Data.Manifold.PseudoAffine

This is the second prototype of a manifold class. It appears to give considerable advantages over Manifold, so that class will probably soon be replaced with the one we define here (though PseudoAffine does not follow the standard notion of a manifold very closely, it should work quite equivalently for pretty much all Haskell types that qualify as manifolds).

Manifolds are interesting as objects of various categories, from continuous to diffeomorphic. At the moment, we mainly focus on region-wise differentiable functions, which are a promising compromise between flexibility of definition and provability of analytic properties. In particular, they are well-suited for visualisation purposes.

The classes in this module are mostly aimed at manifolds without boundary. Manifolds with boundary (which we call MWBound, never manifold!) are more or less treated as a disjoint sum of the interior and the boundary. To understand how this module works, best first forget about boundaries – in this case, Interior x ~ x, fromInterior and toInterior are trivial, and .+~|, |-~. and betweenBounds are irrelevant. The manifold structure of the boundary itself is not considered at all here.

Manifold class

class (PseudoAffine m, LSpace (Needle m)) => Manifold m where

See Semimanifold and PseudoAffine for the methods.

Minimal complete definition

Nothing

Methods

inInterior :: m -> Interior m

inInterior :: m ~ Interior m => m -> Interior m

Instances

(PseudoAffine m, LSpace (Needle m), Interior m ~ m) => Manifold m

Defined in Data.Manifold.PseudoAffine

Methods

boundarylessWitness :: BoundarylessWitness m

inInterior :: m -> Interior m

class AdditiveGroup (Needle x) => Semimanifold x where

Minimal complete definition

((.+~^) | fromInterior), toInterior, translateP

Associated Types

type Needle x :: Type

The space of “natural” ways starting from some reference point and going to some particular target point. Hence, the name: like a compass needle, but also with an actual length. For affine spaces, Needle is simply the space of line segments (aka vectors) between two points, i.e. the same as Diff. The AffineManifold constraint makes that requirement explicit.

This space should be isomorphic to the tangent space (and is in fact used somewhat synonymously).

type Interior x :: Type

Manifolds with boundary are a bit tricky. We support such manifolds, but carry out most calculations only in “the fleshy part” – the interior, which is an “infinite space”, so you can arbitrarily scale paths.

The default implementation is Interior x = x, which corresponds to a manifold that has no boundary to begin with.

Methods

(.+~^) :: Interior x -> Needle x -> x infixl 6

Generalised translation operation. Note that the result will always also be in the interior; scaling up the needle can only get you ever closer to a boundary.

fromInterior :: Interior x -> x

id sans boundary.

toInterior :: x -> Maybe (Interior x)

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

The signature of .+~^ should really be Interior x -> Needle x -> Interior x, only, this is not possible because it only consists of non-injective type families. The solution is this tagged signature, which is of course rather unwieldy. That's why .+~^ has the stronger, but easier usable signature. Without boundary, these functions should be equivalent, i.e. translateP = Tagged (.+~^).

(.-~^) :: Interior x -> Needle x -> x infixl 6

Shorthand for \p v -> p .+~^ negateV v, which should obey the asymptotic law

p .-~^ v .+~^ v ≅ p

Meaning: if v is scaled down with sufficiently small factors η, then the difference (p.-~^v.+~^v) .-~. p should scale down even faster: as O (η²). For large vectors, it will however behave differently, except in flat spaces (where all this should be equivalent to the AffineSpace instance).

semimanifoldWitness :: SemimanifoldWitness x

Instances

Semimanifold Double

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle Double :: Type

type Interior Double :: Type

Methods

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

fromInterior :: Interior Double -> Double

toInterior :: Double -> Maybe (Interior Double)

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

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

semimanifoldWitness :: SemimanifoldWitness Double

Semimanifold Float

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle Float :: Type

type Interior Float :: Type

Methods

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

fromInterior :: Interior Float -> Float

toInterior :: Float -> Maybe (Interior Float)

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

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

semimanifoldWitness :: SemimanifoldWitness Float

Semimanifold Rational

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle Rational :: Type

type Interior Rational :: Type

Methods

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

fromInterior :: Interior Rational -> Rational

toInterior :: Rational -> Maybe (Interior Rational)

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

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

semimanifoldWitness :: SemimanifoldWitness Rational

Semimanifold S⁰

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle S⁰ :: Type

type Interior S⁰ :: Type

Methods

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

fromInterior :: Interior S⁰ -> S⁰

toInterior :: S⁰ -> Maybe (Interior S⁰)

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

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

semimanifoldWitness :: SemimanifoldWitness S⁰

Semimanifold ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle ℝP⁰ :: Type

type Interior ℝP⁰ :: Type

Methods

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

fromInterior :: Interior ℝP⁰ -> ℝP⁰

toInterior :: ℝP⁰ -> Maybe (Interior ℝP⁰)

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

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

semimanifoldWitness :: SemimanifoldWitness ℝP⁰

Semimanifold S¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle S¹ :: Type

type Interior S¹ :: Type

Methods

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

fromInterior :: Interior S¹ -> S¹

toInterior :: S¹ -> Maybe (Interior S¹)

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

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

semimanifoldWitness :: SemimanifoldWitness S¹

Semimanifold ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle ℝP¹ :: Type

type Interior ℝP¹ :: Type

Methods

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

fromInterior :: Interior ℝP¹ -> ℝP¹

toInterior :: ℝP¹ -> Maybe (Interior ℝP¹)

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

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

semimanifoldWitness :: SemimanifoldWitness ℝP¹

Semimanifold S²

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle S² :: Type

type Interior S² :: Type

Methods

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

fromInterior :: Interior S² -> S²

toInterior :: S² -> Maybe (Interior S²)

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

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

semimanifoldWitness :: SemimanifoldWitness S²

Semimanifold ℝP²

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle ℝP² :: Type

type Interior ℝP² :: Type

Methods

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

fromInterior :: Interior ℝP² -> ℝP²

toInterior :: ℝP² -> Maybe (Interior ℝP²)

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

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

semimanifoldWitness :: SemimanifoldWitness ℝP²

Semimanifold D¹

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle D¹ :: Type

type Interior D¹ :: Type

Methods

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

fromInterior :: Interior D¹ -> D¹

toInterior :: D¹ -> Maybe (Interior D¹)

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

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

semimanifoldWitness :: SemimanifoldWitness D¹

AdditiveGroup (DualVector (Needle (VRep m))) => Semimanifold (GenericNeedle' m)

Defined in Math.LinearMap.Category.Class

Associated Types

type Needle (GenericNeedle' m) :: Type

type Interior (GenericNeedle' m) :: Type

Methods

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

fromInterior :: Interior (GenericNeedle' m) -> GenericNeedle' m

toInterior :: GenericNeedle' m -> Maybe (Interior (GenericNeedle' m))

translateP :: Tagged (GenericNeedle' m) (Interior (GenericNeedle' m) -> Needle (GenericNeedle' m) -> Interior (GenericNeedle' m))

(.-~^) :: Interior (GenericNeedle' m) -> Needle (GenericNeedle' m) -> GenericNeedle' m

semimanifoldWitness :: SemimanifoldWitness (GenericNeedle' m)

Semimanifold (ZeroDim k)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ZeroDim k) :: Type

type Interior (ZeroDim k) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (ZeroDim k)

AdditiveGroup (Needle (VRep x)) => Semimanifold (GenericNeedle x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (GenericNeedle x) :: Type

type Interior (GenericNeedle x) :: Type

Methods

(.+~^) :: Interior (GenericNeedle x) -> Needle (GenericNeedle x) -> GenericNeedle x

fromInterior :: Interior (GenericNeedle x) -> GenericNeedle x

toInterior :: GenericNeedle x -> Maybe (Interior (GenericNeedle x))

translateP :: Tagged (GenericNeedle x) (Interior (GenericNeedle x) -> Needle (GenericNeedle x) -> Interior (GenericNeedle x))

(.-~^) :: Interior (GenericNeedle x) -> Needle (GenericNeedle x) -> GenericNeedle x

semimanifoldWitness :: SemimanifoldWitness (GenericNeedle x)

