeroDim k)) = ZeroDim k

type FullSubspace (ZeroDim k)

Defined in Data.Monoid.Additive

type FullSubspace (ZeroDim k) = ZeroDim k

type MirrorJoin (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type MirrorJoin (Cℝay (ZeroDim k)) = k

type MirrorJoin (ZeroDim k)

Defined in Data.Monoid.Additive

type MirrorJoin (ZeroDim k) = ZeroDim k

type Ray (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type Ray (Cℝay (ZeroDim k)) = Cℝay (ZeroDim k)

type Ray (ZeroDim k)

Defined in Data.Monoid.Additive

type Ray (ZeroDim k) = Cℝay (ZeroDim k)

type DualVector (ZeroDim s)

Defined in Math.LinearMap.Category.Class

type DualVector (ZeroDim s) = ZeroDim s

type StaticDimension (ZeroDim s)

Defined in Math.LinearMap.Category.Class

type StaticDimension (ZeroDim s) = 'Just 0

data SubBasis (ZeroDim s)

Defined in Math.VectorSpace.Docile

data SubBasis (ZeroDim s) = ZeroBasis

type ChartIndex (ZeroDim s)

Defined in Data.Manifold.Atlas

type ChartIndex (ZeroDim s) = ()

type Boundary (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type Boundary (CD¹ ℝ⁰) = S⁰

type Boundary (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

type Boundary (ZeroDim k) = EmptyMfd (ZeroDim k)

type HalfNeedle (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type HalfNeedle (CD¹ ℝ⁰) = ℝay

type HalfNeedle (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

type HalfNeedle (ZeroDim k) = ZeroDim k

type Interior (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type Interior (CD¹ ℝ⁰) = ℝ

type Interior (ZeroDim k)

Defined in Data.Manifold.WithBoundary.Class

type Interior (ZeroDim k) = ZeroDim k

type Needle (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ZeroDim k) = ZeroDim k

type Diff (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Diff (ZeroDim s) = ZeroDim s

type Basis (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Basis (ZeroDim s) = Void

type Scalar (ZeroDim s)

Defined in Math.Manifold.VectorSpace.ZeroDimensional

type Scalar (ZeroDim s) = s

type TensorProduct (ZeroDim s) v

Defined in Math.LinearMap.Category.Class

type TensorProduct (ZeroDim s) v = ZeroDim s

Linear manifolds

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)) => 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)

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

Defined in Data.Manifold.Types

Associated Types

type Needle (Stiefel1 v)

Methods

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

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

semimanifoldWitness :: SemimanifoldWitness (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 ℝ²

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ²

Methods

stiefel :: UnitSphere ℝ² -> Stiefel1 ℝ²

unstiefel :: Stiefel1 ℝ² -> UnitSphere ℝ²

HasUnitSphere ℝ³

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ³

Methods

stiefel :: UnitSphere ℝ³ -> Stiefel1 ℝ³

unstiefel :: Stiefel1 ℝ³ -> UnitSphere ℝ³

HasUnitSphere ℝ

Defined in Data.Manifold.Cone

Associated Types

type UnitSphere ℝ

Methods

stiefel :: UnitSphere ℝ -> Stiefel1 ℝ

unstiefel :: Stiefel1 ℝ -> UnitSphere ℝ

type S⁰ = S⁰_ Double

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

data S⁰_ r

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

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

Function S⁰

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S⁰ -> b) -> S⁰ :-> b

Binary S⁰

Defined in Data.Manifold.Types.Primitive

Methods

put :: S⁰ -> Put

get :: Get S⁰

putList :: [S⁰] -> Put

Atlas S⁰

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S⁰

Methods

chartReferencePoint :: ChartIndex S⁰ -> S⁰

lookupAtlas :: S⁰ -> ChartIndex S⁰

Geodesic S⁰

Defined in Data.Manifold.Riemannian

Methods

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

middleBetween :: S⁰ -> S⁰ -> Maybe S⁰

Show S⁰

Defined in Data.Manifold.Types.Primitive

Methods

showsPrec :: Int -> S⁰ -> ShowS

show :: S⁰ -> String

showList :: [S⁰] -> ShowS

Generic (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S⁰_ r) :: Type -> Type

Methods

from :: S⁰_ r -> Rep (S⁰_ r) x

to :: Rep (S⁰_ r) x -> S⁰_ r

Show (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S⁰_ r -> ShowS

show :: S⁰_ r -> String

showList :: [S⁰_ r] -> ShowS

Eq (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S⁰_ r -> S⁰_ r -> Bool

(/=) :: S⁰_ r -> S⁰_ r -> Bool

SemimanifoldWithBoundary (Cℝay S⁰)

Defined in Data.Manifold.Cone

Associated Types

type Interior (Cℝay S⁰)

type Boundary (Cℝay S⁰)

type HalfNeedle (Cℝay S⁰)

Methods

(.+^|) :: Cℝay S⁰ -> Needle (Interior (Cℝay S⁰)) -> Either (Boundary (Cℝay S⁰), Scalar (Needle (Interior (Cℝay S⁰)))) (Interior (Cℝay S⁰))

fromInterior :: Interior (Cℝay S⁰) -> Cℝay S⁰

fromBoundary :: Boundary (Cℝay S⁰) -> Cℝay S⁰

(|+^) :: Boundary (Cℝay S⁰) -> HalfNeedle (Cℝay S⁰) -> Cℝay S⁰

separateInterior :: Cℝay S⁰ -> Either (Boundary (Cℝay S⁰)) (Interior (Cℝay S⁰))

toInterior :: Cℝay S⁰ -> Maybe (Interior (Cℝay S⁰))

extendToBoundary :: Interior (Cℝay S⁰) -> Needle (Interior (Cℝay S⁰)) -> Maybe (Boundary (Cℝay S⁰))

smfdWBoundWitness :: SmfdWBoundWitness (Cℝay S⁰)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (Cℝay S⁰))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (Cℝay S⁰)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (Cℝay S⁰))), Scalar (Needle (Boundary (Cℝay S⁰))) ~ Scalar (Needle (Interior (Cℝay S⁰)))) => r) -> r

RealFloat'' s => SemimanifoldWithBoundary (S⁰_ s)

