Modulelinear-1.22Haskell2010

Linear.Plucker

Plücker coordinates for lines in 3d homogeneous space.

2 types
31 values

Packagelinear-1.22
Exports33
LanguageHaskell2010
LicenceBSD-3-Clause
SourcePlucker.hs

datadata Plucker a

Plücker coordinates for lines in a 3-dimensional space.

Constructors

Plucker !a !a !a !a !a !a

Instances67Monad, Functor, MonadFix, Applicative, Foldable, Traversable, …

Monad PluckerDefined in linear-1.22 · Linear.Plucker
Functor PluckerDefined in linear-1.22 · Linear.Plucker
MonadFix PluckerDefined in linear-1.22 · Linear.Plucker
Applicative PluckerDefined in linear-1.22 · Linear.Plucker
Foldable PluckerDefined in linear-1.22 · Linear.Plucker
Traversable PluckerDefined in linear-1.22 · Linear.Plucker
MonadZip PluckerDefined in linear-1.22 · Linear.Plucker
Foldable1 PluckerDefined in linear-1.22 · Linear.Plucker
Eq1 PluckerDefined in linear-1.22 · Linear.Plucker
Ord1 PluckerDefined in linear-1.22 · Linear.Plucker
Read1 PluckerDefined in linear-1.22 · Linear.Plucker
Show1 PluckerDefined in linear-1.22 · Linear.Plucker
Distributive PluckerDefined in linear-1.22 · Linear.Plucker
Apply PluckerDefined in linear-1.22 · Linear.Plucker
Bind PluckerDefined in linear-1.22 · Linear.Plucker
Traversable1 PluckerDefined in linear-1.22 · Linear.Plucker
Representable PluckerDefined in linear-1.22 · Linear.Plucker
Serial1 PluckerDefined in linear-1.22 · Linear.Plucker
Metric PluckerDefined in linear-1.22 · Linear.Plucker
Trace PluckerDefined in linear-1.22 · Linear.Trace
Additive PluckerDefined in linear-1.22 · Linear.Plucker
Affine PluckerDefined in linear-1.22 · Linear.Affine
Finite PluckerDefined in linear-1.22 · Linear.Plucker
Generic1 PluckerDefined in linear-1.22 · Linear.Plucker
Lift a => Lift (Plucker a)Defined in linear-1.22 · Linear.Plucker
Unbox a => Vector Vector (Plucker a)Defined in linear-1.22 · Linear.Plucker
Unbox a => MVector MVector (Plucker a)Defined in linear-1.22 · Linear.Plucker
Eq a => Eq (Plucker a)Defined in linear-1.22 · Linear.Plucker
Floating a => Floating (Plucker a)Defined in linear-1.22 · Linear.Plucker
Fractional a => Fractional (Plucker a)Defined in linear-1.22 · Linear.Plucker
Num a => Num (Plucker a)Defined in linear-1.22 · Linear.Plucker
Ord a => Ord (Plucker a)Defined in linear-1.22 · Linear.Plucker
Read a => Read (Plucker a)Defined in linear-1.22 · Linear.Plucker
Show a => Show (Plucker a)Defined in linear-1.22 · Linear.Plucker
Ix a => Ix (Plucker a)Defined in linear-1.22 · Linear.Plucker
Generic (Plucker a)Defined in linear-1.22 · Linear.Plucker
Semigroup a => Semigroup (Plucker a)Defined in linear-1.22 · Linear.Plucker
Monoid a => Monoid (Plucker a)Defined in linear-1.22 · Linear.Plucker
Storable a => Storable (Plucker a)Defined in linear-1.22 · Linear.Plucker
NFData a => NFData (Plucker a)Defined in linear-1.22 · Linear.Plucker
Random a => Random (Plucker a)Defined in linear-1.22 · Linear.Plucker
Binary a => Binary (Plucker a)Defined in linear-1.22 · Linear.Plucker
Hashable a => Hashable (Plucker a)Defined in linear-1.22 · Linear.Plucker
Unbox a => Unbox (Plucker a)Defined in linear-1.22 · Linear.Plucker
Serialize a => Serialize (Plucker a)Defined in linear-1.22 · Linear.Plucker
Ixed (Plucker a)Defined in linear-1.22 · Linear.Plucker
Serial a => Serial (Plucker a)Defined in linear-1.22 · Linear.Plucker
Epsilon a => Epsilon (Plucker a)Defined in linear-1.22 · Linear.Plucker
FoldableWithIndex (E Plucker) PluckerDefined in linear-1.22 · Linear.Plucker
FunctorWithIndex (E Plucker) PluckerDefined in linear-1.22 · Linear.Plucker
TraversableWithIndex (E Plucker) PluckerDefined in linear-1.22 · Linear.Plucker
Each (Plucker a) (Plucker b) a bDefined in linear-1.22 · Linear.Plucker
Field1 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
Field2 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
Field3 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
Field4 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
Field5 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
Field6 (Plucker a) (Plucker a) a aDefined in linear-1.22 · Linear.Plucker
type Rep (Plucker a) = D1 ('MetaData "Plucker" "Linear.Plucker" "linear-1.22-3d3RJcba25gDknnFKeY2an" 'False) (C1 ('MetaCons "Plucker" 'PrefixI 'False) ((S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a) :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a) :*: S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a))) :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a) :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a) :*: S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 a)))))Defined in linear-1.22 · Linear.Plucker
type Rep1 Plucker = D1 ('MetaData "Plucker" "Linear.Plucker" "linear-1.22-3d3RJcba25gDknnFKeY2an" 'False) (C1 ('MetaCons "Plucker" 'PrefixI 'False) ((S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1 :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1 :*: S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1)) :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1 :*: (S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1 :*: S1 ('MetaSel 'Nothing 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) Par1))))Defined in linear-1.22 · Linear.Plucker
data MVector s (Plucker a)
- MV_Plucker !Int (MVector s a)
Defined in linear-1.22 · Linear.Plucker
data Vector (Plucker a)
- V_Plucker !Int (Vector a)
Defined in linear-1.22 · Linear.Plucker
type Rep Plucker = E PluckerDefined in linear-1.22 · Linear.Plucker
type Index (Plucker a) = E PluckerDefined in linear-1.22 · Linear.Plucker
type IxValue (Plucker a) = aDefined in linear-1.22 · Linear.Plucker
type Diff Plucker = PluckerDefined in linear-1.22 · Linear.Affine
type Size Plucker = 6Defined in linear-1.22 · Linear.Plucker

