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

data EmptyMfd v

The empty space can be considered a manifold with any sort of tangent space.

Instances

Empty (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

eliminate :: EmptyMfd v -> x

Eq (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: EmptyMfd v -> EmptyMfd v -> Bool

(/=) :: EmptyMfd v -> EmptyMfd v -> Bool

Ord (EmptyMfd v)

Defined in Math.Manifold.Core.Types.Internal

Methods

compare :: EmptyMfd v -> EmptyMfd v -> Ordering

(<) :: EmptyMfd v -> EmptyMfd v -> Bool

(<=) :: EmptyMfd v -> EmptyMfd v -> Bool

(>) :: EmptyMfd v -> EmptyMfd v -> Bool

(>=) :: EmptyMfd v -> EmptyMfd v -> Bool

max :: EmptyMfd v -> EmptyMfd v -> EmptyMfd v

min :: EmptyMfd v -> EmptyMfd v -> EmptyMfd v

type ℝ⁰ = ZeroDim ℝ

type ℝ = Double

type S⁰ = S⁰_ Double

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

data S⁰_ r

Constructors

PositiveHalfSphere

NegativeHalfSphere

Instances

Generic (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S⁰_ r) :: Type -> Type

Methods

from :: S⁰_ r -> Rep (S⁰_ r) x

to :: Rep (S⁰_ r) x -> S⁰_ r

Show (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S⁰_ r -> ShowS

show :: S⁰_ r -> String

showList :: [S⁰_ r] -> ShowS

Eq (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S⁰_ r -> S⁰_ r -> Bool

(/=) :: S⁰_ r -> S⁰_ r -> Bool

type Rep (S⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S⁰_ r) = D1 ('MetaData "S\8304_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "PositiveHalfSphere" 'PrefixI 'False) (U1 :: Type -> Type) :+: C1 ('MetaCons "NegativeHalfSphere" 'PrefixI 'False) (U1 :: Type -> Type))

otherHalfSphere :: S⁰ -> S⁰

type S¹ = S¹_ Double

The unit circle.

newtype S¹_ r

Constructors

S¹Polar

Fields

• φParamS¹ :: r

  Must be in range [-π, π[.

Instances

Generic (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S¹_ r) :: Type -> Type

Methods

from :: S¹_ r -> Rep (S¹_ r) x

to :: Rep (S¹_ r) x -> S¹_ r

Show r => Show (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S¹_ r -> ShowS

show :: S¹_ r -> String

showList :: [S¹_ r] -> ShowS

(Eq r, RealFloat r) => Eq (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S¹_ r -> S¹_ r -> Bool

(/=) :: S¹_ r -> S¹_ r -> Bool

type Rep (S¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S¹_ r) = D1 ('MetaData "S\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'True) (C1 ('MetaCons "S\185Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\966ParamS\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

pattern S¹ :: Double -> S¹

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

type S² = S²_ Double

The ordinary unit sphere.

data S²_ r

Constructors

S²Polar

Fields

• ϑParamS² :: !r

  Range [0, π[.

• φParamS² :: !r

  Range [-π, π[.

Instances

Generic (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (S²_ r) :: Type -> Type

Methods

from :: S²_ r -> Rep (S²_ r) x

to :: Rep (S²_ r) x -> S²_ r

Show r => Show (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> S²_ r -> ShowS

show :: S²_ r -> String

showList :: [S²_ r] -> ShowS

(Eq r, RealFloat r) => Eq (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: S²_ r -> S²_ r -> Bool

(/=) :: S²_ r -> S²_ r -> Bool

type Rep (S²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (S²_ r) = D1 ('MetaData "S\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "S\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\977ParamS\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966ParamS\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

pattern S² :: Double -> Double -> S²

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

type D¹ = D¹_ Double

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.

newtype D¹_ r

Constructors

D¹

Fields

• xParamD¹ :: r

  Range [-1, 1].

