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

Morphism (:-->)

PreArrow (:-->)

CartesianAgent (:-->)

Category (:-->)

Cartesian (:-->)

HasAgent (:-->)

EnhancedCat (->) (:-->)

PointAgent CntnFuncValue (:-->) d c

type UnitObject (:-->) = ()

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

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

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

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)

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

asinh__ :: CntnRealFunction

tanh__ :: CntnRealFunction

cosh__ :: CntnRealFunction

sinh__ :: CntnRealFunction

exp__ :: CntnRealFunction

atan__ :: CntnRealFunction

cos__ :: 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)

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

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

data GraphWindowSpec

Constructors

GraphWindowSpec

Fields

lBound :: Double

rBound :: Double

bBound :: Double

tBound :: Double

xResolution :: Int

yResolution :: Int

data ZeroDim k

Constructors

Origin

Instances

Eq (ZeroDim k)

Show (ZeroDim k)

Monoid (ZeroDim k)

HasBasis (ZeroDim k)

VectorSpace (ZeroDim k)

AdditiveGroup (ZeroDim k)

VectorSpace k => HasMetric (ZeroDim k)

PseudoAffine (ZeroDim k)

type Basis (ZeroDim k) = Void

type Scalar (ZeroDim k) = k

type DualSpace (ZeroDim k) = ZeroDim k

type PseudoDiff (ZeroDim k) = ZeroDim k

data S⁰

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

Eq S⁰

Show S⁰

NaturallyEmbedded S⁰ ℝ

newtype S¹

Constructors

S¹

Fields

φParamS¹ :: Double

Instances

Show S¹

PseudoAffine S¹

NaturallyEmbedded S¹ ℝ²

type PseudoDiff S¹ = ℝ

data S²

Constructors

S²

Fields

ϑParamS² :: !Double

φParamS² :: !Double

Instances

Show S²

PseudoAffine S²

NaturallyEmbedded S² ℝ³

type PseudoDiff S² = ℝ²

class NaturallyEmbedded m v where

Methods

embed :: m -> v

coEmbed :: v -> m

Instances

NaturallyEmbedded S² ℝ³

NaturallyEmbedded S¹ ℝ²

NaturallyEmbedded S⁰ ℝ

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

VectorSpace y => NaturallyEmbedded x (x, y)

type Endomorphism a = a -> a

type ℝ = Double

type ℝ² = (ℝ, ℝ)

type ℝ³ = (ℝ², ℝ)

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