valuesquaredError :: Num a => Plucker a -> a

Valid Plücker coordinates p will have squaredError p == 0

That said, floating point makes a mockery of this claim, so you may want to use nearZero.

valueisotropic :: Epsilon a => Plucker a -> Bool

Checks if the line is near-isotropic (isotropic vectors in this quadratic space represent lines in real 3d space).

value(><) :: Num a => Plucker a -> Plucker a -> a

This isn't th actual metric because this bilinear form gives rise to an isotropic quadratic space

valueplucker :: Num a => V4 a -> V4 a -> Plucker a

Given a pair of points represented by homogeneous coordinates generate Plücker coordinates for the line through them, directed from the second towards the first.

valueplucker3D :: Num a => V3 a -> V3 a -> Plucker a

Given a pair of 3D points, generate Plücker coordinates for the line through them, directed from the second towards the first.

Operations on lines

9 declarations

valueparallel :: Epsilon a => Plucker a -> Plucker a -> Bool

Checks if two lines are parallel.

valueintersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool

Checks if two lines intersect (or nearly intersect).

datadata LinePass

Describe how two lines pass each other.

Constructors

Coplanar
The lines are coplanar (parallel or intersecting).
Clockwise
The lines pass each other clockwise (right-handed screw)
Counterclockwise
The lines pass each other counterclockwise (left-handed screw).

Instances4Eq, Show, Generic, Rep

Eq LinePassDefined in linear-1.22 · Linear.Plucker
Show LinePassDefined in linear-1.22 · Linear.Plucker
Generic LinePassDefined in linear-1.22 · Linear.Plucker
type Rep LinePass = D1 ('MetaData "LinePass" "Linear.Plucker" "linear-1.22-3d3RJcba25gDknnFKeY2an" 'False) (C1 ('MetaCons "Coplanar" 'PrefixI 'False) U1 :+: (C1 ('MetaCons "Clockwise" 'PrefixI 'False) U1 :+: C1 ('MetaCons "Counterclockwise" 'PrefixI 'False) U1))Defined in linear-1.22 · Linear.Plucker

valuepasses :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> LinePass

Check how two lines pass each other. passes l1 l2 describes l2 when looking down l1.

valuequadranceToOrigin :: Fractional a => Plucker a -> a

The minimum squared distance of a line from the origin.

valueclosestToOrigin :: Fractional a => Plucker a -> V3 a

The point where a line is closest to the origin.

valueisLine :: Epsilon a => Plucker a -> Bool

Not all 6-dimensional points correspond to a line in 3D. This predicate tests that a Plücker coordinate lies on the Grassmann manifold, and does indeed represent a 3D line.

valuecoincides :: (Epsilon a, Fractional a) => Plucker a -> Plucker a -> Bool

Checks if two lines coincide in space. In other words, undirected equality.

valuecoincides'

:: (Epsilon a, Fractional a, Ord a)
=> Plucker a
-> Plucker a
-> Bool

Checks if two lines coincide in space, and have the same orientation.

Basis elements

18 declarations

valuep01 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

valuep02 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

valuep03 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)

These elements form a basis for the Plücker space, or the Grassmanian manifold Gr(2,V4).

p01 :: Lens' (Plucker a) a
p02 :: Lens' (Plucker a) a
p03 :: Lens' (Plucker a) a
p23 :: Lens' (Plucker a) a
p31 :: Lens' (Plucker a) a
p12 :: Lens' (Plucker a) a

valuep10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)