Defined in Data.Manifold.WithBoundary

Associated Types

type Interior (S⁰_ s)

type Boundary (S⁰_ s)

type HalfNeedle (S⁰_ s)

Methods

(.+^|) :: S⁰_ s -> Needle (Interior (S⁰_ s)) -> Either (Boundary (S⁰_ s), Scalar (Needle (Interior (S⁰_ s)))) (Interior (S⁰_ s))

fromInterior :: Interior (S⁰_ s) -> S⁰_ s

fromBoundary :: Boundary (S⁰_ s) -> S⁰_ s

(|+^) :: Boundary (S⁰_ s) -> HalfNeedle (S⁰_ s) -> S⁰_ s

separateInterior :: S⁰_ s -> Either (Boundary (S⁰_ s)) (Interior (S⁰_ s))

toInterior :: S⁰_ s -> Maybe (Interior (S⁰_ s))

extendToBoundary :: Interior (S⁰_ s) -> Needle (Interior (S⁰_ s)) -> Maybe (Boundary (S⁰_ s))

smfdWBoundWitness :: SmfdWBoundWitness (S⁰_ s)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (S⁰_ s))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (S⁰_ s)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (S⁰_ s))), Scalar (Needle (Boundary (S⁰_ s))) ~ Scalar (Needle (Interior (S⁰_ s)))) => r) -> r

RealFloat' r => PseudoAffine (S⁰_ r)

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: S⁰_ r -> S⁰_ r -> Maybe (Needle (S⁰_ r))

(.-~!) :: S⁰_ r -> S⁰_ r -> Needle (S⁰_ r)

pseudoAffineWitness :: PseudoAffineWitness (S⁰_ r)

RealFloat' r => Semimanifold (S⁰_ r)

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle (S⁰_ r)

Methods

(.+~^) :: S⁰_ r -> Needle (S⁰_ r) -> S⁰_ r

(.-~^) :: S⁰_ r -> Needle (S⁰_ r) -> S⁰_ r

semimanifoldWitness :: SemimanifoldWitness (S⁰_ r)

(RealFloat s, VectorSpace s, s' ~ s) => NaturallyEmbedded (S⁰_ s) s'

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S⁰_ s -> s'

coEmbed :: s' -> S⁰_ s

type ChartIndex S⁰

Defined in Data.Manifold.Atlas

type ChartIndex S⁰ = S⁰

type Rep (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S⁰_ r) = D1 ('MetaData "S\8304_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "PositiveHalfSphere" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "NegativeHalfSphere" 'PrefixI 'False) (U1 :: Type -> Type))

type Boundary (Cℝay S⁰)

Defined in Data.Manifold.Cone

type Boundary (Cℝay S⁰) = EmptyMfd ℝ⁰

type Boundary (S⁰_ s)

Defined in Data.Manifold.WithBoundary

type Boundary (S⁰_ s) = EmptyMfd (ZeroDim s)

type HalfNeedle (Cℝay S⁰)

Defined in Data.Manifold.Cone

type HalfNeedle (Cℝay S⁰) = ℝay

type HalfNeedle (S⁰_ s)

Defined in Data.Manifold.WithBoundary

type HalfNeedle (S⁰_ s) = ZeroDim s

type Interior (Cℝay S⁰)

Defined in Data.Manifold.Cone

type Interior (Cℝay S⁰) = ℝ

type Interior (S⁰_ s)

Defined in Data.Manifold.WithBoundary

type Interior (S⁰_ s) = S⁰_ s

type Needle (S⁰_ r)

Defined in Data.Manifold.PseudoAffine

type Needle (S⁰_ r) = ZeroDim r

type S¹ = S¹_ Double

The unit circle.

newtype S¹_ r

Constructors

S¹Polar

Fields

• φParamS¹ :: r

  Must be in range [-π, π[.

Instances

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

Function S¹

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S¹ -> b) -> S¹ :-> b

Binary S¹

Defined in Data.Manifold.Types.Primitive

Methods

put :: S¹ -> Put

get :: Get S¹

putList :: [S¹] -> Put

Atlas S¹

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S¹

Methods

chartReferencePoint :: ChartIndex S¹ -> S¹

lookupAtlas :: S¹ -> ChartIndex S¹

Connected S¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S¹ -> S¹ -> Needle S¹

Geodesic S¹

Defined in Data.Manifold.Riemannian

Methods

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

middleBetween :: S¹ -> S¹ -> Maybe S¹

CoordDifferential S¹

Defined in Math.Manifold.Real.Coordinates

Methods

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

HasAzimuth S¹

Defined in Math.Manifold.Real.Coordinates

Methods

azimuth :: Coordinate S¹

HasCoordinates S¹

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier S¹

Methods

coordinateAsLens :: CoordinateIdentifier S¹ -> Lens' S¹ ℝ

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

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

Methods

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

Arbitrary (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

arbitrary :: Gen (CoordinateIdentifier S¹)

shrink :: CoordinateIdentifier S¹ -> [CoordinateIdentifier S¹]

Generic (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S¹_ r) :: Type -> Type

Methods

from :: S¹_ r -> Rep (S¹_ r) x

to :: Rep (S¹_ r) x -> S¹_ r

Show (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

showsPrec :: Int -> CoordinateIdentifier S¹ -> ShowS

show :: CoordinateIdentifier S¹ -> String

showList :: [CoordinateIdentifier S¹] -> ShowS

Show r => Show (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S¹_ r -> ShowS

show :: S¹_ r -> String

showList :: [S¹_ r] -> ShowS

Eq (CoordinateIdentifier S¹)

Defined in Math.Manifold.Real.Coordinates

Methods

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

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

(Eq r, RealFloat r) => Eq (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S¹_ r -> S¹_ r -> Bool

(/=) :: S¹_ r -> S¹_ r -> Bool

RealFloat'' s => ProjectableBoundary (S¹_ s)

Defined in Data.Manifold.WithBoundary

Methods

projectToBoundary :: S¹_ s -> Boundary (S¹_ s) -> Maybe (Needle (Boundary (S¹_ s)), Scalar (Needle (Interior (S¹_ s))))

marginFromBoundary :: Boundary (S¹_ s) -> Scalar (Needle (Interior (S¹_ s))) -> S¹_ s

needleBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Needle (Interior (S¹_ s))) => r) -> r

scalarBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Scalar (Needle (Interior (S¹_ s)))) => r) -> r

