Data.Manifold

This is something of a first attempt at formalising manifolds and continuous mappings thereon. They work (check out http://hackage.haskell.org/package/dynamic-plot-0.1.0.0 for a use case), but aren't very efficient. The interface might well change considerably in the future.

Documentation

data domain :--> codomain where

Continuous mapping.

Constructors

Continuous :: (Manifold d, Manifold c, v ~ TangentSpace d, u ~ TangentSpace c, δ ~ Metric v, ε ~ Metric u) => (Chart d -> v -> (Chart c, u, ε -> Option δ)) -> d :--> c

Fields

runContinuous :: Chart d -> v -> (Chart c, u, ε -> Option δ)

Instances

HasAgent (:-->)

Category (:-->)

Cartesian (:-->)

PreArrow (:-->)

Morphism (:-->)

CartesianAgent (:-->)

EnhancedCat (->) (:-->)

PointAgent CntnFuncValue (:-->) d c

type UnitObject (:-->) = ()

type Object (:-->) t = Manifold t

type AgentVal (:-->) d c = CntnFuncValue d c

type PairObjects (:-->) a b = (FlatManifold a, FlatManifold b, Manifold (a, b))

continuous_id' :: Manifold m => m :--> m

const__ :: (Manifold c, Manifold d) => c -> d :--> c

flatContinuous :: (FlatManifold v, FlatManifold w, δ ~ Metric v, ε ~ Metric w) => (v -> (w, ε -> Option δ)) -> v :--> w

runFlatContinuous :: (FlatManifold v, FlatManifold w, δ ~ Metric v, ε ~ Metric w) => (v :--> w) -> v -> (w, ε -> Option δ)

data Chart :: * -> * where

A chart is a homeomorphism from a connected, open subset Q ⊂ M of an n-manifold M to either the open unit disk Dⁿ ⊂ V ≃ ℝⁿ, or the half-disk Hⁿ = {x ∊ Dⁿ: x₀≥0}. In e.g. the former case, chartInMap is thus defined ∀ v ∊ V : |v| < 1, while 'chartOutMap p' will yield Just x with x ∊ Dⁿ provided p is in Q, and Nothing otherwise. Obviously, fromJust . chartOutMap . chartInMap should be equivalent to id on Dⁿ, and chartInMap . fromJust . chartOutMap to id on Q.

Constructors

IdChart :: FlatManifold v => Chart v

Chart :: (Manifold m, v ~ TangentSpace m, FlatManifold v) => (v :--> m) -> (m -> Maybe (m :--> v)) -> ChartKind -> Chart m

Fields

chartInMap :: v :--> m

chartOutMap :: m -> Maybe (m :--> v)

chartKind :: ChartKind

data ChartKind

Constructors

LandlockedChart

A M ⇆ Dⁿ chart, for ordinary manifolds

RimChart

A M ⇆ Hⁿ chart, for manifolds with a rim

type FlatManifold v = (MetricSpace v, Manifold v, v ~ TangentSpace v)

type EuclidSpace v = (HasBasis v, EqFloating (Scalar v), Eq v)

isInUpperHemi :: EuclidSpace v => v -> Bool

type Atlas m = [Chart m]

class (MetricSpace (TangentSpace m), Metric (TangentSpace m) ~ ℝ) => Manifold m where

Associated Types

type TangentSpace m :: *

Methods

localAtlas :: m -> Atlas m

Instances

Manifold Double

Manifold ()

