Math.Manifold.Core.Types

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

Documentation

type ℝ⁰ = ZeroDim ℝ

type ℝ = Double

data S⁰

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

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

Eq S⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S⁰ -> S⁰ -> Bool

(/=) :: S⁰ -> S⁰ -> Bool

Show S⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S⁰ -> ShowS

show :: S⁰ -> String

showList :: [S⁰] -> ShowS

Generic S⁰

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S⁰ :: Type -> Type

Methods

from :: S⁰ -> Rep S⁰ x

to :: Rep S⁰ x -> S⁰

PseudoAffine S⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: S⁰ -> S⁰ -> Needle S⁰

pseudoAffineWitness :: PseudoAffineWitness S⁰

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

type Rep S⁰

Defined in Math.Manifold.Core.Types.Internal

type Rep S⁰ = D1 (MetaData "S\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "PositiveHalfSphere" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "NegativeHalfSphere" PrefixI False) (U1 :: Type -> Type))

type Needle S⁰

Defined in Math.Manifold.Core.PseudoAffine

type Needle S⁰ = ZeroDim ℝ

type Interior S⁰

Defined in Math.Manifold.Core.PseudoAffine

type Interior S⁰ = S⁰

otherHalfSphere :: S⁰ -> S⁰

newtype S¹

The unit circle.

Constructors

S¹Polar

Fields

• φParamS¹ :: Double

  Must be in range [-π, π[.

Instances

Eq S¹

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S¹ -> S¹ -> Bool

(/=) :: S¹ -> S¹ -> Bool

Show S¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S¹ -> ShowS

show :: S¹ -> String

showList :: [S¹] -> ShowS

Generic S¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S¹ :: Type -> Type

Methods

from :: S¹ -> Rep S¹ x

to :: Rep S¹ x -> S¹

PseudoAffine S¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: S¹ -> S¹ -> Needle S¹

pseudoAffineWitness :: PseudoAffineWitness S¹

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

type Rep S¹

Defined in Math.Manifold.Core.Types.Internal

type Rep S¹ = D1 (MetaData "S\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" True) (C1 (MetaCons "S\185Polar" PrefixI True) (S1 (MetaSel (Just "\966ParamS\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Needle S¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle S¹ = ℝ

type Interior S¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior S¹ = S¹

pattern S¹ :: Double -> S¹

Deprecated: Use Math.Manifold.Core.Types.S¹Polar

data S²

The ordinary unit sphere.

Constructors

S²Polar

Fields

• ϑParamS² :: !Double

  Range [0, π[.

• φParamS² :: !Double

  Range [-π, π[.

Instances

Eq S²

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S² -> S² -> Bool

(/=) :: S² -> S² -> Bool

Show S²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S² -> ShowS

show :: S² -> String

showList :: [S²] -> ShowS

Generic S²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep S² :: Type -> Type

Methods

from :: S² -> Rep S² x

to :: Rep S² x -> S²

type Rep S²

Defined in Math.Manifold.Core.Types.Internal

type Rep S² = D1 (MetaData "S\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "S\178Polar" PrefixI True) (S1 (MetaSel (Just "\977ParamS\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966ParamS\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

pattern S² :: Double -> Double -> S²

Deprecated: Use Math.Manifold.Core.Types.S²Polar

newtype D¹

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

Constructors

D¹

Fields

• xParamD¹ :: Double

  Range [-1, 1].

Instances

Show D¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D¹ -> ShowS

show :: D¹ -> String

showList :: [D¹] -> ShowS

Generic D¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep D¹ :: Type -> Type

Methods

from :: D¹ -> Rep D¹ x

to :: Rep D¹ x -> D¹

PseudoAffine D¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: D¹ -> D¹ -> Needle D¹

pseudoAffineWitness :: PseudoAffineWitness D¹

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

type Rep D¹

Defined in Math.Manifold.Core.Types.Internal

type Rep D¹ = D1 (MetaData "D\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" True) (C1 (MetaCons "D\185" PrefixI True) (S1 (MetaSel (Just "xParamD\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Needle D¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle D¹ = ℝ

type Interior D¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior D¹ = ℝ

fromIntv0to1 :: ℝ -> D¹

data D²

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

Constructors

D²Polar

Fields

• rParamD² :: !Double

  Range [0, 1].

• φParamD² :: !Double

  Range [-π, π[.

