Copyright	(c) Justus Sagemüller 2016
License	GPL v3
Maintainer	(@) jsag $ hvl.no
Stability	experimental
Portability	portable
Safe Haskell	Safe
Language	Haskell2010

Math.Manifold.Core.Types

Contents

Orphan instances

Description

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

Synopsis

type ℝ⁰ = ZeroDim ℝ
type ℝ = Double
data S⁰
- = PositiveHalfSphere
- | NegativeHalfSphere
otherHalfSphere :: S⁰ -> S⁰
newtype S¹ = S¹Polar {
- φParamS¹ :: Double
}
pattern S¹ :: Double -> S¹
data S² = S²Polar {
- ϑParamS² :: !Double
- φParamS² :: !Double
}
pattern S² :: Double -> Double -> S²
newtype D¹ = D¹ {
- xParamD¹ :: Double
}
fromIntv0to1 :: ℝ -> D¹
data D² = D²Polar {
- rParamD² :: !Double
- φParamD² :: !Double
}
pattern D² :: Double -> Double -> D²
data ℝP⁰ = ℝPZero
newtype ℝP¹ = HemisphereℝP¹Polar {
- φParamℝP¹ :: Double
}
pattern ℝP¹ :: Double -> ℝP¹
data ℝP² = HemisphereℝP²Polar {
- ϑParamℝP² :: !Double
- φParamℝP² :: !Double
}
pattern ℝP² :: Double -> Double -> ℝP²
data Cℝay x = Cℝay {
- hParamCℝay :: !Double
- pParamCℝay :: !x
}
data CD¹ x = CD¹ {
- hParamCD¹ :: !Double
- pParamCD¹ :: !x
}

Documentation

type ℝ⁰ = ZeroDim ℝ Source #

type ℝ = Double Source #

data S⁰ Source #

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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods (==) :: S⁰ -> S⁰ -> Bool # (/=) :: S⁰ -> S⁰ -> Bool #
Show S⁰ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> S⁰ -> ShowS # show :: S⁰ -> String # showList :: [S⁰] -> ShowS #
Generic S⁰ Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: S⁰ -> S⁰ -> Maybe (Needle S⁰) Source # (.-~!) :: S⁰ -> S⁰ -> Needle S⁰ Source # pseudoAffineWitness :: PseudoAffineWitness S⁰ Source #
Semimanifold S⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle S⁰ :: Type Source # type Interior S⁰ :: Type Source # Methods (.+~^) :: Interior S⁰ -> Needle S⁰ -> S⁰ Source # fromInterior :: Interior S⁰ -> S⁰ Source # toInterior :: S⁰ -> Maybe (Interior S⁰) Source # translateP :: Tagged S⁰ (Interior S⁰ -> Needle S⁰ -> Interior S⁰) Source # (.-~^) :: Interior S⁰ -> Needle S⁰ -> S⁰ Source # semimanifoldWitness :: SemimanifoldWitness S⁰ Source #
type Rep S⁰ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep S⁰ = D1 (MetaData "S\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "PositiveHalfSphere" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "NegativeHalfSphere" PrefixI False) (U1 :: Type -> Type))
type Needle S⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle S⁰ = ZeroDim ℝ
type Interior S⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Interior S⁰ = S⁰

otherHalfSphere :: S⁰ -> S⁰ Source #

newtype S¹ Source #

The unit circle.

Constructors

S¹Polar
Fields φParamS¹ :: Double Must be in range `[-π, π[`.