Instances

Generic (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (D¹_ r) :: Type -> Type

Methods

from :: D¹_ r -> Rep (D¹_ r) x

to :: Rep (D¹_ r) x -> D¹_ r

Show r => Show (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D¹_ r -> ShowS

show :: D¹_ r -> String

showList :: [D¹_ r] -> ShowS

type Rep (D¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (D¹_ r) = D1 ('MetaData "D\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'True) (C1 ('MetaCons "D\185" 'PrefixI 'True) (S1 ('MetaSel ('Just "xParamD\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

fromIntv0to1 :: ℝ -> D¹

type D² = D²_ Double

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!)

data D²_ r

Constructors

D²Polar

Fields

• rParamD² :: !r

  Range [0, 1].

• φParamD² :: !r

  Range [-π, π[.

Instances

Generic (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (D²_ r) :: Type -> Type

Methods

from :: D²_ r -> Rep (D²_ r) x

to :: Rep (D²_ r) x -> D²_ r

Show r => Show (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> D²_ r -> ShowS

show :: D²_ r -> String

showList :: [D²_ r] -> ShowS

type Rep (D²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (D²_ r) = D1 ('MetaData "D\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "D\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "rParamD\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966ParamD\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

pattern D² :: Double -> Double -> D²

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

type ℝP⁰ = ℝP⁰_ Double

data ℝP⁰_ r

Constructors

ℝPZero

Instances

Generic (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP⁰_ r) :: Type -> Type

Methods

from :: ℝP⁰_ r -> Rep (ℝP⁰_ r) x

to :: Rep (ℝP⁰_ r) x -> ℝP⁰_ r

Show (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP⁰_ r -> ShowS

show :: ℝP⁰_ r -> String

showList :: [ℝP⁰_ r] -> ShowS

Eq (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

(==) :: ℝP⁰_ r -> ℝP⁰_ r -> Bool

(/=) :: ℝP⁰_ r -> ℝP⁰_ r -> Bool

PseudoAffine (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP⁰_ r -> ℝP⁰_ r -> Maybe (Needle (ℝP⁰_ r))

(.-~!) :: ℝP⁰_ r -> ℝP⁰_ r -> Needle (ℝP⁰_ r)

pseudoAffineWitness :: PseudoAffineWitness (ℝP⁰_ r)

Semimanifold (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ℝP⁰_ r)

Methods

(.+~^) :: ℝP⁰_ r -> Needle (ℝP⁰_ r) -> ℝP⁰_ r

(.-~^) :: ℝP⁰_ r -> Needle (ℝP⁰_ r) -> ℝP⁰_ r

semimanifoldWitness :: SemimanifoldWitness (ℝP⁰_ r)

type Rep (ℝP⁰_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP⁰_ r) = D1 ('MetaData "\8477P\8304_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "\8477PZero" 'PrefixI 'False) (U1 :: Type -> Type))

type Needle (ℝP⁰_ r)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ℝP⁰_ r) = ZeroDim r

type ℝP¹ = ℝP¹_ Double

newtype ℝP¹_ r

Constructors

HemisphereℝP¹Polar

Fields

• φParamℝP¹ :: r

  Range [-π/2,π/2[.

Instances

Generic (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP¹_ r) :: Type -> Type

Methods

from :: ℝP¹_ r -> Rep (ℝP¹_ r) x

to :: Rep (ℝP¹_ r) x -> ℝP¹_ r

Show r => Show (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP¹_ r -> ShowS

show :: ℝP¹_ r -> String

showList :: [ℝP¹_ r] -> ShowS

ℝeal r => PseudoAffine (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

Methods

(.-~.) :: ℝP¹_ r -> ℝP¹_ r -> Maybe (Needle (ℝP¹_ r))

(.-~!) :: ℝP¹_ r -> ℝP¹_ r -> Needle (ℝP¹_ r)

pseudoAffineWitness :: PseudoAffineWitness (ℝP¹_ r)

ℝeal r => Semimanifold (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Needle (ℝP¹_ r)