RealFloat'' s => PseudoAffineWithBoundary (S¹_ s)

Defined in Data.Manifold.WithBoundary

Methods

(.--!) :: S¹_ s -> S¹_ s -> Needle (Interior (S¹_ s))

(.-|) :: S¹_ s -> Boundary (S¹_ s) -> Maybe (HalfNeedle (S¹_ s))

(!-|) :: S¹_ s -> Boundary (S¹_ s) -> HalfNeedle (S¹_ s)

(.--.) :: S¹_ s -> S¹_ s -> Maybe (Needle (Interior (S¹_ s)))

RealFloat'' s => SemimanifoldWithBoundary (S¹_ s)

Defined in Data.Manifold.WithBoundary

Associated Types

type Interior (S¹_ s)

type Boundary (S¹_ s)

type HalfNeedle (S¹_ s)

Methods

(.+^|) :: S¹_ s -> Needle (Interior (S¹_ s)) -> Either (Boundary (S¹_ s), Scalar (Needle (Interior (S¹_ s)))) (Interior (S¹_ s))

fromInterior :: Interior (S¹_ s) -> S¹_ s

fromBoundary :: Boundary (S¹_ s) -> S¹_ s

(|+^) :: Boundary (S¹_ s) -> HalfNeedle (S¹_ s) -> S¹_ s

separateInterior :: S¹_ s -> Either (Boundary (S¹_ s)) (Interior (S¹_ s))

toInterior :: S¹_ s -> Maybe (Interior (S¹_ s))

extendToBoundary :: Interior (S¹_ s) -> Needle (Interior (S¹_ s)) -> Maybe (Boundary (S¹_ s))

smfdWBoundWitness :: SmfdWBoundWitness (S¹_ s)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (S¹_ s))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (S¹_ s)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (S¹_ s))), Scalar (Needle (Boundary (S¹_ s))) ~ Scalar (Needle (Interior (S¹_ s)))) => r) -> r

RealFloat' r => PseudoAffine (S¹_ r)

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: S¹_ r -> S¹_ r -> Maybe (Needle (S¹_ r))

(.-~!) :: S¹_ r -> S¹_ r -> Needle (S¹_ r)

pseudoAffineWitness :: PseudoAffineWitness (S¹_ r)

RealFloat' r => Semimanifold (S¹_ r)

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle (S¹_ r)

Methods

(.+~^) :: S¹_ r -> Needle (S¹_ r) -> S¹_ r

(.-~^) :: S¹_ r -> Needle (S¹_ r) -> S¹_ r

semimanifoldWitness :: SemimanifoldWitness (S¹_ r)

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

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S¹_ s -> V2 s'

coEmbed :: V2 s' -> S¹_ s

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

Defined in Data.Manifold.FibreBundle

Methods

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

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

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

type AxisSpace S¹

Defined in Math.Rotations.Class

type AxisSpace S¹ = ℝP⁰

type Rep (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S¹_ r) = D1 ('MetaData "S\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'True) (C1 ('MetaCons "S\185Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\966ParamS\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

type Boundary (S¹_ s)

Defined in Data.Manifold.WithBoundary

type Boundary (S¹_ s) = EmptyMfd (ZeroDim s)

type HalfNeedle (S¹_ s)

Defined in Data.Manifold.WithBoundary

type HalfNeedle (S¹_ s)

type Interior (S¹_ s)

Defined in Data.Manifold.WithBoundary

type Interior (S¹_ s) = S¹_ s

type Needle (S¹_ r)

Defined in Data.Manifold.PseudoAffine

type Needle (S¹_ r) = r

pattern S¹ :: Double -> S¹

type S² = S²_ Double

The ordinary unit sphere.

data S²_ r

Constructors

S²Polar

Fields

• ϑParamS² :: !r

  Range [0, π[.

• φParamS² :: !r

  Range [-π, π[.

Instances

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

Function S²

Defined in Data.Manifold.Types.Primitive

Methods

function :: (S² -> b) -> S² :-> b

Binary S²

Defined in Data.Manifold.Types.Primitive

Methods

put :: S² -> Put

get :: Get S²

putList :: [S²] -> Put

Atlas S²

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex S²

Methods

chartReferencePoint :: ChartIndex S² -> S²

lookupAtlas :: S² -> ChartIndex S²

Connected S²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S² -> S² -> Needle S²

CoordDifferential S²

Defined in Math.Manifold.Real.Coordinates

Methods

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

HasAzimuth S²

Defined in Math.Manifold.Real.Coordinates

Methods

azimuth :: Coordinate S²

HasCoordinates S²

Defined in Math.Manifold.Real.Coordinates

Associated Types

data CoordinateIdentifier S²

Methods

coordinateAsLens :: CoordinateIdentifier S² -> Lens' S² ℝ

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

HasZenithDistance S²

Defined in Math.Manifold.Real.Coordinates

Methods

zenithAngle :: Coordinate 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

Methods

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

Arbitrary (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

arbitrary :: Gen (CoordinateIdentifier S²)

shrink :: CoordinateIdentifier S² -> [CoordinateIdentifier S²]

Generic (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S²_ r) :: Type -> Type

Methods

from :: S²_ r -> Rep (S²_ r) x

to :: Rep (S²_ r) x -> S²_ r

Show (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

showsPrec :: Int -> CoordinateIdentifier S² -> ShowS

show :: CoordinateIdentifier S² -> String

showList :: [CoordinateIdentifier S²] -> ShowS

Show r => Show (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S²_ r -> ShowS

show :: S²_ r -> String

showList :: [S²_ r] -> ShowS

Eq (CoordinateIdentifier S²)

Defined in Math.Manifold.Real.Coordinates

Methods

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

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

(Eq r, RealFloat r) => Eq (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S²_ r -> S²_ r -> Bool

(/=) :: S²_ r -> S²_ r -> Bool

RealFloat'' s => ProjectableBoundary (S²_ s)