(FlatManifold v₁, FlatManifold v₂, (~) * (Scalar v₁) (Scalar v₂), MetricSpace (Scalar v₁), (~) * (Metric (Scalar v₁)) ℝ, VectorSpace (v₁, v₂), (~) * (Scalar (v₁, v₂)) (Scalar v₁)) => Manifold (v₁, v₂)

vectorSpaceAtlas :: FlatManifold v => v -> Atlas v

type Representsℝ r = (EqFloating r, FlatManifold r, r ~ Scalar r, r ~ Metric r)

continuousFlatFunction :: (FlatManifold d, FlatManifold c, ε ~ Metric c, δ ~ Metric d) => (d -> (c, ε -> Option δ)) -> d :--> c

type CntnRealFunction = Representsℝ r => r :--> r

sin__ :: CntnRealFunction

cos__ :: CntnRealFunction

atan__ :: CntnRealFunction

exp__ :: CntnRealFunction

sinh__ :: CntnRealFunction

cosh__ :: CntnRealFunction

tanh__ :: CntnRealFunction

asinh__ :: CntnRealFunction

cntnFuncsCombine :: forall d v c c' c'' ε ε' ε''. (FlatManifold c, FlatManifold c', FlatManifold c'', ε ~ Metric c, ε' ~ Metric c', ε'' ~ Metric c'', ε ~ ε', ε ~ ε'') => (c' -> c'' -> (c, ε -> (ε', ε''))) -> (d :--> c') -> (d :--> c'') -> d :--> c

data CntnFuncValue d c

Constructors

CntnFuncValue

Fields

runCntnFuncValue :: d :--> c

CntnFuncConst c

Instances

PointAgent CntnFuncValue (:-->) d c

(Representsℝ r, Manifold d) => Floating (CntnFuncValue d r)

(Representsℝ r, Manifold d) => Fractional (CntnFuncValue d r)

(Representsℝ r, Manifold d) => Num (CntnFuncValue d r)

(FlatManifold v, MetricSpace v, (~) * (Metric v) ℝ, FlatManifold (Scalar v), MetricSpace (Scalar v), (~) * (Metric (Scalar v)) ℝ, Manifold d) => VectorSpace (CntnFuncValue d v)

(FlatManifold v, Manifold d) => AdditiveGroup (CntnFuncValue d v)

type Scalar (CntnFuncValue d v) = CntnFuncValue d (Scalar v)

cntnFnValsFunc :: (FlatManifold c, FlatManifold c', Manifold d, ε ~ Metric c, ε ~ Metric c') => (c' -> (c, ε -> Option ε)) -> CntnFuncValue d c' -> CntnFuncValue d c

cntnFnValsCombine :: forall d c c' c'' ε ε' ε''. (FlatManifold c, FlatManifold c', FlatManifold c'', Manifold d, ε ~ Metric c, ε' ~ Metric c', ε'' ~ Metric c'', ε ~ ε', ε ~ ε'') => (c' -> c'' -> (c, ε -> (ε', (ε', ε''), ε''))) -> CntnFuncValue d c' -> CntnFuncValue d c'' -> CntnFuncValue d c

finiteGraphContinℝtoℝ :: GraphWindowSpec -> (Double :--> Double) -> [(Double, Double)]

finiteGraphContinℝtoℝ² :: GraphWindowSpec -> (Double :--> (Double, Double)) -> [[(Double, Double)]]

midBetween :: (VectorSpace v, Fractional (Scalar v)) => [v] -> v

(.:) :: (c -> d) -> (a -> b -> c) -> a -> b -> d

just :: a -> Option a

nothing :: Option a

class (RealFloat (Metric v), InnerSpace v) => MetricSpace v where

Minimal complete definition

metricToScalar

Associated Types

type Metric v :: *

Methods

metric :: v -> Metric v

metricSq :: v -> Metric v

(|*^) :: Metric v -> v -> v

metricToScalar :: v -> Metric v -> Scalar v

Instances

MetricSpace ()

MetricSpace ℝ

(RealFloat r, MetricSpace r, (~) * (Scalar (Complex r)) (Metric r)) => MetricSpace (Complex r)

(MetricSpace v, MetricSpace (Scalar v), MetricSpace w, (~) * (Scalar v) (Scalar w), (~) * (Metric v) (Metric (Scalar v)), (~) * (Metric w) (Metric v), (~) * (Metric (Scalar w)) (Metric v), RealFloat (Metric v)) => MetricSpace (v, w)

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 Projective1 = ℝP¹

type Projective2 = ℝP²

type Disk1 = D¹

type Disk2 = D²

type Cone = CD¹

type OpenCone = Cℝay

Linear manifolds

data ZeroDim k

A single point. Can be considered a zero-dimensional vector space, WRT any scalar.