Semimanifold (VRep x) => Semimanifold (GenericInterior x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (GenericInterior x) :: Type

type Interior (GenericInterior x) :: Type

Methods

(.+~^) :: Interior (GenericInterior x) -> Needle (GenericInterior x) -> GenericInterior x

fromInterior :: Interior (GenericInterior x) -> GenericInterior x

toInterior :: GenericInterior x -> Maybe (Interior (GenericInterior x))

translateP :: Tagged (GenericInterior x) (Interior (GenericInterior x) -> Needle (GenericInterior x) -> Interior (GenericInterior x))

(.-~^) :: Interior (GenericInterior x) -> Needle (GenericInterior x) -> GenericInterior x

semimanifoldWitness :: SemimanifoldWitness (GenericInterior x)

ConeSemimfd m => Semimanifold (CD¹ m)

Defined in Data.Manifold.Cone

Associated Types

type Needle (CD¹ m) :: Type

type Interior (CD¹ m) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (CD¹ m)

ConeSemimfd m => Semimanifold (Cℝay m)

Defined in Data.Manifold.Cone

Associated Types

type Needle (Cℝay m) :: Type

type Interior (Cℝay m) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (Cℝay m)

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

Defined in Data.Manifold.Types

Associated Types

type Needle (Stiefel1 v) :: Type

type Interior (Stiefel1 v) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (Stiefel1 v)

AffineManifold x => Semimanifold (Shade' x)

Defined in Data.Manifold.Shade

Associated Types

type Needle (Shade' x) :: Type

type Interior (Shade' x) :: Type

Methods

(.+~^) :: Interior (Shade' x) -> Needle (Shade' x) -> Shade' x

fromInterior :: Interior (Shade' x) -> Shade' x

toInterior :: Shade' x -> Maybe (Interior (Shade' x))

translateP :: Tagged (Shade' x) (Interior (Shade' x) -> Needle (Shade' x) -> Interior (Shade' x))

(.-~^) :: Interior (Shade' x) -> Needle (Shade' x) -> Shade' x

semimanifoldWitness :: SemimanifoldWitness (Shade' x)

PseudoAffine x => Semimanifold (Shade x)

Defined in Data.Manifold.Shade

Associated Types

type Needle (Shade x) :: Type

type Interior (Shade x) :: Type

Methods

(.+~^) :: Interior (Shade x) -> Needle (Shade x) -> Shade x

fromInterior :: Interior (Shade x) -> Shade x

toInterior :: Shade x -> Maybe (Interior (Shade x))

translateP :: Tagged (Shade x) (Interior (Shade x) -> Needle (Shade x) -> Interior (Shade x))

(.-~^) :: Interior (Shade x) -> Needle (Shade x) -> Shade x

semimanifoldWitness :: SemimanifoldWitness (Shade x)

(Semimanifold a, Semimanifold b) => Semimanifold (a, b)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (a, b) :: Type

type Interior (a, b) :: Type

Methods

(.+~^) :: Interior (a, b) -> Needle (a, b) -> (a, b)

fromInterior :: Interior (a, b) -> (a, b)

toInterior :: (a, b) -> Maybe (Interior (a, b))

translateP :: Tagged (a, b) (Interior (a, b) -> Needle (a, b) -> Interior (a, b))

(.-~^) :: Interior (a, b) -> Needle (a, b) -> (a, b)

semimanifoldWitness :: SemimanifoldWitness (a, b)

(LinearSpace (a n), Needle (a n) ~ a n, Interior (a n) ~ a n) => Semimanifold (Point a n)

Defined in Data.Manifold.PseudoAffine

Associated Types

type Needle (Point a n) :: Type

type Interior (Point a n) :: Type

Methods

(.+~^) :: Interior (Point a n) -> Needle (Point a n) -> Point a n

fromInterior :: Interior (Point a n) -> Point a n

toInterior :: Point a n -> Maybe (Interior (Point a n))

translateP :: Tagged (Point a n) (Interior (Point a n) -> Needle (Point a n) -> Interior (Point a n))

(.-~^) :: Interior (Point a n) -> Needle (Point a n) -> Point a n

semimanifoldWitness :: SemimanifoldWitness (Point a n)

(Generic1 f, TensorSpace y, TensorSpace (f y), Scalar (f y) ~ Scalar y, Monoidal f (LinearFunction (Scalar y)) (LinearFunction (Scalar y))) => Semimanifold (LinearApplicativeSpace f y)

Defined in Math.LinearMap.Category.Instances

Associated Types

type Needle (LinearApplicativeSpace f y) :: Type

type Interior (LinearApplicativeSpace f y) :: Type

Methods

(.+~^) :: Interior (LinearApplicativeSpace f y) -> Needle (LinearApplicativeSpace f y) -> LinearApplicativeSpace f y

fromInterior :: Interior (LinearApplicativeSpace f y) -> LinearApplicativeSpace f y

toInterior :: LinearApplicativeSpace f y -> Maybe (Interior (LinearApplicativeSpace f y))

translateP :: Tagged (LinearApplicativeSpace f y) (Interior (LinearApplicativeSpace f y) -> Needle (LinearApplicativeSpace f y) -> Interior (LinearApplicativeSpace f y))

(.-~^) :: Interior (LinearApplicativeSpace f y) -> Needle (LinearApplicativeSpace f y) -> LinearApplicativeSpace f y

semimanifoldWitness :: SemimanifoldWitness (LinearApplicativeSpace f y)

(TensorSpace v, Scalar v ~ s) => Semimanifold (SymmetricTensor s v)

Defined in Math.LinearMap.Category.Instances

Associated Types

type Needle (SymmetricTensor s v) :: Type

type Interior (SymmetricTensor s v) :: Type

Methods

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

fromInterior :: Interior (SymmetricTensor s v) -> SymmetricTensor s v

toInterior :: SymmetricTensor s v -> Maybe (Interior (SymmetricTensor s v))

translateP :: Tagged (SymmetricTensor s v) (Interior (SymmetricTensor s v) -> Needle (SymmetricTensor s v) -> Interior (SymmetricTensor s v))

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

semimanifoldWitness :: SemimanifoldWitness (SymmetricTensor s v)

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

Defined in Data.Manifold.FibreBundle

Associated Types

type Needle (FibreBundle m f) :: Type

type Interior (FibreBundle m f) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (FibreBundle m f)

Semimanifold x => Semimanifold (WithAny x y)

Defined in Data.Manifold.Shade

Associated Types

type Needle (WithAny x y) :: Type

type Interior (WithAny x y) :: Type

Methods

(.+~^) :: Interior (WithAny x y) -> Needle (WithAny x y) -> WithAny x y

fromInterior :: Interior (WithAny x y) -> WithAny x y

toInterior :: WithAny x y -> Maybe (Interior (WithAny x y))

translateP :: Tagged (WithAny x y) (Interior (WithAny x y) -> Needle (WithAny x y) -> Interior (WithAny x y))

(.-~^) :: Interior (WithAny x y) -> Needle (WithAny x y) -> WithAny x y

semimanifoldWitness :: SemimanifoldWitness (WithAny x y)

Semimanifold a => Semimanifold (Rec0 a s)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (Rec0 a s) :: Type

type Interior (Rec0 a s) :: Type

Methods

(.+~^) :: Interior (Rec0 a s) -> Needle (Rec0 a s) -> Rec0 a s

fromInterior :: Interior (Rec0 a s) -> Rec0 a s

toInterior :: Rec0 a s -> Maybe (Interior (Rec0 a s))

translateP :: Tagged (Rec0 a s) (Interior (Rec0 a s) -> Needle (Rec0 a s) -> Interior (Rec0 a s))

(.-~^) :: Interior (Rec0 a s) -> Needle (Rec0 a s) -> Rec0 a s

semimanifoldWitness :: SemimanifoldWitness (Rec0 a s)

(Semimanifold a, Semimanifold b, Semimanifold c) => Semimanifold (a, b, c)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (a, b, c) :: Type

type Interior (a, b, c) :: Type

Methods

(.+~^) :: Interior (a, b, c) -> Needle (a, b, c) -> (a, b, c)

fromInterior :: Interior (a, b, c) -> (a, b, c)

toInterior :: (a, b, c) -> Maybe (Interior (a, b, c))

translateP :: Tagged (a, b, c) (Interior (a, b, c) -> Needle (a, b, c) -> Interior (a, b, c))