Defined in Data.Manifold.WithBoundary

Methods

projectToBoundary :: S²_ s -> Boundary (S²_ s) -> Maybe (Needle (Boundary (S²_ s)), Scalar (Needle (Interior (S²_ s))))

marginFromBoundary :: Boundary (S²_ s) -> Scalar (Needle (Interior (S²_ s))) -> S²_ s

needleBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Needle (Interior (S²_ s))) => r) -> r

scalarBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Scalar (Needle (Interior (S²_ s)))) => r) -> r

RealFloat'' s => PseudoAffineWithBoundary (S²_ s)

Defined in Data.Manifold.WithBoundary

Methods

(.--!) :: S²_ s -> S²_ s -> Needle (Interior (S²_ s))

(.-|) :: S²_ s -> Boundary (S²_ s) -> Maybe (HalfNeedle (S²_ s))

(!-|) :: S²_ s -> Boundary (S²_ s) -> HalfNeedle (S²_ s)

(.--.) :: S²_ s -> S²_ s -> Maybe (Needle (Interior (S²_ s)))

RealFloat'' s => SemimanifoldWithBoundary (S²_ s)

Defined in Data.Manifold.WithBoundary

Associated Types

type Interior (S²_ s)

type Boundary (S²_ s)

type HalfNeedle (S²_ s)

Methods

(.+^|) :: S²_ s -> Needle (Interior (S²_ s)) -> Either (Boundary (S²_ s), Scalar (Needle (Interior (S²_ s)))) (Interior (S²_ s))

fromInterior :: Interior (S²_ s) -> S²_ s

fromBoundary :: Boundary (S²_ s) -> S²_ s

(|+^) :: Boundary (S²_ s) -> HalfNeedle (S²_ s) -> S²_ s

separateInterior :: S²_ s -> Either (Boundary (S²_ s)) (Interior (S²_ s))

toInterior :: S²_ s -> Maybe (Interior (S²_ s))

extendToBoundary :: Interior (S²_ s) -> Needle (Interior (S²_ s)) -> Maybe (Boundary (S²_ s))

smfdWBoundWitness :: SmfdWBoundWitness (S²_ s)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (S²_ s))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (S²_ s)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (S²_ s))), Scalar (Needle (Boundary (S²_ s))) ~ Scalar (Needle (Interior (S²_ s)))) => r) -> r

RealFloat' s => PseudoAffine (S²_ s)

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: S²_ s -> S²_ s -> Maybe (Needle (S²_ s))

(.-~!) :: S²_ s -> S²_ s -> Needle (S²_ s)

pseudoAffineWitness :: PseudoAffineWitness (S²_ s)

RealFloat' s => Semimanifold (S²_ s)

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle (S²_ s)

Methods

(.+~^) :: S²_ s -> Needle (S²_ s) -> S²_ s

(.-~^) :: S²_ s -> Needle (S²_ s) -> S²_ s

semimanifoldWitness :: SemimanifoldWitness (S²_ s)

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

Defined in Data.Manifold.Types.Primitive

Methods

embed :: S²_ s -> V3 s'

coEmbed :: V3 s' -> S²_ s

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

Defined in Data.Manifold.FibreBundle

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')

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

Defined in Data.Manifold.FibreBundle

Methods

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

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

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 S²

Defined in Math.Rotations.Class

type AxisSpace S² = ℝP²

type Rep (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S²_ r) = D1 ('MetaData "S\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "S\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\977ParamS\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966ParamS\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

type Boundary (S²_ s)

Defined in Data.Manifold.WithBoundary

type Boundary (S²_ s) = EmptyMfd s

type HalfNeedle (S²_ s)

Defined in Data.Manifold.WithBoundary

type HalfNeedle (S²_ s)

type Interior (S²_ s)

Defined in Data.Manifold.WithBoundary

type Interior (S²_ s) = S²_ s

type Needle (S²_ s)

Defined in Data.Manifold.PseudoAffine

type Needle (S²_ s) = V2 s

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

Defined in Data.Manifold.FibreBundle

type AxisSpace (FibreBundle (S²_ s) (V2 s')) = ℝP²_ s

pattern S² :: Double -> Double -> S²

Projective spaces

type ℝP⁰ = ℝP⁰_ Double

data ℝP⁰_ r

Constructors

ℝPZero

Instances

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

Connected ℝP⁰

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

Generic (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP⁰_ r) :: Type -> Type

Methods

from :: ℝP⁰_ r -> Rep (ℝP⁰_ r) x

to :: Rep (ℝP⁰_ r) x -> ℝP⁰_ r

Show (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP⁰_ r -> ShowS

show :: ℝP⁰_ r -> String

showList :: [ℝP⁰_ r] -> ShowS

Eq (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: ℝP⁰_ r -> ℝP⁰_ r -> Bool

(/=) :: ℝP⁰_ r -> ℝP⁰_ r -> Bool

PseudoAffine (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP⁰_ r -> ℝP⁰_ r -> Maybe (Needle (ℝP⁰_ r))

(.-~!) :: ℝP⁰_ r -> ℝP⁰_ r -> Needle (ℝP⁰_ r)

pseudoAffineWitness :: PseudoAffineWitness (ℝP⁰_ r)

Semimanifold (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ℝP⁰_ r)

Methods

(.+~^) :: ℝP⁰_ r -> Needle (ℝP⁰_ r) -> ℝP⁰_ r

(.-~^) :: ℝP⁰_ r -> Needle (ℝP⁰_ r) -> ℝP⁰_ r

semimanifoldWitness :: SemimanifoldWitness (ℝP⁰_ r)

type Rep (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP⁰_ r) = D1 ('MetaData "\8477P\8304_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "\8477PZero" 'PrefixI 'False) (U1 :: Type -> Type))

type Needle (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ℝP⁰_ r) = ZeroDim r

type ℝP¹ = ℝP¹_ Double

newtype ℝP¹_ r

Constructors

HemisphereℝP¹Polar

Fields

• φParamℝP¹ :: r

  Range [-π/2,π/2[.