Instances

Eq S¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods (==) :: S¹ -> S¹ -> Bool # (/=) :: S¹ -> S¹ -> Bool #
Show S¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> S¹ -> ShowS # show :: S¹ -> String # showList :: [S¹] -> ShowS #
Generic S¹ Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: S¹ -> S¹ -> Maybe (Needle S¹) Source # (.-~!) :: S¹ -> S¹ -> Needle S¹ Source # pseudoAffineWitness :: PseudoAffineWitness S¹ Source #
Semimanifold S¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle S¹ :: Type Source # type Interior S¹ :: Type Source # Methods (.+~^) :: Interior S¹ -> Needle S¹ -> S¹ Source # fromInterior :: Interior S¹ -> S¹ Source # toInterior :: S¹ -> Maybe (Interior S¹) Source # translateP :: Tagged S¹ (Interior S¹ -> Needle S¹ -> Interior S¹) Source # (.-~^) :: Interior S¹ -> Needle S¹ -> S¹ Source # semimanifoldWitness :: SemimanifoldWitness S¹ Source #
type Rep S¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep S¹ = D1 (MetaData "S\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "S\185Polar" PrefixI True) (S1 (MetaSel (Just "\966ParamS\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))
type Needle S¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle S¹ = ℝ
type Interior S¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Interior S¹ = S¹

pattern S¹ :: Double -> S¹ Source #

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

data S² Source #

The ordinary unit sphere.

Constructors

S²Polar
Fields ϑParamS² :: !Double Range `[0, π[`. φParamS² :: !Double Range `[-π, π[`.

Instances

Eq S² Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods (==) :: S² -> S² -> Bool # (/=) :: S² -> S² -> Bool #
Show S² Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> S² -> ShowS # show :: S² -> String # showList :: [S²] -> ShowS #
Generic S² Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep S² = D1 (MetaData "S\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" 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 Source #

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

newtype D¹ Source #

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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> D¹ -> ShowS # show :: D¹ -> String # showList :: [D¹] -> ShowS #
Generic D¹ Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: D¹ -> D¹ -> Maybe (Needle D¹) Source # (.-~!) :: D¹ -> D¹ -> Needle D¹ Source # pseudoAffineWitness :: PseudoAffineWitness D¹ Source #
Semimanifold D¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle D¹ :: Type Source # type Interior D¹ :: Type Source # Methods (.+~^) :: Interior D¹ -> Needle D¹ -> D¹ Source # fromInterior :: Interior D¹ -> D¹ Source # toInterior :: D¹ -> Maybe (Interior D¹) Source # translateP :: Tagged D¹ (Interior D¹ -> Needle D¹ -> Interior D¹) Source # (.-~^) :: Interior D¹ -> Needle D¹ -> D¹ Source # semimanifoldWitness :: SemimanifoldWitness D¹ Source #
type Rep D¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep D¹ = D1 (MetaData "D\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "D\185" PrefixI True) (S1 (MetaSel (Just "xParamD\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))
type Needle D¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle D¹ = ℝ
type Interior D¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Interior D¹ = ℝ

fromIntv0to1 :: ℝ -> D¹ Source #

data D² Source #

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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> D² -> ShowS # show :: D² -> String # showList :: [D²] -> ShowS #
Generic D² Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep D² = D1 (MetaData "D\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" 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 Source #

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

data ℝP⁰ Source #

Constructors

ℝPZero

Instances

Eq ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods (==) :: ℝP⁰ -> ℝP⁰ -> Bool # (/=) :: ℝP⁰ -> ℝP⁰ -> Bool #
Show ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> ℝP⁰ -> ShowS # show :: ℝP⁰ -> String # showList :: [ℝP⁰] -> ShowS #
Generic ℝP⁰ Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: ℝP⁰ -> ℝP⁰ -> Maybe (Needle ℝP⁰) Source # (.-~!) :: ℝP⁰ -> ℝP⁰ -> Needle ℝP⁰ Source # pseudoAffineWitness :: PseudoAffineWitness ℝP⁰ Source #
Semimanifold ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle ℝP⁰ :: Type Source # type Interior ℝP⁰ :: Type Source # Methods (.+~^) :: Interior ℝP⁰ -> Needle ℝP⁰ -> ℝP⁰ Source # fromInterior :: Interior ℝP⁰ -> ℝP⁰ Source # toInterior :: ℝP⁰ -> Maybe (Interior ℝP⁰) Source # translateP :: Tagged ℝP⁰ (Interior ℝP⁰ -> Needle ℝP⁰ -> Interior ℝP⁰) Source # (.-~^) :: Interior ℝP⁰ -> Needle ℝP⁰ -> ℝP⁰ Source # semimanifoldWitness :: SemimanifoldWitness ℝP⁰ Source #
type Rep ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep ℝP⁰ = D1 (MetaData "\8477P\8304" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" False) (C1 (MetaCons "\8477PZero" PrefixI False) (U1 :: Type -> Type))
type Needle ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle ℝP⁰ = ZeroDim ℝ
type Interior ℝP⁰ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Interior ℝP⁰ = ℝP⁰