Instances

Show D²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D² -> ShowS

show :: D² -> String

showList :: [D²] -> ShowS

Generic D²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep D² :: Type -> Type

Methods

from :: D² -> Rep D² x

to :: Rep D² x -> D²

type Rep D²

Defined in Math.Manifold.Core.Types.Internal

type Rep D² = D1 (MetaData "D\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "D\178Polar" PrefixI True) (S1 (MetaSel (Just "rParamD\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966ParamD\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

pattern D² :: Double -> Double -> D²

Deprecated: Use Math.Manifold.Core.Types.D²Polar

data ℝP⁰

Constructors

ℝPZero

Instances

Eq ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: ℝP⁰ -> ℝP⁰ -> Bool

(/=) :: ℝP⁰ -> ℝP⁰ -> Bool

Show ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP⁰ -> ShowS

show :: ℝP⁰ -> String

showList :: [ℝP⁰] -> ShowS

Generic ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP⁰ :: Type -> Type

Methods

from :: ℝP⁰ -> Rep ℝP⁰ x

to :: Rep ℝP⁰ x -> ℝP⁰

PseudoAffine ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰

pseudoAffineWitness :: PseudoAffineWitness ℝP⁰

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

type Rep ℝP⁰

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP⁰ = D1 (MetaData "\8477P\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "\8477PZero" PrefixI False) (U1 :: Type -> Type))

type Needle ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

type Needle ℝP⁰ = ZeroDim ℝ

type Interior ℝP⁰

Defined in Math.Manifold.Core.PseudoAffine

type Interior ℝP⁰ = ℝP⁰

newtype ℝP¹

Constructors

HemisphereℝP¹Polar

Fields

• φParamℝP¹ :: Double

  Range [-π/2,π/2[.

Instances

Show ℝP¹

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP¹ -> ShowS

show :: ℝP¹ -> String

showList :: [ℝP¹] -> ShowS

Generic ℝP¹

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP¹ :: Type -> Type

Methods

from :: ℝP¹ -> Rep ℝP¹ x

to :: Rep ℝP¹ x -> ℝP¹

PseudoAffine ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

Methods

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

(.-~!) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹

pseudoAffineWitness :: PseudoAffineWitness ℝP¹

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

type Rep ℝP¹

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP¹ = D1 (MetaData "\8477P\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" True) (C1 (MetaCons "Hemisphere\8477P\185Polar" PrefixI True) (S1 (MetaSel (Just "\966Param\8477P\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))

type Needle ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

type Needle ℝP¹ = ℝ

type Interior ℝP¹

Defined in Math.Manifold.Core.PseudoAffine

type Interior ℝP¹ = ℝP¹

pattern ℝP¹ :: Double -> ℝP¹

Deprecated: Use Math.Manifold.Core.Types.HemisphereℝP¹Polar (notice: different range)

data ℝP²

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

Constructors

HemisphereℝP²Polar

Fields

• ϑParamℝP² :: !Double

  Range [0, π/2].

• φParamℝP² :: !Double

  Range [-π, π[.

Instances

Show ℝP²

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP² -> ShowS

show :: ℝP² -> String

showList :: [ℝP²] -> ShowS

Generic ℝP²

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep ℝP² :: Type -> Type

Methods

from :: ℝP² -> Rep ℝP² x

to :: Rep ℝP² x -> ℝP²

type Rep ℝP²

Defined in Math.Manifold.Core.Types.Internal

type Rep ℝP² = D1 (MetaData "\8477P\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "Hemisphere\8477P\178Polar" PrefixI True) (S1 (MetaSel (Just "\977Param\8477P\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "\966Param\8477P\178") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double)))

pattern ℝP² :: Double -> Double -> ℝP²

Deprecated: Use Math.Manifold.Core.Types.HemisphereℝP²Polar (notice: different range)

data Cℝay x

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

Constructors

Cℝay

Fields

• hParamCℝay :: !Double

  Range [0, ∞[

• pParamCℝay :: !x

  Irrelevant at h = 0.

Instances

Show x => Show (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> Cℝay x -> ShowS

show :: Cℝay x -> String

showList :: [Cℝay x] -> ShowS

Generic (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (Cℝay x) :: Type -> Type

Methods

from :: Cℝay x -> Rep (Cℝay x) x0

to :: Rep (Cℝay x) x0 -> Cℝay x

type Rep (Cℝay x)

Defined in Math.Manifold.Core.Types.Internal

type Rep (Cℝay x) = D1 (MetaData "C\8477ay" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "C\8477ay" PrefixI True) (S1 (MetaSel (Just "hParamC\8477ay") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "pParamC\8477ay") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 x)))

data CD¹ x

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

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

Constructors

CD¹

Fields

• hParamCD¹ :: !Double

  Range [0, 1]

• pParamCD¹ :: !x

  Irrelevant at h = 0.

Instances

Show x => Show (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> CD¹ x -> ShowS

show :: CD¹ x -> String

showList :: [CD¹ x] -> ShowS

Generic (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (CD¹ x) :: Type -> Type

Methods

from :: CD¹ x -> Rep (CD¹ x) x0

to :: Rep (CD¹ x) x0 -> CD¹ x

type Rep (CD¹ x)

Defined in Math.Manifold.Core.Types.Internal

type Rep (CD¹ x) = D1 (MetaData "CD\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.1.0-7erFyQcGCyw3Wj1JfXnH10" False) (C1 (MetaCons "CD\185" PrefixI True) (S1 (MetaSel (Just "hParamCD\185") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 Double) :*: S1 (MetaSel (Just "pParamCD\185") NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 x)))

Orphan instances

HasBasis ()

Associated Types

type Basis () :: Type

Methods

basisValue :: Basis () -> ()

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

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

VectorSpace ()

Associated Types

type Scalar () :: Type

Methods

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

InnerSpace ()

Methods

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