(.-~^) :: Interior (a, b, c) -> Needle (a, b, c) -> (a, b, c)

semimanifoldWitness :: SemimanifoldWitness (a, b, c)

(AdditiveGroup (DualVector (f p)), AdditiveGroup (DualVector (g p))) => Semimanifold (GenericTupleDual f g p)

Defined in Math.LinearMap.Category.Class

Associated Types

type Needle (GenericTupleDual f g p) :: Type

type Interior (GenericTupleDual f g p) :: Type

Methods

(.+~^) :: Interior (GenericTupleDual f g p) -> Needle (GenericTupleDual f g p) -> GenericTupleDual f g p

fromInterior :: Interior (GenericTupleDual f g p) -> GenericTupleDual f g p

toInterior :: GenericTupleDual f g p -> Maybe (Interior (GenericTupleDual f g p))

translateP :: Tagged (GenericTupleDual f g p) (Interior (GenericTupleDual f g p) -> Needle (GenericTupleDual f g p) -> Interior (GenericTupleDual f g p))

(.-~^) :: Interior (GenericTupleDual f g p) -> Needle (GenericTupleDual f g p) -> GenericTupleDual f g p

semimanifoldWitness :: SemimanifoldWitness (GenericTupleDual f g p)

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

Defined in Math.LinearMap.Category.Class

Associated Types

type Needle (LinearMap s v w) :: Type

type Interior (LinearMap s v w) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (LinearMap s v w)

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

Defined in Math.LinearMap.Category.Class

Associated Types

type Needle (Tensor s v w) :: Type

type Interior (Tensor s v w) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (Tensor s v w)

VectorSpace w => Semimanifold (LinearFunction s v w)

Defined in Math.LinearMap.Asserted

Associated Types

type Needle (LinearFunction s v w) :: Type

type Interior (LinearFunction s v w) :: Type

Methods

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

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

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

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

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

semimanifoldWitness :: SemimanifoldWitness (LinearFunction s v w)

(Semimanifold (f p), Semimanifold (g p)) => Semimanifold (NeedleProductSpace f g p)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (NeedleProductSpace f g p) :: Type

type Interior (NeedleProductSpace f g p) :: Type

Methods

(.+~^) :: Interior (NeedleProductSpace f g p) -> Needle (NeedleProductSpace f g p) -> NeedleProductSpace f g p

fromInterior :: Interior (NeedleProductSpace f g p) -> NeedleProductSpace f g p

toInterior :: NeedleProductSpace f g p -> Maybe (Interior (NeedleProductSpace f g p))

translateP :: Tagged (NeedleProductSpace f g p) (Interior (NeedleProductSpace f g p) -> Needle (NeedleProductSpace f g p) -> Interior (NeedleProductSpace f g p))

(.-~^) :: Interior (NeedleProductSpace f g p) -> Needle (NeedleProductSpace f g p) -> NeedleProductSpace f g p

semimanifoldWitness :: SemimanifoldWitness (NeedleProductSpace f g p)

(Semimanifold (f p), Semimanifold (g p)) => Semimanifold (InteriorProductSpace f g p)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (InteriorProductSpace f g p) :: Type

type Interior (InteriorProductSpace f g p) :: Type

Methods

(.+~^) :: Interior (InteriorProductSpace f g p) -> Needle (InteriorProductSpace f g p) -> InteriorProductSpace f g p

fromInterior :: Interior (InteriorProductSpace f g p) -> InteriorProductSpace f g p

toInterior :: InteriorProductSpace f g p -> Maybe (Interior (InteriorProductSpace f g p))

translateP :: Tagged (InteriorProductSpace f g p) (Interior (InteriorProductSpace f g p) -> Needle (InteriorProductSpace f g p) -> Interior (InteriorProductSpace f g p))

(.-~^) :: Interior (InteriorProductSpace f g p) -> Needle (InteriorProductSpace f g p) -> InteriorProductSpace f g p

semimanifoldWitness :: SemimanifoldWitness (InteriorProductSpace f g p)

(Atlas x, HasTrie (ChartIndex x), LinearSpace (Needle x), Scalar (Needle x) ~ s, Manifold y, Scalar (Needle y) ~ s) => Semimanifold (Affine s x y)

Defined in Data.Function.Affine

Associated Types

type Needle (Affine s x y) :: Type

type Interior (Affine s x y) :: Type

Methods

(.+~^) :: Interior (Affine s x y) -> Needle (Affine s x y) -> Affine s x y

fromInterior :: Interior (Affine s x y) -> Affine s x y

toInterior :: Affine s x y -> Maybe (Interior (Affine s x y))

translateP :: Tagged (Affine s x y) (Interior (Affine s x y) -> Needle (Affine s x y) -> Interior (Affine s x y))

(.-~^) :: Interior (Affine s x y) -> Needle (Affine s x y) -> Affine s x y

semimanifoldWitness :: SemimanifoldWitness (Affine s x y)

(Semimanifold (f p), Semimanifold (g p)) => Semimanifold ((f :*: g) p)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle ((f :*: g) p) :: Type

type Interior ((f :*: g) p) :: Type

Methods

(.+~^) :: Interior ((f :*: g) p) -> Needle ((f :*: g) p) -> (f :*: g) p

fromInterior :: Interior ((f :*: g) p) -> (f :*: g) p

toInterior :: (f :*: g) p -> Maybe (Interior ((f :*: g) p))

translateP :: Tagged ((f :*: g) p) (Interior ((f :*: g) p) -> Needle ((f :*: g) p) -> Interior ((f :*: g) p))

(.-~^) :: Interior ((f :*: g) p) -> Needle ((f :*: g) p) -> (f :*: g) p

semimanifoldWitness :: SemimanifoldWitness ((f :*: g) p)

Semimanifold (f p) => Semimanifold (M1 i c f p)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (M1 i c f p) :: Type

type Interior (M1 i c f p) :: Type

Methods

(.+~^) :: Interior (M1 i c f p) -> Needle (M1 i c f p) -> M1 i c f p

fromInterior :: Interior (M1 i c f p) -> M1 i c f p

toInterior :: M1 i c f p -> Maybe (Interior (M1 i c f p))

translateP :: Tagged (M1 i c f p) (Interior (M1 i c f p) -> Needle (M1 i c f p) -> Interior (M1 i c f p))

(.-~^) :: Interior (M1 i c f p) -> Needle (M1 i c f p) -> M1 i c f p

semimanifoldWitness :: SemimanifoldWitness (M1 i c f p)

type Needle' x = DualVector (Needle x)

A co-needle can be understood as a “paper stack”, with which you can measure the length that a needle reaches in a given direction by counting the number of holes punched through them.

class Semimanifold x => PseudoAffine x where

This is the class underlying manifolds. (Manifold only precludes boundaries and adds an extra constraint that would be circular if it was in a single class. You can always just use Manifold as a constraint in your signatures, but you must define only PseudoAffine for manifold types – the Manifold instance follows universally from this, if 'Interior x ~ x.)

The interface is (boundaries aside) almost identical to the better-known AffineSpace class, but we don't require associativity of .+~^ with ^+^ – except in an asymptotic sense for small vectors.

That innocent-looking change makes the class applicable to vastly more general types: while an affine space is basically nothing but a vector space without particularly designated origin, a pseudo-affine space can have nontrivial topology on the global scale, and yet be used in practically the same way as an affine space. At least the usual spheres and tori make good instances, perhaps the class is in fact equivalent to manifolds in their usual maths definition (with an atlas of charts: a family of overlapping regions of the topological space, each homeomorphic to the Needle vector space or some simply-connected subset thereof).

Minimal complete definition

(.-~.) | (.-~!)

Methods

(.-~.) :: x -> x -> Maybe (Needle x) infix 6

The path reaching from one point to another. Should only yield Nothing if

• The points are on disjoint segments of a non–path-connected space.
• Either of the points is on the boundary. Use |-~. to deal with this.