Instances

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

Connected ℝP¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

Generic (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP¹_ r) :: Type -> Type

Methods

from :: ℝP¹_ r -> Rep (ℝP¹_ r) x

to :: Rep (ℝP¹_ r) x -> ℝP¹_ r

Show r => Show (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP¹_ r -> ShowS

show :: ℝP¹_ r -> String

showList :: [ℝP¹_ r] -> ShowS

ℝeal r => PseudoAffine (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP¹_ r -> ℝP¹_ r -> Maybe (Needle (ℝP¹_ r))

(.-~!) :: ℝP¹_ r -> ℝP¹_ r -> Needle (ℝP¹_ r)

pseudoAffineWitness :: PseudoAffineWitness (ℝP¹_ r)

ℝeal r => Semimanifold (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ℝP¹_ r)

Methods

(.+~^) :: ℝP¹_ r -> Needle (ℝP¹_ r) -> ℝP¹_ r

(.-~^) :: ℝP¹_ r -> Needle (ℝP¹_ r) -> ℝP¹_ r

semimanifoldWitness :: SemimanifoldWitness (ℝP¹_ r)

type Rep (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP¹_ r) = D1 ('MetaData "\8477P\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'True) (C1 ('MetaCons "Hemisphere\8477P\185Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\966Param\8477P\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

type Needle (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ℝP¹_ r) = r

pattern ℝP¹ :: Double -> ℝP¹

type ℝP² = ℝP²_ Double

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.

data ℝP²_ r

Constructors

HemisphereℝP²Polar

Fields

• ϑParamℝP² :: !r

  Range [0, π/2].

• φParamℝP² :: !r

  Range [-π, π[.

Instances

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

Connected ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP² -> ℝP² -> Needle ℝP²

PseudoAffine ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

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

(.-~!) :: ℝP² -> ℝP² -> Needle ℝP²

pseudoAffineWitness :: PseudoAffineWitness ℝP²

Semimanifold ℝP²

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle ℝP²

Methods

(.+~^) :: ℝP² -> Needle ℝP² -> ℝP²

(.-~^) :: ℝP² -> Needle ℝP² -> ℝP²

semimanifoldWitness :: SemimanifoldWitness ℝP²

Generic (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP²_ r) :: Type -> Type

Methods

from :: ℝP²_ r -> Rep (ℝP²_ r) x

to :: Rep (ℝP²_ r) x -> ℝP²_ r

Show r => Show (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP²_ r -> ShowS

show :: ℝP²_ r -> String

showList :: [ℝP²_ r] -> ShowS

(RealFloat s, s' ~ s) => NaturallyEmbedded (ℝP²_ s) (V3 s')

Defined in Data.Manifold.Types.Primitive

Methods

embed :: ℝP²_ s -> V3 s'

coEmbed :: V3 s' -> ℝP²_ s

type Needle ℝP²

Defined in Data.Manifold.PseudoAffine

type Needle ℝP² = ℝ²

type Rep (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP²_ r) = D1 ('MetaData "\8477P\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "Hemisphere\8477P\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\977Param\8477P\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966Param\8477P\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

pattern ℝP² :: Double -> Double -> ℝP²

Intervals/disks/cones

type D¹ = D¹_ Double

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.

newtype D¹_ r

Constructors

D¹

Fields

• xParamD¹ :: r

  Range [-1, 1].

Instances

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

IntervalLike D¹

Defined in Data.Manifold.Riemannian

Methods

toClosedInterval :: D¹ -> D¹

Generic (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (D¹_ r) :: Type -> Type

Methods

from :: D¹_ r -> Rep (D¹_ r) x

to :: Rep (D¹_ r) x -> D¹_ r

Show r => Show (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D¹_ r -> ShowS

show :: D¹_ r -> String

showList :: [D¹_ r] -> ShowS

RealFloat'' s => SemimanifoldWithBoundary (D¹_ s)

Defined in Data.Manifold.WithBoundary

Associated Types

type Interior (D¹_ s)

type Boundary (D¹_ s)

type HalfNeedle (D¹_ s)

Methods

(.+^|) :: D¹_ s -> Needle (Interior (D¹_ s)) -> Either (Boundary (D¹_ s), Scalar (Needle (Interior (D¹_ s)))) (Interior (D¹_ s))

fromInterior :: Interior (D¹_ s) -> D¹_ s

fromBoundary :: Boundary (D¹_ s) -> D¹_ s

(|+^) :: Boundary (D¹_ s) -> HalfNeedle (D¹_ s) -> D¹_ s

separateInterior :: D¹_ s -> Either (Boundary (D¹_ s)) (Interior (D¹_ s))

toInterior :: D¹_ s -> Maybe (Interior (D¹_ s))

extendToBoundary :: Interior (D¹_ s) -> Needle (Interior (D¹_ s)) -> Maybe (Boundary (D¹_ s))

smfdWBoundWitness :: SmfdWBoundWitness (D¹_ s)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (D¹_ s))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (D¹_ s)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (D¹_ s))), Scalar (Needle (Boundary (D¹_ s))) ~ Scalar (Needle (Interior (D¹_ s)))) => r) -> r

(RealFloat s, VectorSpace s, s' ~ s) => NaturallyEmbedded (D¹_ s) s'

Defined in Data.Manifold.Types.Primitive

Methods

embed :: D¹_ s -> s'

coEmbed :: s' -> D¹_ s

type Rep (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (D¹_ r) = D1 ('MetaData "D\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'True) (C1 ('MetaCons "D\185" 'PrefixI 'True) (S1 ('MetaSel ('Just "xParamD\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

type Boundary (D¹_ s)

Defined in Data.Manifold.WithBoundary

type Boundary (D¹_ s) = S⁰_ s

type HalfNeedle (D¹_ s)

Defined in Data.Manifold.WithBoundary

type HalfNeedle (D¹_ s)

type Interior (D¹_ s)

Defined in Data.Manifold.WithBoundary

type Interior (D¹_ s) = s

type D² = D²_ Double

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!)

data D²_ r

Constructors

D²Polar

Fields

• rParamD² :: !r

  Range [0, 1].

• φParamD² :: !r

  Range [-π, π[.

Instances

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

Generic (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (D²_ r) :: Type -> Type

Methods

from :: D²_ r -> Rep (D²_ r) x

to :: Rep (D²_ r) x -> D²_ r

Show r => Show (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D²_ r -> ShowS

show :: D²_ r -> String

showList :: [D²_ r] -> ShowS

type Rep (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (D²_ r) = D1 ('MetaData "D\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "D\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "rParamD\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966ParamD\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

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 :: !(Scalar (Needle x))

  Range [0, 1]

• pParamCD¹ :: !x

  Irrelevant at h = 0.