Constructors

Origin

Instances

Eq (ZeroDim k)

Show (ZeroDim k)

Monoid (ZeroDim k)

VectorSpace (ZeroDim k)

AdditiveGroup (ZeroDim k)

AffineSpace (ZeroDim k)

HasBasis (ZeroDim k)

SmoothScalar k => FiniteDimensional (ZeroDim k)

MetricScalar k => HasMetric' (ZeroDim k)

PseudoAffine (ZeroDim k)

Semimanifold (ZeroDim k)

IntervalLike (Cℝay ℝ⁰)

IntervalLike (CD¹ ℝ⁰)

Geodesic (Cℝay ℝ⁰)

Geodesic (CD¹ ℝ⁰)

Geodesic (ZeroDim ℝ)

type Scalar (ZeroDim k) = k

type Diff (ZeroDim k) = ZeroDim k

type Basis (ZeroDim k) = Void

type DualSpace (ZeroDim k) = ZeroDim k

type Needle (ZeroDim k) = ZeroDim k

type Interior (ZeroDim k) = ZeroDim k

isoAttachZeroDim :: (WellPointed c, UnitObject c ~ (), ObjectPair c a (), Object c (ZeroDim k), ObjectPair c a (ZeroDim k), PointObject c (ZeroDim k)) => Isomorphism c a (a, ZeroDim k)

type ℝ⁰ = ZeroDim ℝ

type ℝ = Double

type ℝ² = (ℝ, ℝ)

type ℝ³ = (ℝ², ℝ)

Hyperspheres

data S⁰

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

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

Eq S⁰

Show S⁰

PseudoAffine S⁰

Semimanifold S⁰

Geodesic S⁰

NaturallyEmbedded S⁰ ℝ

IntervalLike (Cℝay S⁰)

IntervalLike (CD¹ S⁰)

Geodesic (Cℝay S⁰)

Geodesic (CD¹ S⁰)

type Needle S⁰ = ℝ⁰

type Interior S⁰ = S⁰

otherHalfSphere :: S⁰ -> S⁰

newtype S¹

The unit circle.

Constructors

S¹

Fields

φParamS¹ :: Double

Must be in range [-π, π[.

Instances

Show S¹

PseudoAffine S¹

Semimanifold S¹

Geodesic S¹

NaturallyEmbedded S¹ ℝ²

Geodesic (Cℝay S¹)

Geodesic (CD¹ S¹)

type Needle S¹ = ℝ

type Interior S¹ = S¹

data S²

The ordinary unit sphere.

Constructors

S²

Fields

ϑParamS² :: !Double

Range [0, π[.

φParamS² :: !Double

Range [-π, π[.

Instances

Show S²

PseudoAffine S²

Semimanifold S²

NaturallyEmbedded S² ℝ³

Geodesic (Cℝay S²)

Geodesic (CD¹ S²)

type Needle S² = ℝ²

type Interior S² = S²

Projective spaces

type ℝP¹ = S¹

data ℝP²

The two-dimensional real projective space, implemented as a unit disk with opposing points on the rim glued together.

Constructors

ℝP²

Fields

rParamℝP² :: !Double

Range [0, 1].

φParamℝP² :: !Double

Range [-π, π[.

Instances

Show ℝP²

PseudoAffine ℝP²

Semimanifold ℝP²

NaturallyEmbedded ℝP² ℝ³

type Needle ℝP² = ℝ²

type Interior ℝP² = ℝP²

Intervals/disks/cones

newtype D¹

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

Constructors

D¹

Fields

xParamD¹ :: Double

Range [-1, 1].