On manifolds, the identity

p .+~^ (q.-~.p) ≡ q

should hold, at least save for floating-point precision limits etc..

.-~. and .+~^ only really work in manifolds without boundary. If you consider the path between two points, one of which lies on the boundary, it can't really be possible to scale this path any longer – it would have to reach “out of the manifold”. To adress this problem, these functions basically consider only the interior of the space.

(.-~!) :: x -> x -> Needle x infix 6

Unsafe version of .-~.. If the two points lie in disjoint regions, the behaviour is undefined.

pseudoAffineWitness :: PseudoAffineWitness x

Instances

PseudoAffine Double

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: Double -> Double -> Needle Double

pseudoAffineWitness :: PseudoAffineWitness Double

PseudoAffine Float

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: Float -> Float -> Needle Float

pseudoAffineWitness :: PseudoAffineWitness Float

PseudoAffine Rational

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: Rational -> Rational -> Needle Rational

pseudoAffineWitness :: PseudoAffineWitness Rational

PseudoAffine S⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: S⁰ -> S⁰ -> Needle S⁰

pseudoAffineWitness :: PseudoAffineWitness S⁰

PseudoAffine ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

pseudoAffineWitness :: PseudoAffineWitness ℝP⁰

PseudoAffine S¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: S¹ -> S¹ -> Needle S¹

pseudoAffineWitness :: PseudoAffineWitness S¹

PseudoAffine ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

pseudoAffineWitness :: PseudoAffineWitness ℝP¹

PseudoAffine S²

Defined in Data.Manifold.PseudoAffine

Methods

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

(.-~!) :: S² -> S² -> Needle S²

pseudoAffineWitness :: PseudoAffineWitness S²

PseudoAffine ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

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

(.-~!) :: ℝP² -> ℝP² -> Needle ℝP²

pseudoAffineWitness :: PseudoAffineWitness ℝP²

PseudoAffine D¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: D¹ -> D¹ -> Needle D¹

pseudoAffineWitness :: PseudoAffineWitness D¹

AdditiveGroup (DualVector (Needle (VRep m))) => PseudoAffine (GenericNeedle' m)

Defined in Math.LinearMap.Category.Class

Methods

(.-~.) :: GenericNeedle' m -> GenericNeedle' m -> Maybe (Needle (GenericNeedle' m))

(.-~!) :: GenericNeedle' m -> GenericNeedle' m -> Needle (GenericNeedle' m)

pseudoAffineWitness :: PseudoAffineWitness (GenericNeedle' m)

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)

AdditiveGroup (Needle (VRep x)) => PseudoAffine (GenericNeedle x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: GenericNeedle x -> GenericNeedle x -> Maybe (Needle (GenericNeedle x))

(.-~!) :: GenericNeedle x -> GenericNeedle x -> Needle (GenericNeedle x)

pseudoAffineWitness :: PseudoAffineWitness (GenericNeedle x)

PseudoAffine (VRep x) => PseudoAffine (GenericInterior x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: GenericInterior x -> GenericInterior x -> Maybe (Needle (GenericInterior x))

(.-~!) :: GenericInterior x -> GenericInterior x -> Needle (GenericInterior x)

pseudoAffineWitness :: PseudoAffineWitness (GenericInterior x)

(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)

(PseudoAffine a, PseudoAffine b) => PseudoAffine (a, b)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: (a, b) -> (a, b) -> Maybe (Needle (a, b))

(.-~!) :: (a, b) -> (a, b) -> Needle (a, b)

pseudoAffineWitness :: PseudoAffineWitness (a, b)

(LinearSpace (a n), Needle (a n) ~ a n, Interior (a n) ~ a n) => PseudoAffine (Point a n)

Defined in Data.Manifold.PseudoAffine

Methods

(.-~.) :: Point a n -> Point a n -> Maybe (Needle (Point a n))

(.-~!) :: Point a n -> Point a n -> Needle (Point a n)

pseudoAffineWitness :: PseudoAffineWitness (Point a n)

(Generic1 f, TensorSpace y, TensorSpace (f y), Scalar (f y) ~ Scalar y, Monoidal f (LinearFunction (Scalar y)) (LinearFunction (Scalar y))) => PseudoAffine (LinearApplicativeSpace f y)

Defined in Math.LinearMap.Category.Instances

Methods

(.-~.) :: LinearApplicativeSpace f y -> LinearApplicativeSpace f y -> Maybe (Needle (LinearApplicativeSpace f y))

(.-~!) :: LinearApplicativeSpace f y -> LinearApplicativeSpace f y -> Needle (LinearApplicativeSpace f y)

pseudoAffineWitness :: PseudoAffineWitness (LinearApplicativeSpace f y)

(TensorSpace v, Scalar v ~ s) => PseudoAffine (SymmetricTensor s v)

Defined in Math.LinearMap.Category.Instances

Methods

(.-~.) :: SymmetricTensor s v -> SymmetricTensor s v -> Maybe (Needle (SymmetricTensor s v))

(.-~!) :: SymmetricTensor s v -> SymmetricTensor s v -> Needle (SymmetricTensor s v)

pseudoAffineWitness :: PseudoAffineWitness (SymmetricTensor s v)

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

Defined in Data.Manifold.FibreBundle

Methods

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

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

pseudoAffineWitness :: PseudoAffineWitness (FibreBundle m f)

PseudoAffine x => PseudoAffine (WithAny x y)

Defined in Data.Manifold.Shade

Methods

(.-~.) :: WithAny x y -> WithAny x y -> Maybe (Needle (WithAny x y))

(.-~!) :: WithAny x y -> WithAny x y -> Needle (WithAny x y)

pseudoAffineWitness :: PseudoAffineWitness (WithAny x y)

PseudoAffine a => PseudoAffine (Rec0 a s)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: Rec0 a s -> Rec0 a s -> Maybe (Needle (Rec0 a s))

(.-~!) :: Rec0 a s -> Rec0 a s -> Needle (Rec0 a s)

pseudoAffineWitness :: PseudoAffineWitness (Rec0 a s)

(PseudoAffine a, PseudoAffine b, PseudoAffine c) => PseudoAffine (a, b, c)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: (a, b, c) -> (a, b, c) -> Maybe (Needle (a, b, c))

(.-~!) :: (a, b, c) -> (a, b, c) -> Needle (a, b, c)

pseudoAffineWitness :: PseudoAffineWitness (a, b, c)

(AdditiveGroup (DualVector (f p)), AdditiveGroup (DualVector (g p))) => PseudoAffine (GenericTupleDual f g p)

Defined in Math.LinearMap.Category.Class

Methods

(.-~.) :: GenericTupleDual f g p -> GenericTupleDual f g p -> Maybe (Needle (GenericTupleDual f g p))

(.-~!) :: GenericTupleDual f g p -> GenericTupleDual f g p -> Needle (GenericTupleDual f g p)

pseudoAffineWitness :: PseudoAffineWitness (GenericTupleDual f g p)

(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)

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

Defined in Math.LinearMap.Category.Class

Methods

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

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

pseudoAffineWitness :: PseudoAffineWitness (Tensor s v w)

VectorSpace w => PseudoAffine (LinearFunction s v w)

Defined in Math.LinearMap.Asserted

Methods

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

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

pseudoAffineWitness :: PseudoAffineWitness (LinearFunction s v w)

(PseudoAffine (f p), PseudoAffine (g p)) => PseudoAffine (NeedleProductSpace f g p)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: NeedleProductSpace f g p -> NeedleProductSpace f g p -> Maybe (Needle (NeedleProductSpace f g p))

(.-~!) :: NeedleProductSpace f g p -> NeedleProductSpace f g p -> Needle (NeedleProductSpace f g p)

pseudoAffineWitness :: PseudoAffineWitness (NeedleProductSpace f g p)

(PseudoAffine (f p), PseudoAffine (g p)) => PseudoAffine (InteriorProductSpace f g p)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: InteriorProductSpace f g p -> InteriorProductSpace f g p -> Maybe (Needle (InteriorProductSpace f g p))