newtype ℝP¹ Source #

Constructors

HemisphereℝP¹Polar
Fields φParamℝP¹ :: Double Range `[-π/2,π/2[`.

Instances

Show ℝP¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> ℝP¹ -> ShowS # show :: ℝP¹ -> String # showList :: [ℝP¹] -> ShowS #
Generic ℝP¹ Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Methods (.-~.) :: ℝP¹ -> ℝP¹ -> Maybe (Needle ℝP¹) Source # (.-~!) :: ℝP¹ -> ℝP¹ -> Needle ℝP¹ Source # pseudoAffineWitness :: PseudoAffineWitness ℝP¹ Source #
Semimanifold ℝP¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine Associated Types type Needle ℝP¹ :: Type Source # type Interior ℝP¹ :: Type Source # Methods (.+~^) :: Interior ℝP¹ -> Needle ℝP¹ -> ℝP¹ Source # fromInterior :: Interior ℝP¹ -> ℝP¹ Source # toInterior :: ℝP¹ -> Maybe (Interior ℝP¹) Source # translateP :: Tagged ℝP¹ (Interior ℝP¹ -> Needle ℝP¹ -> Interior ℝP¹) Source # (.-~^) :: Interior ℝP¹ -> Needle ℝP¹ -> ℝP¹ Source # semimanifoldWitness :: SemimanifoldWitness ℝP¹ Source #
type Rep ℝP¹ Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep ℝP¹ = D1 (MetaData "\8477P\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" True) (C1 (MetaCons "Hemisphere\8477P\185Polar" PrefixI True) (S1 (MetaSel (Just "\966Param\8477P\185") NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 Double)))
type Needle ℝP¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Needle ℝP¹ = ℝ
type Interior ℝP¹ Source #
Instance details Defined in Math.Manifold.Core.PseudoAffine type Interior ℝP¹ = ℝP¹

pattern ℝP¹ :: Double -> ℝP¹ Source #

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

data ℝP² Source #

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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> ℝP² -> ShowS # show :: ℝP² -> String # showList :: [ℝP²] -> ShowS #
Generic ℝP² Source #
Instance details 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 Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep ℝP² = D1 (MetaData "\8477P\178" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" 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 Source #

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

data Cℝay x Source #

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) Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> Cℝay x -> ShowS # show :: Cℝay x -> String # showList :: [Cℝay x] -> ShowS #
Generic (Cℝay x) Source #
Instance details 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) Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep (Cℝay x) = D1 (MetaData "C\8477ay" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" 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 Source #

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) Source #
Instance details Defined in Math.Manifold.Core.Types.Internal Methods showsPrec :: Int -> CD¹ x -> ShowS # show :: CD¹ x -> String # showList :: [CD¹ x] -> ShowS #
Generic (CD¹ x) Source #
Instance details 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) Source #
Instance details Defined in Math.Manifold.Core.Types.Internal type Rep (CD¹ x) = D1 (MetaData "CD\185" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.5.0.4-B8Kp6L8TW711kRSgoVKNgU" 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 () Source #
Instance details Associated Types type Basis () :: Type # Methods basisValue :: Basis () -> () # decompose :: () -> [(Basis (), Scalar ())] # decompose' :: () -> Basis () -> Scalar () #
VectorSpace () Source #
Instance details Associated Types type Scalar () :: Type # Methods (*^) :: Scalar () -> () -> () #
InnerSpace () Source #
Instance details Methods (<.>) :: () -> () -> Scalar () #