Methods

(.+~^) :: ℝP¹_ r -> Needle (ℝP¹_ r) -> ℝP¹_ r

(.-~^) :: ℝP¹_ r -> Needle (ℝP¹_ r) -> ℝP¹_ r

semimanifoldWitness :: SemimanifoldWitness (ℝP¹_ r)

type Rep (ℝP¹_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP¹_ r) = D1 ('MetaData "\8477P\185_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'True) (C1 ('MetaCons "Hemisphere\8477P\185Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\966Param\8477P\185") 'NoSourceUnpackedness 'NoSourceStrictness 'DecidedLazy) (Rec0 r)))

type Needle (ℝP¹_ r)

Defined in Math.Manifold.Core.PseudoAffine

type Needle (ℝP¹_ r) = r

pattern ℝP¹ :: Double -> ℝP¹

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

type ℝP² = ℝP²_ Double

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.

data ℝP²_ r

Constructors

HemisphereℝP²Polar

Fields

• ϑParamℝP² :: !r

  Range [0, π/2].

• φParamℝP² :: !r

  Range [-π, π[.

Instances

Generic (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

Associated Types

type Rep (ℝP²_ r) :: Type -> Type

Methods

from :: ℝP²_ r -> Rep (ℝP²_ r) x

to :: Rep (ℝP²_ r) x -> ℝP²_ r

Show r => Show (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

Methods

showsPrec :: Int -> ℝP²_ r -> ShowS

show :: ℝP²_ r -> String

showList :: [ℝP²_ r] -> ShowS

type Rep (ℝP²_ r)

Defined in Math.Manifold.Core.Types.Internal

type Rep (ℝP²_ r) = D1 ('MetaData "\8477P\178_" "Math.Manifold.Core.Types.Internal" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "Hemisphere\8477P\178Polar" 'PrefixI 'True) (S1 ('MetaSel ('Just "\977Param\8477P\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r) :*: S1 ('MetaSel ('Just "\966Param\8477P\178") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 r)))

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 :: !(Scalar (Needle x))

  Range [0, ∞[

• pParamCℝay :: !x

  Irrelevant at h = 0.

Instances

Generic (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Rep (Cℝay x) :: Type -> Type

Methods

from :: Cℝay x -> Rep (Cℝay x) x0

to :: Rep (Cℝay x) x0 -> Cℝay x

(Show x, Show (Scalar (Needle x))) => Show (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

showsPrec :: Int -> Cℝay x -> ShowS

show :: Cℝay x -> String

showList :: [Cℝay x] -> ShowS

type Rep (Cℝay x)

Defined in Math.Manifold.Core.PseudoAffine

type Rep (Cℝay x) = D1 ('MetaData "C\8477ay" "Math.Manifold.Core.PseudoAffine" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "C\8477ay" 'PrefixI 'True) (S1 ('MetaSel ('Just "hParamC\8477ay") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 (Scalar (Needle x))) :*: 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 :: !(Scalar (Needle x))

  Range [0, 1]

• pParamCD¹ :: !x

  Irrelevant at h = 0.

Instances

Generic (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

Associated Types

type Rep (CD¹ x) :: Type -> Type

Methods

from :: CD¹ x -> Rep (CD¹ x) x0

to :: Rep (CD¹ x) x0 -> CD¹ x

(Show x, Show (Scalar (Needle x))) => Show (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

Methods

showsPrec :: Int -> CD¹ x -> ShowS

show :: CD¹ x -> String

showList :: [CD¹ x] -> ShowS

type Rep (CD¹ x)

Defined in Math.Manifold.Core.PseudoAffine

type Rep (CD¹ x) = D1 ('MetaData "CD\185" "Math.Manifold.Core.PseudoAffine" "manifolds-core-0.6.1.0-7JLeNpkgg097AjSvrWgCob" 'False) (C1 ('MetaCons "CD\185" 'PrefixI 'True) (S1 ('MetaSel ('Just "hParamCD\185") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 (Scalar (Needle x))) :*: S1 ('MetaSel ('Just "pParamCD\185") 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 x)))

Orphan instances

HasBasis ()

Associated Types

type Basis ()

Methods

basisValue :: Basis () -> ()

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

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

InnerSpace ()

Methods

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

VectorSpace ()

Associated Types

type Scalar ()

Methods

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