(.-~!) :: InteriorProductSpace f g p -> InteriorProductSpace f g p -> Needle (InteriorProductSpace f g p)

pseudoAffineWitness :: PseudoAffineWitness (InteriorProductSpace f g p)

(Atlas x, HasTrie (ChartIndex x), LinearSpace (Needle x), Scalar (Needle x) ~ s, Manifold y, Scalar (Needle y) ~ s) => PseudoAffine (Affine s x y)

Defined in Data.Function.Affine

Methods

(.-~.) :: Affine s x y -> Affine s x y -> Maybe (Needle (Affine s x y))

(.-~!) :: Affine s x y -> Affine s x y -> Needle (Affine s x y)

pseudoAffineWitness :: PseudoAffineWitness (Affine s x y)

(PseudoAffine (f p), PseudoAffine (g p)) => PseudoAffine ((f :*: g) p)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: (f :*: g) p -> (f :*: g) p -> Maybe (Needle ((f :*: g) p))

(.-~!) :: (f :*: g) p -> (f :*: g) p -> Needle ((f :*: g) p)

pseudoAffineWitness :: PseudoAffineWitness ((f :*: g) p)

PseudoAffine (f p) => PseudoAffine (M1 i c f p)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: M1 i c f p -> M1 i c f p -> Maybe (Needle (M1 i c f p))

(.-~!) :: M1 i c f p -> M1 i c f p -> Needle (M1 i c f p)

pseudoAffineWitness :: PseudoAffineWitness (M1 i c f p)

Type definitions

Needles

newtype Local x

A point on a manifold, as seen from a nearby reference point.

Constructors

Local

Fields

• getLocalOffset :: Needle x

Instances

Show (Needle x) => Show (Local x)

Defined in Data.Manifold.PseudoAffine

Methods

showsPrec :: Int -> Local x -> ShowS

show :: Local x -> String

showList :: [Local x] -> ShowS

(⊙+^) :: forall x proxy. Semimanifold x => Interior x -> Needle x -> proxy x -> Interior x infix 6

Proxy-version of translateP.

(!+~^) :: forall x. (Semimanifold x, HasCallStack) => x -> Needle x -> x infixl 6

Boundary-unsafe version of .+~^.

Metrics

type Metric x = Norm (Needle x)

The word “metric” is used in the sense as in general relativity. Actually this is just the type of scalar products on the tangent space. The actual metric is the function x -> x -> Scalar (Needle x) defined by

\p q -> m |$| (p.-~!q)

type Metric' x = Variance (Needle x)

type RieMetric x = x -> Metric x

A Riemannian metric assigns each point on a manifold a scalar product on the tangent space. Note that this association is not continuous, because the charts/tangent spaces in the bundle are a priori disjoint. However, for a proper Riemannian metric, all arising expressions of scalar products from needles between points on the manifold ought to be differentiable.

type RieMetric' x = x -> Metric' x

Constraints

data SemimanifoldWitness x where

This is the reified form of the property that the interior of a semimanifold is a manifold. These constraints would ideally be expressed directly as superclass constraints, but that would require the UndecidableSuperclasses extension, which is not reliable yet.

Also, if all those equality constraints are in scope, GHC tends to infer needlessly complicated types like Interior (Interior (Needle (Interior x))), which is the same as just Needle x.

Constructors

SemimanifoldWitness :: forall x. (Semimanifold (Needle x), Needle (Interior x) ~ Needle x, Needle (Needle x) ~ Needle x, Interior (Needle x) ~ Needle x) => BoundarylessWitness (Interior x) -> SemimanifoldWitness x

data PseudoAffineWitness x where

Constructors

PseudoAffineWitness :: forall x. (PseudoAffine (Interior x), PseudoAffine (Needle x)) => SemimanifoldWitness x -> PseudoAffineWitness x

data BoundarylessWitness m where

Constructors

BoundarylessWitness :: forall m. (Semimanifold m, Interior m ~ m) => BoundarylessWitness m

boundarylessWitness :: Manifold m => BoundarylessWitness m

type DualNeedleWitness x = DualSpaceWitness (Needle x)

type WithField s c x = (c x, s ~ Scalar (Needle x), s ~ Scalar (Needle' x))

Require some constraint on a manifold, and also fix the type of the manifold's underlying field. For example, WithField ℝ HilbertManifold v constrains v to be a real (i.e., Double-) Hilbert space. Note that for this to compile, you will in general need the -XLiberalTypeSynonyms extension (except if the constraint is an actual type class (like Manifold): only those can always be partially applied, for type constraints this is by default not allowed).

type LocallyScalable s x = (PseudoAffine x, LSpace (Needle x), s ~ Scalar (Needle x), s ~ Scalar (Needle' x), Num' s)

Local functions

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

type LocalBilinear x y = LinearMap (Scalar (Needle x)) (SymmetricTensor (Scalar (Needle x)) (Needle x)) (Needle y)

type LocalAffine x y = (Needle y, LocalLinear x y)

Misc

alerpB :: (AffineSpace x, VectorSpace (Diff x), Scalar (Diff x) ~ ℝ) => x -> x -> D¹ -> x

Like alerp, but actually restricted to the interval between the points.

palerp :: (PseudoAffine x, VectorSpace (Needle x)) => x -> x -> Maybe (Scalar (Needle x) -> x)

Interpolate between points, approximately linearly. For points that aren't close neighbours (i.e. lie in an almost flat region), the pathway is basically undefined – save for its end points.

A proper, really well-defined (on global scales) interpolation only makes sense on a Riemannian manifold, as Geodesic.

palerpB :: (PseudoAffine x, VectorSpace (Needle x), Scalar (Needle x) ~ ℝ) => x -> x -> Maybe (D¹ -> x)

Like palerp, but actually restricted to the interval between the points, with a signature like geodesicBetween rather than alerp.

class (Semimanifold x, Semimanifold ξ, LSpace (Needle x), LSpace (Needle ξ), Scalar (Needle x) ~ Scalar (Needle ξ)) => LocallyCoercible x ξ where

Instances of this class must be diffeomorphic manifolds, and even have canonically isomorphic tangent spaces, so that fromPackedVector . asPackedVector :: Needle x -> Needle ξ defines a meaningful “representational identity“ between these spaces.

Minimal complete definition

locallyTrivialDiffeomorphism, coerceNeedle, coerceNeedle'

Methods

locallyTrivialDiffeomorphism :: x -> ξ

Must be compatible with the isomorphism on the tangent spaces, i.e. locallyTrivialDiffeomorphism (p .+~^ v) ≡ locallyTrivialDiffeomorphism p .+~^ coerceNeedle v

coerceNeedle :: Functor p => p (x, ξ) -> Needle x -+> Needle ξ

coerceNeedle' :: Functor p => p (x, ξ) -> Needle' x -+> Needle' ξ

coerceNorm :: Functor p => p (x, ξ) -> Metric x -> Metric ξ

coerceVariance :: Functor p => p (x, ξ) -> Metric' x -> Metric' ξ

oppositeLocalCoercion :: CanonicalDiffeomorphism ξ x

oppositeLocalCoercion :: LocallyCoercible ξ x => CanonicalDiffeomorphism ξ x

interiorLocalCoercion :: Functor p (->) (->) => p (x, ξ) -> CanonicalDiffeomorphism (Interior x) (Interior ξ)

interiorLocalCoercion :: LocallyCoercible (Interior x) (Interior ξ) => p (x, ξ) -> CanonicalDiffeomorphism (Interior x) (Interior ξ)

Instances

LocallyCoercible ℝ ℝ

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ℝ -> ℝ

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

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

coerceNorm :: Functor p => p (ℝ, ℝ) -> Metric ℝ -> Metric ℝ

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

oppositeLocalCoercion :: CanonicalDiffeomorphism ℝ ℝ

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

LocallyCoercible ℝ (V1 ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ℝ -> V1 ℝ

coerceNeedle :: Functor p => p (ℝ, V1 ℝ) -> Needle ℝ -+> Needle (V1 ℝ)

coerceNeedle' :: Functor p => p (ℝ, V1 ℝ) -> Needle' ℝ -+> Needle' (V1 ℝ)

coerceNorm :: Functor p => p (ℝ, V1 ℝ) -> Metric ℝ -> Metric (V1 ℝ)

coerceVariance :: Functor p => p (ℝ, V1 ℝ) -> Metric' ℝ -> Metric' (V1 ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V1 ℝ) ℝ

interiorLocalCoercion :: Functor p (->) (->) => p (ℝ, V1 ℝ) -> CanonicalDiffeomorphism (Interior ℝ) (Interior (V1 ℝ))