Instances

Generic (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Rep (CD¹ x) :: Type -> Type

Methods

from :: CD¹ x -> Rep (CD¹ x) x0

to :: Rep (CD¹ x) x0 -> CD¹ x

(Show x, Show (Scalar (Needle x))) => Show (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

showsPrec :: Int -> CD¹ x -> ShowS

show :: CD¹ x -> String

showList :: [CD¹ x] -> ShowS

(Binary y, Binary (Scalar (Needle y))) => Binary (CD¹ y)

Defined in Data.Manifold.Types.Primitive

Methods

put :: CD¹ y -> Put

get :: Get (CD¹ y)

putList :: [CD¹ y] -> Put

SemimanifoldWithBoundary (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

Associated Types

type Interior (CD¹ ℝ⁰)

type Boundary (CD¹ ℝ⁰)

type HalfNeedle (CD¹ ℝ⁰)

Methods

(.+^|) :: CD¹ ℝ⁰ -> Needle (Interior (CD¹ ℝ⁰)) -> Either (Boundary (CD¹ ℝ⁰), Scalar (Needle (Interior (CD¹ ℝ⁰)))) (Interior (CD¹ ℝ⁰))

fromInterior :: Interior (CD¹ ℝ⁰) -> CD¹ ℝ⁰

fromBoundary :: Boundary (CD¹ ℝ⁰) -> CD¹ ℝ⁰

(|+^) :: Boundary (CD¹ ℝ⁰) -> HalfNeedle (CD¹ ℝ⁰) -> CD¹ ℝ⁰

separateInterior :: CD¹ ℝ⁰ -> Either (Boundary (CD¹ ℝ⁰)) (Interior (CD¹ ℝ⁰))

toInterior :: CD¹ ℝ⁰ -> Maybe (Interior (CD¹ ℝ⁰))

extendToBoundary :: Interior (CD¹ ℝ⁰) -> Needle (Interior (CD¹ ℝ⁰)) -> Maybe (Boundary (CD¹ ℝ⁰))

smfdWBoundWitness :: SmfdWBoundWitness (CD¹ ℝ⁰)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (CD¹ ℝ⁰))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (CD¹ ℝ⁰)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (CD¹ ℝ⁰))), Scalar (Needle (Boundary (CD¹ ℝ⁰))) ~ Scalar (Needle (Interior (CD¹ ℝ⁰)))) => r) -> r

type Rep (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

type Rep (CD¹ x) = D1 ('MetaData "CD\185" "Math.Manifold.Core.PseudoAffine" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "CD\185" 'PrefixI 'True) (S1 ('MetaSel ('Just "hParamCD\185") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 (Scalar (Needle x))) :*: S1 ('MetaSel ('Just "pParamCD\185") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 x)))

type Boundary (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type Boundary (CD¹ ℝ⁰) = S⁰

type HalfNeedle (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type HalfNeedle (CD¹ ℝ⁰) = ℝay

type Interior (CD¹ ℝ⁰)

Defined in Data.Manifold.Cone

type Interior (CD¹ ℝ⁰) = ℝ

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 :: !(Scalar (Needle x))

  Range [0, ∞[

• pParamCℝay :: !x

  Irrelevant at h = 0.

Instances

SemimanifoldWithBoundary ℝay

Defined in Data.Manifold.Cone

Associated Types

type Interior ℝay

type Boundary ℝay

type HalfNeedle ℝay

Methods

(.+^|) :: ℝay -> Needle (Interior ℝay) -> Either (Boundary ℝay, Scalar (Needle (Interior ℝay))) (Interior ℝay)

fromInterior :: Interior ℝay -> ℝay

fromBoundary :: Boundary ℝay -> ℝay

(|+^) :: Boundary ℝay -> HalfNeedle ℝay -> ℝay

separateInterior :: ℝay -> Either (Boundary ℝay) (Interior ℝay)

toInterior :: ℝay -> Maybe (Interior ℝay)

extendToBoundary :: Interior ℝay -> Needle (Interior ℝay) -> Maybe (Boundary ℝay)

smfdWBoundWitness :: SmfdWBoundWitness ℝay

needleIsOpenMfd :: (OpenManifold (Needle (Interior ℝay)) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior ℝay))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary ℝay)), Scalar (Needle (Boundary ℝay)) ~ Scalar (Needle (Interior ℝay))) => r) -> r

Generic (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Rep (Cℝay x) :: Type -> Type

Methods

from :: Cℝay x -> Rep (Cℝay x) x0

to :: Rep (Cℝay x) x0 -> Cℝay x

(Show x, Show (Scalar (Needle x))) => Show (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

showsPrec :: Int -> Cℝay x -> ShowS

show :: Cℝay x -> String

showList :: [Cℝay x] -> ShowS

(Binary y, Binary (Scalar (Needle y))) => Binary (Cℝay y)

Defined in Data.Manifold.Types.Primitive

Methods

put :: Cℝay y -> Put

get :: Get (Cℝay y)

putList :: [Cℝay y] -> Put

Num k => AdditiveMonoid (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

Methods

zeroHV :: Cℝay (ZeroDim k)

addHVs :: Cℝay (ZeroDim k) -> Cℝay (ZeroDim k) -> Cℝay (ZeroDim k)

(Num k, VectorSpace k, Scalar k ~ k) => HalfSpace (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

Associated Types

type FullSubspace (Cℝay (ZeroDim k))

type Ray (Cℝay (ZeroDim k))

type MirrorJoin (Cℝay (ZeroDim k))