Instances

PseudoAffine D¹

Semimanifold D¹

IntervalLike D¹

NaturallyEmbedded D¹ ℝ

type Needle D¹ = ℝ

type Interior D¹ = ℝ

data D²

The standard, closed unit disk. Homeomorphic to the cone over 'S¹', but not in the the obvious, “flat” way. (And not at all, despite the identical ADT definition, to the projective space 'ℝP²'!)

Constructors

D²

Fields

rParamD² :: !Double

Range [0, 1].

φParamD² :: !Double

Range [-π, π[.

Instances

Show D²

type ℝay = Cℝay ℝ⁰

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

data CD¹ x

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

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

Constructors

CD¹

Fields

hParamCD¹ :: !Double

Range [0, 1]

pParamCD¹ :: !x

Irrelevant at h = 0.

Instances

IntervalLike (CD¹ ℝ⁰)

IntervalLike (CD¹ S⁰)

(WithField ℝ HilbertSpace a, WithField ℝ HilbertSpace b, Geodesic (a, b)) => Geodesic (CD¹ (a, b))

Geodesic (CD¹ ℝ)

Geodesic (CD¹ ℝ⁰)

Geodesic (CD¹ S²)

Geodesic (CD¹ S¹)

Geodesic (CD¹ S⁰)

type Needle (CD¹ m)

type Interior (CD¹ m)

data Cℝay x

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

Constructors

Cℝay

Fields

hParamCℝay :: !Double

Range [0, ∞[

pParamCℝay :: !x

Irrelevant at h = 0.

Instances

IntervalLike (Cℝay ℝ⁰)

IntervalLike (Cℝay S⁰)

(WithField ℝ HilbertSpace a, WithField ℝ HilbertSpace b, Geodesic (a, b)) => Geodesic (Cℝay (a, b))

Geodesic (Cℝay ℝ)

Geodesic (Cℝay ℝ⁰)

Geodesic (Cℝay S²)

Geodesic (Cℝay S¹)

Geodesic (Cℝay S⁰)

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

type Needle (Cℝay m)

type Interior (Cℝay m)

Tensor products

newtype x ⊗ y

Dense tensor product of two vector spaces.

Constructors

DensTensProd

Fields

getDensTensProd :: Matrix (Scalar y)

Instances

(FiniteDimensional a, FiniteDimensional b, (~) * (Scalar a) (Scalar b)) => FiniteDimensional ((⊗) a b)

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) (Scalar b)) => PseudoAffine ((⊗) a b)

(HasMetric a, FiniteDimensional b, (~) * (Scalar a) (Scalar b)) => Semimanifold ((⊗) a b)

type Scalar ((⊗) x y) = Scalar y

type Basis ((⊗) x y) = (Basis x, Basis y)

type Needle ((⊗) a b) = (⊗) a b

type Interior ((⊗) a b) = (⊗) a b

Utility (deprecated)

class NaturallyEmbedded m v where

Methods

embed :: m -> v

coEmbed :: v -> m

Instances

NaturallyEmbedded D¹ ℝ

NaturallyEmbedded ℝP² ℝ³

NaturallyEmbedded S² ℝ³

NaturallyEmbedded S¹ ℝ²

NaturallyEmbedded S⁰ ℝ

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

VectorSpace y => NaturallyEmbedded x (x, y)

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

data GraphWindowSpec

Constructors

GraphWindowSpec

Fields

lBound, rBound, bBound, tBound :: Double

xResolution, yResolution :: Int

type Endomorphism a = a -> a

(^) :: Num a => a -> Int -> a infixr 8

(^.) :: s -> (forall f. Functor f => (a -> f a) -> s -> f s) -> a infixl 8

type EqFloating f = (Eq f, Ord f, Floating f)

empty :: Alternative f => forall a. f a

The identity of <|>