LocallyCoercible (V1 ℝ) ℝ

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V1 ℝ -> ℝ

coerceNeedle :: Functor p => p (V1 ℝ, ℝ) -> Needle (V1 ℝ) -+> Needle ℝ

coerceNeedle' :: Functor p => p (V1 ℝ, ℝ) -> Needle' (V1 ℝ) -+> Needle' ℝ

coerceNorm :: Functor p => p (V1 ℝ, ℝ) -> Metric (V1 ℝ) -> Metric ℝ

coerceVariance :: Functor p => p (V1 ℝ, ℝ) -> Metric' (V1 ℝ) -> Metric' ℝ

oppositeLocalCoercion :: CanonicalDiffeomorphism ℝ (V1 ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (V1 ℝ, ℝ) -> CanonicalDiffeomorphism (Interior (V1 ℝ)) (Interior ℝ)

NumPrime s => LocallyCoercible (V1 s) (V1 s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V1 s -> V1 s

coerceNeedle :: Functor p => p (V1 s, V1 s) -> Needle (V1 s) -+> Needle (V1 s)

coerceNeedle' :: Functor p => p (V1 s, V1 s) -> Needle' (V1 s) -+> Needle' (V1 s)

coerceNorm :: Functor p => p (V1 s, V1 s) -> Metric (V1 s) -> Metric (V1 s)

coerceVariance :: Functor p => p (V1 s, V1 s) -> Metric' (V1 s) -> Metric' (V1 s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V1 s) (V1 s)

interiorLocalCoercion :: Functor p (->) (->) => p (V1 s, V1 s) -> CanonicalDiffeomorphism (Interior (V1 s)) (Interior (V1 s))

NumPrime s => LocallyCoercible (V2 s) (V2 s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V2 s -> V2 s

coerceNeedle :: Functor p => p (V2 s, V2 s) -> Needle (V2 s) -+> Needle (V2 s)

coerceNeedle' :: Functor p => p (V2 s, V2 s) -> Needle' (V2 s) -+> Needle' (V2 s)

coerceNorm :: Functor p => p (V2 s, V2 s) -> Metric (V2 s) -> Metric (V2 s)

coerceVariance :: Functor p => p (V2 s, V2 s) -> Metric' (V2 s) -> Metric' (V2 s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V2 s) (V2 s)

interiorLocalCoercion :: Functor p (->) (->) => p (V2 s, V2 s) -> CanonicalDiffeomorphism (Interior (V2 s)) (Interior (V2 s))

NumPrime s => LocallyCoercible (V3 s) (V3 s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V3 s -> V3 s

coerceNeedle :: Functor p => p (V3 s, V3 s) -> Needle (V3 s) -+> Needle (V3 s)

coerceNeedle' :: Functor p => p (V3 s, V3 s) -> Needle' (V3 s) -+> Needle' (V3 s)

coerceNorm :: Functor p => p (V3 s, V3 s) -> Metric (V3 s) -> Metric (V3 s)

coerceVariance :: Functor p => p (V3 s, V3 s) -> Metric' (V3 s) -> Metric' (V3 s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V3 s) (V3 s)

interiorLocalCoercion :: Functor p (->) (->) => p (V3 s, V3 s) -> CanonicalDiffeomorphism (Interior (V3 s)) (Interior (V3 s))

NumPrime s => LocallyCoercible (V4 s) (V4 s)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V4 s -> V4 s

coerceNeedle :: Functor p => p (V4 s, V4 s) -> Needle (V4 s) -+> Needle (V4 s)

coerceNeedle' :: Functor p => p (V4 s, V4 s) -> Needle' (V4 s) -+> Needle' (V4 s)

coerceNorm :: Functor p => p (V4 s, V4 s) -> Metric (V4 s) -> Metric (V4 s)

coerceVariance :: Functor p => p (V4 s, V4 s) -> Metric' (V4 s) -> Metric' (V4 s)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V4 s) (V4 s)

interiorLocalCoercion :: Functor p (->) (->) => p (V4 s, V4 s) -> CanonicalDiffeomorphism (Interior (V4 s)) (Interior (V4 s))

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

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V0 s -> ZeroDim s

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

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

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

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

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

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

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

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V0 s -> V0 s

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

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

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

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

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

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

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

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

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ZeroDim s -> ZeroDim s

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

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

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

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

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

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

LocallyCoercible (V2 ℝ) (ℝ, ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V2 ℝ -> (ℝ, ℝ)

coerceNeedle :: Functor p => p (V2 ℝ, (ℝ, ℝ)) -> Needle (V2 ℝ) -+> Needle (ℝ, ℝ)

coerceNeedle' :: Functor p => p (V2 ℝ, (ℝ, ℝ)) -> Needle' (V2 ℝ) -+> Needle' (ℝ, ℝ)

coerceNorm :: Functor p => p (V2 ℝ, (ℝ, ℝ)) -> Metric (V2 ℝ) -> Metric (ℝ, ℝ)

coerceVariance :: Functor p => p (V2 ℝ, (ℝ, ℝ)) -> Metric' (V2 ℝ) -> Metric' (ℝ, ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (ℝ, ℝ) (V2 ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (V2 ℝ, (ℝ, ℝ)) -> CanonicalDiffeomorphism (Interior (V2 ℝ)) (Interior (ℝ, ℝ))

LocallyCoercible (V3 ℝ) ((ℝ, ℝ), ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V3 ℝ -> ((ℝ, ℝ), ℝ)

coerceNeedle :: Functor p => p (V3 ℝ, ((ℝ, ℝ), ℝ)) -> Needle (V3 ℝ) -+> Needle ((ℝ, ℝ), ℝ)

coerceNeedle' :: Functor p => p (V3 ℝ, ((ℝ, ℝ), ℝ)) -> Needle' (V3 ℝ) -+> Needle' ((ℝ, ℝ), ℝ)

coerceNorm :: Functor p => p (V3 ℝ, ((ℝ, ℝ), ℝ)) -> Metric (V3 ℝ) -> Metric ((ℝ, ℝ), ℝ)

coerceVariance :: Functor p => p (V3 ℝ, ((ℝ, ℝ), ℝ)) -> Metric' (V3 ℝ) -> Metric' ((ℝ, ℝ), ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism ((ℝ, ℝ), ℝ) (V3 ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (V3 ℝ, ((ℝ, ℝ), ℝ)) -> CanonicalDiffeomorphism (Interior (V3 ℝ)) (Interior ((ℝ, ℝ), ℝ))

LocallyCoercible (V3 ℝ) (ℝ, (ℝ, ℝ))

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V3 ℝ -> (ℝ, (ℝ, ℝ))

coerceNeedle :: Functor p => p (V3 ℝ, (ℝ, (ℝ, ℝ))) -> Needle (V3 ℝ) -+> Needle (ℝ, (ℝ, ℝ))

coerceNeedle' :: Functor p => p (V3 ℝ, (ℝ, (ℝ, ℝ))) -> Needle' (V3 ℝ) -+> Needle' (ℝ, (ℝ, ℝ))

coerceNorm :: Functor p => p (V3 ℝ, (ℝ, (ℝ, ℝ))) -> Metric (V3 ℝ) -> Metric (ℝ, (ℝ, ℝ))

coerceVariance :: Functor p => p (V3 ℝ, (ℝ, (ℝ, ℝ))) -> Metric' (V3 ℝ) -> Metric' (ℝ, (ℝ, ℝ))

oppositeLocalCoercion :: CanonicalDiffeomorphism (ℝ, (ℝ, ℝ)) (V3 ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (V3 ℝ, (ℝ, (ℝ, ℝ))) -> CanonicalDiffeomorphism (Interior (V3 ℝ)) (Interior (ℝ, (ℝ, ℝ)))