Methods

scaleNonNeg :: Ray (Cℝay (ZeroDim k)) -> Cℝay (ZeroDim k) -> Cℝay (ZeroDim k)

fromFullSubspace :: FullSubspace (Cℝay (ZeroDim k)) -> Cℝay (ZeroDim k)

projectToFullSubspace :: Cℝay (ZeroDim k) -> FullSubspace (Cℝay (ZeroDim k))

fullSubspaceIsVectorSpace :: ((VectorSpace (FullSubspace (Cℝay (ZeroDim k))), ScalarSpace (Scalar (FullSubspace (Cℝay (ZeroDim k)))), Scalar (FullSubspace (Cℝay (ZeroDim k))) ~ MirrorJoin (Ray (Cℝay (ZeroDim k)))) => r) -> r

rayIsHalfSpace :: (HalfSpace (Ray (Cℝay (ZeroDim k))) => r) -> r

mirrorJoinIsVectorSpace :: ((VectorSpace (MirrorJoin (Cℝay (ZeroDim k))), Scalar (MirrorJoin (Cℝay (ZeroDim k))) ~ MirrorJoin (Ray (Cℝay (ZeroDim k)))) => r) -> r

fromPositiveHalf :: Cℝay (ZeroDim k) -> MirrorJoin (Cℝay (ZeroDim k))

fromNegativeHalf :: Cℝay (ZeroDim k) -> MirrorJoin (Cℝay (ZeroDim k))

SemimanifoldWithBoundary (Cℝay S⁰)

Defined in Data.Manifold.Cone

Associated Types

type Interior (Cℝay S⁰)

type Boundary (Cℝay S⁰)

type HalfNeedle (Cℝay S⁰)

Methods

(.+^|) :: Cℝay S⁰ -> Needle (Interior (Cℝay S⁰)) -> Either (Boundary (Cℝay S⁰), Scalar (Needle (Interior (Cℝay S⁰)))) (Interior (Cℝay S⁰))

fromInterior :: Interior (Cℝay S⁰) -> Cℝay S⁰

fromBoundary :: Boundary (Cℝay S⁰) -> Cℝay S⁰

(|+^) :: Boundary (Cℝay S⁰) -> HalfNeedle (Cℝay S⁰) -> Cℝay S⁰

separateInterior :: Cℝay S⁰ -> Either (Boundary (Cℝay S⁰)) (Interior (Cℝay S⁰))

toInterior :: Cℝay S⁰ -> Maybe (Interior (Cℝay S⁰))

extendToBoundary :: Interior (Cℝay S⁰) -> Needle (Interior (Cℝay S⁰)) -> Maybe (Boundary (Cℝay S⁰))

smfdWBoundWitness :: SmfdWBoundWitness (Cℝay S⁰)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (Cℝay S⁰))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (Cℝay S⁰)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (Cℝay S⁰))), Scalar (Needle (Boundary (Cℝay S⁰))) ~ Scalar (Needle (Interior (Cℝay S⁰)))) => r) -> r

(Real s, NaturallyEmbedded x p, s ~ Scalar (Needle x)) => NaturallyEmbedded (Cℝay x) (p, s)

Defined in Data.Manifold.Types.Primitive

Methods

embed :: Cℝay x -> (p, s)

coEmbed :: (p, s) -> Cℝay x

type Boundary ℝay

Defined in Data.Manifold.Cone

type Boundary ℝay = ℝ⁰

type HalfNeedle ℝay

Defined in Data.Manifold.Cone

type HalfNeedle ℝay = ℝay

type Interior ℝay

Defined in Data.Manifold.Cone

type Interior ℝay = ℝ

type Rep (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

type Rep (Cℝay x) = D1 ('MetaData "C\8477ay" "Math.Manifold.Core.PseudoAffine" "manifolds-core-0.6.1.0-KuNCXrWMCuy7Hkv7Og2ego" 'False) (C1 ('MetaCons "C\8477ay" 'PrefixI 'True) (S1 ('MetaSel ('Just "hParamC\8477ay") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 (Scalar (Needle x))) :*: S1 ('MetaSel ('Just "pParamC\8477ay") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 x)))

type FullSubspace (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type FullSubspace (Cℝay (ZeroDim k)) = ZeroDim k

type MirrorJoin (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type MirrorJoin (Cℝay (ZeroDim k)) = k

type Ray (Cℝay (ZeroDim k))

Defined in Data.Monoid.Additive

type Ray (Cℝay (ZeroDim k)) = Cℝay (ZeroDim k)

type Boundary (Cℝay S⁰)

Defined in Data.Manifold.Cone

type Boundary (Cℝay S⁰) = EmptyMfd ℝ⁰

type HalfNeedle (Cℝay S⁰)

Defined in Data.Manifold.Cone

type HalfNeedle (Cℝay S⁰) = ℝay

type Interior (Cℝay S⁰)

Defined in Data.Manifold.Cone

type Interior (Cℝay S⁰) = ℝ

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

:: forall x. (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

Category (LinearMap s :: Type -> Type -> TYPE LiftedRep)

Defined in Math.LinearMap.Category.Class

Associated Types

type Object (LinearMap s) o

Methods

id :: forall (a :: κ). Object (LinearMap s) a => LinearMap s a a

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

(Dimensional n u, Dimensional m v, LinearSpace u, TensorSpace v, Scalar u ~ s, Scalar v ~ s, nm ~ (n * m)) => Dimensional nm (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

Methods

knownDimensionalitySing :: Sing nm

unsafeFromArrayWithOffset :: Vector α (Scalar (LinearMap s u v)) => Int -> α (Scalar (LinearMap s u v)) -> LinearMap s u v

unsafeWriteArrayWithOffset :: forall (α :: Type -> Type) σ. Vector α (Scalar (LinearMap s u v)) => Mutable α σ (Scalar (LinearMap s u v)) -> Int -> LinearMap s u v -> ST σ ()

(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

Num' s => Cartesian (LinearMap s)

Defined in Math.LinearMap.Category.Class

Associated Types

type PairObjects (LinearMap s) a b

type UnitObject (LinearMap s)

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 (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

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

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 (->) (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) (VSCCoercion s) (VSCCoercion s)

Defined in Math.LinearMap.Category.Class

Methods

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

(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)

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 :: VSCCoercion (Scalar (LinearMap s u v)) (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

useTupleLinearSpaceComponents :: LinearMap s u v ~ (x, y) => ((LinearSpace x, LinearSpace y, Scalar x ~ Scalar y) => φ) -> φ