LocallyCoercible (V4 ℝ) ((ℝ, ℝ), (ℝ, ℝ))

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: V4 ℝ -> ((ℝ, ℝ), (ℝ, ℝ))

coerceNeedle :: Functor p => p (V4 ℝ, ((ℝ, ℝ), (ℝ, ℝ))) -> Needle (V4 ℝ) -+> Needle ((ℝ, ℝ), (ℝ, ℝ))

coerceNeedle' :: Functor p => p (V4 ℝ, ((ℝ, ℝ), (ℝ, ℝ))) -> Needle' (V4 ℝ) -+> Needle' ((ℝ, ℝ), (ℝ, ℝ))

coerceNorm :: Functor p => p (V4 ℝ, ((ℝ, ℝ), (ℝ, ℝ))) -> Metric (V4 ℝ) -> Metric ((ℝ, ℝ), (ℝ, ℝ))

coerceVariance :: Functor p => p (V4 ℝ, ((ℝ, ℝ), (ℝ, ℝ))) -> Metric' (V4 ℝ) -> Metric' ((ℝ, ℝ), (ℝ, ℝ))

oppositeLocalCoercion :: CanonicalDiffeomorphism ((ℝ, ℝ), (ℝ, ℝ)) (V4 ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (V4 ℝ, ((ℝ, ℝ), (ℝ, ℝ))) -> CanonicalDiffeomorphism (Interior (V4 ℝ)) (Interior ((ℝ, ℝ), (ℝ, ℝ)))

LocallyCoercible ((ℝ, ℝ), (ℝ, ℝ)) (V4 ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ((ℝ, ℝ), (ℝ, ℝ)) -> V4 ℝ

coerceNeedle :: Functor p => p (((ℝ, ℝ), (ℝ, ℝ)), V4 ℝ) -> Needle ((ℝ, ℝ), (ℝ, ℝ)) -+> Needle (V4 ℝ)

coerceNeedle' :: Functor p => p (((ℝ, ℝ), (ℝ, ℝ)), V4 ℝ) -> Needle' ((ℝ, ℝ), (ℝ, ℝ)) -+> Needle' (V4 ℝ)

coerceNorm :: Functor p => p (((ℝ, ℝ), (ℝ, ℝ)), V4 ℝ) -> Metric ((ℝ, ℝ), (ℝ, ℝ)) -> Metric (V4 ℝ)

coerceVariance :: Functor p => p (((ℝ, ℝ), (ℝ, ℝ)), V4 ℝ) -> Metric' ((ℝ, ℝ), (ℝ, ℝ)) -> Metric' (V4 ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V4 ℝ) ((ℝ, ℝ), (ℝ, ℝ))

interiorLocalCoercion :: Functor p (->) (->) => p (((ℝ, ℝ), (ℝ, ℝ)), V4 ℝ) -> CanonicalDiffeomorphism (Interior ((ℝ, ℝ), (ℝ, ℝ))) (Interior (V4 ℝ))

LocallyCoercible ((ℝ, ℝ), ℝ) (V3 ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ((ℝ, ℝ), ℝ) -> V3 ℝ

coerceNeedle :: Functor p => p (((ℝ, ℝ), ℝ), V3 ℝ) -> Needle ((ℝ, ℝ), ℝ) -+> Needle (V3 ℝ)

coerceNeedle' :: Functor p => p (((ℝ, ℝ), ℝ), V3 ℝ) -> Needle' ((ℝ, ℝ), ℝ) -+> Needle' (V3 ℝ)

coerceNorm :: Functor p => p (((ℝ, ℝ), ℝ), V3 ℝ) -> Metric ((ℝ, ℝ), ℝ) -> Metric (V3 ℝ)

coerceVariance :: Functor p => p (((ℝ, ℝ), ℝ), V3 ℝ) -> Metric' ((ℝ, ℝ), ℝ) -> Metric' (V3 ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V3 ℝ) ((ℝ, ℝ), ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p (((ℝ, ℝ), ℝ), V3 ℝ) -> CanonicalDiffeomorphism (Interior ((ℝ, ℝ), ℝ)) (Interior (V3 ℝ))

LocallyCoercible (ℝ, (ℝ, ℝ)) (V3 ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: (ℝ, (ℝ, ℝ)) -> V3 ℝ

coerceNeedle :: Functor p => p ((ℝ, (ℝ, ℝ)), V3 ℝ) -> Needle (ℝ, (ℝ, ℝ)) -+> Needle (V3 ℝ)

coerceNeedle' :: Functor p => p ((ℝ, (ℝ, ℝ)), V3 ℝ) -> Needle' (ℝ, (ℝ, ℝ)) -+> Needle' (V3 ℝ)

coerceNorm :: Functor p => p ((ℝ, (ℝ, ℝ)), V3 ℝ) -> Metric (ℝ, (ℝ, ℝ)) -> Metric (V3 ℝ)

coerceVariance :: Functor p => p ((ℝ, (ℝ, ℝ)), V3 ℝ) -> Metric' (ℝ, (ℝ, ℝ)) -> Metric' (V3 ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V3 ℝ) (ℝ, (ℝ, ℝ))

interiorLocalCoercion :: Functor p (->) (->) => p ((ℝ, (ℝ, ℝ)), V3 ℝ) -> CanonicalDiffeomorphism (Interior (ℝ, (ℝ, ℝ))) (Interior (V3 ℝ))

LocallyCoercible (ℝ, ℝ) (V2 ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: (ℝ, ℝ) -> V2 ℝ

coerceNeedle :: Functor p => p ((ℝ, ℝ), V2 ℝ) -> Needle (ℝ, ℝ) -+> Needle (V2 ℝ)

coerceNeedle' :: Functor p => p ((ℝ, ℝ), V2 ℝ) -> Needle' (ℝ, ℝ) -+> Needle' (V2 ℝ)

coerceNorm :: Functor p => p ((ℝ, ℝ), V2 ℝ) -> Metric (ℝ, ℝ) -> Metric (V2 ℝ)

coerceVariance :: Functor p => p ((ℝ, ℝ), V2 ℝ) -> Metric' (ℝ, ℝ) -> Metric' (V2 ℝ)

oppositeLocalCoercion :: CanonicalDiffeomorphism (V2 ℝ) (ℝ, ℝ)

interiorLocalCoercion :: Functor p (->) (->) => p ((ℝ, ℝ), V2 ℝ) -> CanonicalDiffeomorphism (Interior (ℝ, ℝ)) (Interior (V2 ℝ))

(Semimanifold a, Semimanifold b, Semimanifold c, LSpace (Needle a), LSpace (Needle b), LSpace (Needle c), Scalar (Needle a) ~ Scalar (Needle b), Scalar (Needle b) ~ Scalar (Needle c), Scalar (Needle' a) ~ Scalar (Needle a), Scalar (Needle' b) ~ Scalar (Needle b), Scalar (Needle' c) ~ Scalar (Needle c)) => LocallyCoercible ((a, b), c) (a, (b, c))

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ((a, b), c) -> (a, (b, c))

coerceNeedle :: Functor p => p (((a, b), c), (a, (b, c))) -> Needle ((a, b), c) -+> Needle (a, (b, c))

coerceNeedle' :: Functor p => p (((a, b), c), (a, (b, c))) -> Needle' ((a, b), c) -+> Needle' (a, (b, c))

coerceNorm :: Functor p => p (((a, b), c), (a, (b, c))) -> Metric ((a, b), c) -> Metric (a, (b, c))

coerceVariance :: Functor p => p (((a, b), c), (a, (b, c))) -> Metric' ((a, b), c) -> Metric' (a, (b, c))

oppositeLocalCoercion :: CanonicalDiffeomorphism (a, (b, c)) ((a, b), c)

interiorLocalCoercion :: Functor p (->) (->) => p (((a, b), c), (a, (b, c))) -> CanonicalDiffeomorphism (Interior ((a, b), c)) (Interior (a, (b, c)))

LocallyCoercible ((ℝ, ℝ), ℝ) ((ℝ, ℝ), ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: ((ℝ, ℝ), ℝ) -> ((ℝ, ℝ), ℝ)

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

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

coerceNorm :: Functor p => p (((ℝ, ℝ), ℝ), ((ℝ, ℝ), ℝ)) -> Metric ((ℝ, ℝ), ℝ) -> Metric ((ℝ, ℝ), ℝ)

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

oppositeLocalCoercion :: CanonicalDiffeomorphism ((ℝ, ℝ), ℝ) ((ℝ, ℝ), ℝ)

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

(Semimanifold a, Semimanifold b, Semimanifold c, LSpace (Needle a), LSpace (Needle b), LSpace (Needle c), Scalar (Needle a) ~ Scalar (Needle b), Scalar (Needle b) ~ Scalar (Needle c), Scalar (Needle' a) ~ Scalar (Needle a), Scalar (Needle' b) ~ Scalar (Needle b), Scalar (Needle' c) ~ Scalar (Needle c)) => LocallyCoercible (a, (b, c)) ((a, b), c)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: (a, (b, c)) -> ((a, b), c)

coerceNeedle :: Functor p => p ((a, (b, c)), ((a, b), c)) -> Needle (a, (b, c)) -+> Needle ((a, b), c)

coerceNeedle' :: Functor p => p ((a, (b, c)), ((a, b), c)) -> Needle' (a, (b, c)) -+> Needle' ((a, b), c)

coerceNorm :: Functor p => p ((a, (b, c)), ((a, b), c)) -> Metric (a, (b, c)) -> Metric ((a, b), c)

coerceVariance :: Functor p => p ((a, (b, c)), ((a, b), c)) -> Metric' (a, (b, c)) -> Metric' ((a, b), c)

oppositeLocalCoercion :: CanonicalDiffeomorphism ((a, b), c) (a, (b, c))

interiorLocalCoercion :: Functor p (->) (->) => p ((a, (b, c)), ((a, b), c)) -> CanonicalDiffeomorphism (Interior (a, (b, c))) (Interior ((a, b), c))

LocallyCoercible (ℝ, (ℝ, ℝ)) (ℝ, (ℝ, ℝ))

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: (ℝ, (ℝ, ℝ)) -> (ℝ, (ℝ, ℝ))

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

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

coerceNorm :: Functor p => p ((ℝ, (ℝ, ℝ)), (ℝ, (ℝ, ℝ))) -> Metric (ℝ, (ℝ, ℝ)) -> Metric (ℝ, (ℝ, ℝ))

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

oppositeLocalCoercion :: CanonicalDiffeomorphism (ℝ, (ℝ, ℝ)) (ℝ, (ℝ, ℝ))

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

LocallyCoercible (ℝ, ℝ) (ℝ, ℝ)

Defined in Data.Manifold.PseudoAffine

Methods

locallyTrivialDiffeomorphism :: (ℝ, ℝ) -> (ℝ, ℝ)

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

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

coerceNorm :: Functor p => p ((ℝ, ℝ), (ℝ, ℝ)) -> Metric (ℝ, ℝ) -> Metric (ℝ, ℝ)

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

oppositeLocalCoercion :: CanonicalDiffeomorphism (ℝ, ℝ) (ℝ, ℝ)

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

data CanonicalDiffeomorphism a b where

Constructors

CanonicalDiffeomorphism :: LocallyCoercible a b => CanonicalDiffeomorphism a b

class ImpliesMetric s where

Associated Types

type MetricRequirement s x :: Constraint

Methods

inferMetric :: (MetricRequirement s x, LSpace (Needle x)) => s x -> Metric x

inferMetric' :: (MetricRequirement s x, LSpace (Needle x)) => s x -> Metric' x

Instances

ImpliesMetric Norm

Defined in Data.Manifold.PseudoAffine

Associated Types

type MetricRequirement Norm x :: Constraint

Methods

inferMetric :: (MetricRequirement Norm x, LSpace (Needle x)) => Norm x -> Metric x

inferMetric' :: (MetricRequirement Norm x, LSpace (Needle x)) => Norm x -> Metric' x

ImpliesMetric Shade'

Defined in Data.Manifold.Shade

Associated Types

type MetricRequirement Shade' x :: Constraint

Methods

inferMetric :: (MetricRequirement Shade' x, LSpace (Needle x)) => Shade' x -> Metric x

inferMetric' :: (MetricRequirement Shade' x, LSpace (Needle x)) => Shade' x -> Metric' x

ImpliesMetric Shade

Defined in Data.Manifold.Shade

Associated Types

type MetricRequirement Shade x :: Constraint

Methods

inferMetric :: (MetricRequirement Shade x, LSpace (Needle x)) => Shade x -> Metric x

inferMetric' :: (MetricRequirement Shade x, LSpace (Needle x)) => Shade x -> Metric' x

coerceMetric :: forall x ξ. (LocallyCoercible x ξ, LSpace (Needle ξ)) => RieMetric ξ -> RieMetric x

coerceMetric' :: forall x ξ. (LocallyCoercible x ξ, LSpace (Needle ξ)) => RieMetric' ξ -> RieMetric' x

class PseudoAffine m => Connected m where

A connected manifold is one where any point can be reached by translation from any other point.

Minimal complete definition

Nothing

Methods

(.−.) :: m -> m -> Needle m infix 6

Safe version of '(.-~.)'.

Instances

Connected ℝP⁰

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

Connected S¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S¹ -> S¹ -> Needle S¹

Connected ℝP¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

Connected S²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: S² -> S² -> Needle S²

Connected ℝP²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝP² -> ℝP² -> Needle ℝP²

Connected ℝ

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ -> ℝ -> Needle ℝ

Connected ℝ⁰

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ⁰ -> ℝ⁰ -> Needle ℝ⁰

Connected ℝ⁴

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ⁴ -> ℝ⁴ -> Needle ℝ⁴

Connected ℝ³

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ³ -> ℝ³ -> Needle ℝ³

Connected ℝ²

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ² -> ℝ² -> Needle ℝ²

Connected ℝ¹

Defined in Data.Manifold.PseudoAffine

Methods

(.−.) :: ℝ¹ -> ℝ¹ -> Needle ℝ¹

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

Defined in Data.Manifold.PseudoAffine

Methods

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

(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)

Orphan instances

Semimanifold S²

Associated Types

type Needle S² :: Type

type Interior S² :: Type

Methods

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

fromInterior :: Interior S² -> S²

toInterior :: S² -> Maybe (Interior S²)

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

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

semimanifoldWitness :: SemimanifoldWitness S²

Semimanifold ℝP²

Associated Types

type Needle ℝP² :: Type

type Interior ℝP² :: Type

Methods

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

fromInterior :: Interior ℝP² -> ℝP²

toInterior :: ℝP² -> Maybe (Interior ℝP²)

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

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

semimanifoldWitness :: SemimanifoldWitness ℝP²

PseudoAffine S²

Methods

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

(.-~!) :: S² -> S² -> Needle S²

pseudoAffineWitness :: PseudoAffineWitness S²

PseudoAffine ℝP²

Methods

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

(.-~!) :: ℝP² -> ℝP² -> Needle ℝP²

pseudoAffineWitness :: PseudoAffineWitness ℝP²

(LinearSpace (a n), Needle (a n) ~ a n, Interior (a n) ~ a n) => Semimanifold (Point a n)

Associated Types

type Needle (Point a n) :: Type

type Interior (Point a n) :: Type

Methods

(.+~^) :: Interior (Point a n) -> Needle (Point a n) -> Point a n

fromInterior :: Interior (Point a n) -> Point a n

toInterior :: Point a n -> Maybe (Interior (Point a n))

translateP :: Tagged (Point a n) (Interior (Point a n) -> Needle (Point a n) -> Interior (Point a n))

(.-~^) :: Interior (Point a n) -> Needle (Point a n) -> Point a n

semimanifoldWitness :: SemimanifoldWitness (Point a n)

(LinearSpace (a n), Needle (a n) ~ a n, Interior (a n) ~ a n) => PseudoAffine (Point a n)

Methods

(.-~.) :: Point a n -> Point a n -> Maybe (Needle (Point a n))

(.-~!) :: Point a n -> Point a n -> Needle (Point a n)

pseudoAffineWitness :: PseudoAffineWitness (Point a n)