(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

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)

tensorUnsafeFromArrayWithOffset :: forall w α (n :: Nat) (m :: Nat). (Dimensional n (LinearMap s u v), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (LinearMap s u v), Vector α (Scalar (LinearMap s u v))) => Int -> α (Scalar (LinearMap s u v)) -> LinearMap s u v ⊗ w

tensorUnsafeWriteArrayWithOffset :: forall w (α :: Type -> Type) σ (n :: Nat) (m :: Nat). (Dimensional n (LinearMap s u v), TensorSpace w, Dimensional m w, Scalar w ~ Scalar (LinearMap s u v), Vector α (Scalar (LinearMap s u v))) => Mutable α σ (Scalar (LinearMap s u v)) -> Int -> (LinearMap s u v ⊗ w) -> ST σ ()

coerceFmapTensorProduct :: (Functor p, TensorSpace a, Scalar a ~ Scalar (LinearMap s u v), TensorSpace b, Scalar b ~ Scalar (LinearMap s u v)) => p (LinearMap s u v) -> VSCCoercion (Scalar (LinearMap s u v)) 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, TensorSpace v, Scalar u ~ s, Scalar v ~ s) => DimensionAware (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

Associated Types

type StaticDimension (LinearMap s u v) :: Maybe Nat

Methods

dimensionalityWitness :: DimensionalityWitness (LinearMap s u v)

(LSpace u, FiniteDimensional 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)

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)

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

dualFinitenessWitness :: DualFinitenessWitness (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)

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

Defined in Data.Manifold.Atlas

Associated Types

type ChartIndex (LinearMap s v w)

Methods

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

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

(LinearSpace v, Scalar v ~ ℝ, LinearSpace 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)

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)

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

(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

(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)

(LinearSpace v, LinearSpace w, s ~ Scalar v, s ~ Scalar w, Num' s, OpenManifold s, ProjectableBoundary s) => ProjectableBoundary (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

Methods

projectToBoundary :: LinearMap s v w -> Boundary (LinearMap s v w) -> Maybe (Needle (Boundary (LinearMap s v w)), Scalar (Needle (Interior (LinearMap s v w))))

marginFromBoundary :: Boundary (LinearMap s v w) -> Scalar (Needle (Interior (LinearMap s v w))) -> LinearMap s v w

needleBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Needle (Interior (LinearMap s v w))) => r) -> r

scalarBoundaryIsTriviallyProjectible :: (ProjectableBoundary (Scalar (Needle (Interior (LinearMap s v w)))) => r) -> r

(LinearSpace v, LinearSpace w, s ~ Scalar v, s ~ Scalar w, Num' s, OpenManifold s) => PseudoAffineWithBoundary (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

Methods

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

(.-|) :: LinearMap s v w -> Boundary (LinearMap s v w) -> Maybe (HalfNeedle (LinearMap s v w))

(!-|) :: LinearMap s v w -> Boundary (LinearMap s v w) -> HalfNeedle (LinearMap s v w)

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

(LinearSpace v, LinearSpace w, s ~ Scalar v, s ~ Scalar w, Num' s, OpenManifold s) => SemimanifoldWithBoundary (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

Associated Types

type Interior (LinearMap s v w)

type Boundary (LinearMap s v w)

type HalfNeedle (LinearMap s v w)

Methods

(.+^|) :: LinearMap s v w -> Needle (Interior (LinearMap s v w)) -> Either (Boundary (LinearMap s v w), Scalar (Needle (Interior (LinearMap s v w)))) (Interior (LinearMap s v w))

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

fromBoundary :: Boundary (LinearMap s v w) -> LinearMap s v w

(|+^) :: Boundary (LinearMap s v w) -> HalfNeedle (LinearMap s v w) -> LinearMap s v w

separateInterior :: LinearMap s v w -> Either (Boundary (LinearMap s v w)) (Interior (LinearMap s v w))

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

extendToBoundary :: Interior (LinearMap s v w) -> Needle (Interior (LinearMap s v w)) -> Maybe (Boundary (LinearMap s v w))

smfdWBoundWitness :: SmfdWBoundWitness (LinearMap s v w)

needleIsOpenMfd :: (OpenManifold (Needle (Interior (LinearMap s v w))) => r) -> r

scalarIsOpenMfd :: (OpenManifold (Scalar (Needle (Interior (LinearMap s v w)))) => r) -> r

boundaryHasSameScalar :: ((LinearSpace (Needle (Boundary (LinearMap s v w))), Scalar (Needle (Boundary (LinearMap s v w))) ~ Scalar (Needle (Interior (LinearMap s v w)))) => r) -> r

(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, 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)

Methods

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

(.-~^) :: 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) => 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 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)

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 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)

Methods

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

type Object (LinearMap s :: Type -> Type -> TYPE LiftedRep) (v :: Type)

Defined in Math.LinearMap.Category.Class

type Object (LinearMap s :: Type -> Type -> TYPE LiftedRep) (v :: Type) = (LinearSpace v, Scalar v ~ s)

type UnitObject (LinearMap s)

Defined in Math.LinearMap.Category.Class

type UnitObject (LinearMap s) = ZeroDim s

type PairObjects (LinearMap s) a b

Defined in Math.LinearMap.Category.Class

type PairObjects (LinearMap s) a b = ()

type DualVector (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

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

type StaticDimension (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

type StaticDimension (LinearMap s u v) = ZipWithTimes (StaticDimension u) (StaticDimension 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 ChartIndex (LinearMap s v w)

Defined in Data.Manifold.Atlas

type ChartIndex (LinearMap s v w) = ()

type Boundary (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

type Boundary (LinearMap s v w) = EmptyMfd (ZeroDim s)

type HalfNeedle (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

type HalfNeedle (LinearMap s v w) = ℝay

type Interior (LinearMap s v w)

Defined in Data.Manifold.WithBoundary

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 Diff (LinearMap s u v)

Defined in Math.LinearMap.Category.Class

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

type Scalar (LinearMap s v w)

Defined in Math.LinearMap.Category.Class

type Scalar (LinearMap s v w) = s

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)) => PseudoAffine (Stiefel1 v)

Methods

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

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

pseudoAffineWitness :: PseudoAffineWitness (Stiefel1 v)

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

Associated Types

type Needle (Stiefel1 v)

Methods

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

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

semimanifoldWitness :: SemimanifoldWitness (Stiefel1 v)