Linear.Plucker

Plücker coordinates for lines in 3d homogeneous space.

Documentation

data Plucker a

Plücker coordinates for lines in a 3-dimensional space.

Constructors

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

Instances

Monad Plucker

Defined in Linear.Plucker

Methods

(>>=) :: Plucker a -> (a -> Plucker b) -> Plucker b

(>>) :: Plucker a -> Plucker b -> Plucker b

return :: a -> Plucker a

fail :: String -> Plucker a

Functor Plucker

Defined in Linear.Plucker

Methods

fmap :: (a -> b) -> Plucker a -> Plucker b

(<$) :: a -> Plucker b -> Plucker a

MonadFix Plucker

Defined in Linear.Plucker

Methods

mfix :: (a -> Plucker a) -> Plucker a

Applicative Plucker

Defined in Linear.Plucker

Methods

pure :: a -> Plucker a

(<*>) :: Plucker (a -> b) -> Plucker a -> Plucker b

liftA2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c

(*>) :: Plucker a -> Plucker b -> Plucker b

(<*) :: Plucker a -> Plucker b -> Plucker a

Foldable Plucker

Defined in Linear.Plucker

Methods

fold :: Monoid m => Plucker m -> m

foldMap :: Monoid m => (a -> m) -> Plucker a -> m

foldr :: (a -> b -> b) -> b -> Plucker a -> b

foldr' :: (a -> b -> b) -> b -> Plucker a -> b

foldl :: (b -> a -> b) -> b -> Plucker a -> b

foldl' :: (b -> a -> b) -> b -> Plucker a -> b

foldr1 :: (a -> a -> a) -> Plucker a -> a

foldl1 :: (a -> a -> a) -> Plucker a -> a

toList :: Plucker a -> [a]

null :: Plucker a -> Bool

length :: Plucker a -> Int

elem :: Eq a => a -> Plucker a -> Bool

maximum :: Ord a => Plucker a -> a

minimum :: Ord a => Plucker a -> a

sum :: Num a => Plucker a -> a

product :: Num a => Plucker a -> a

Traversable Plucker

Defined in Linear.Plucker

Methods

traverse :: Applicative f => (a -> f b) -> Plucker a -> f (Plucker b)

sequenceA :: Applicative f => Plucker (f a) -> f (Plucker a)

mapM :: Monad m => (a -> m b) -> Plucker a -> m (Plucker b)

sequence :: Monad m => Plucker (m a) -> m (Plucker a)

Distributive Plucker

Defined in Linear.Plucker

Methods

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

collect :: Functor f => (a -> Plucker b) -> f a -> Plucker (f b)

distributeM :: Monad m => m (Plucker a) -> Plucker (m a)

collectM :: Monad m => (a -> Plucker b) -> m a -> Plucker (m b)

Representable Plucker

Defined in Linear.Plucker

Associated Types

type Rep Plucker :: Type

Methods

tabulate :: (Rep Plucker -> a) -> Plucker a

index :: Plucker a -> Rep Plucker -> a

Eq1 Plucker

Defined in Linear.Plucker

Methods

liftEq :: (a -> b -> Bool) -> Plucker a -> Plucker b -> Bool

Ord1 Plucker

Defined in Linear.Plucker

Methods

liftCompare :: (a -> b -> Ordering) -> Plucker a -> Plucker b -> Ordering

Read1 Plucker

Defined in Linear.Plucker

Methods

liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Plucker a)

liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Plucker a]

liftReadPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec (Plucker a)

liftReadListPrec :: ReadPrec a -> ReadPrec [a] -> ReadPrec [Plucker a]

Show1 Plucker

Defined in Linear.Plucker

Methods

liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Plucker a -> ShowS

liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Plucker a] -> ShowS

MonadZip Plucker

Defined in Linear.Plucker

Methods

mzip :: Plucker a -> Plucker b -> Plucker (a, b)

mzipWith :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c

munzip :: Plucker (a, b) -> (Plucker a, Plucker b)

Serial1 Plucker

Defined in Linear.Plucker

Methods

serializeWith :: MonadPut m => (a -> m ()) -> Plucker a -> m ()

deserializeWith :: MonadGet m => m a -> m (Plucker a)

Apply Plucker

Defined in Linear.Plucker

Methods

(<.>) :: Plucker (a -> b) -> Plucker a -> Plucker b

(.>) :: Plucker a -> Plucker b -> Plucker b

(<.) :: Plucker a -> Plucker b -> Plucker a

liftF2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c

Traversable1 Plucker

Defined in Linear.Plucker

Methods

traverse1 :: Apply f => (a -> f b) -> Plucker a -> f (Plucker b)

sequence1 :: Apply f => Plucker (f b) -> f (Plucker b)

Foldable1 Plucker

Defined in Linear.Plucker

Methods

fold1 :: Semigroup m => Plucker m -> m

foldMap1 :: Semigroup m => (a -> m) -> Plucker a -> m

toNonEmpty :: Plucker a -> NonEmpty a

Bind Plucker

Defined in Linear.Plucker

Methods

(>>-) :: Plucker a -> (a -> Plucker b) -> Plucker b

join :: Plucker (Plucker a) -> Plucker a

Additive Plucker

Defined in Linear.Plucker

Methods

zero :: Num a => Plucker a

(^+^) :: Num a => Plucker a -> Plucker a -> Plucker a

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

lerp :: Num a => a -> Plucker a -> Plucker a -> Plucker a

liftU2 :: (a -> a -> a) -> Plucker a -> Plucker a -> Plucker a

liftI2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c

Metric Plucker

Defined in Linear.Plucker

Methods

dot :: Num a => Plucker a -> Plucker a -> a

quadrance :: Num a => Plucker a -> a

qd :: Num a => Plucker a -> Plucker a -> a

distance :: Floating a => Plucker a -> Plucker a -> a

norm :: Floating a => Plucker a -> a

signorm :: Floating a => Plucker a -> Plucker a

Finite Plucker

Defined in Linear.Plucker

Associated Types

type Size Plucker :: Nat

Methods

toV :: Plucker a -> V (Size Plucker) a

fromV :: V (Size Plucker) a -> Plucker a

Trace Plucker

Defined in Linear.Trace

Methods

trace :: Num a => Plucker (Plucker a) -> a

diagonal :: Plucker (Plucker a) -> Plucker a

Affine Plucker

Defined in Linear.Affine

Associated Types

type Diff Plucker :: Type -> Type

Methods

(.-.) :: Num a => Plucker a -> Plucker a -> Diff Plucker a

(.+^) :: Num a => Plucker a -> Diff Plucker a -> Plucker a

(.-^) :: Num a => Plucker a -> Diff Plucker a -> Plucker a

Unbox a => Vector Vector (Plucker a)

Defined in Linear.Plucker

Methods

basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) (Plucker a) -> m (Vector (Plucker a))

basicUnsafeThaw :: PrimMonad m => Vector (Plucker a) -> m (Mutable Vector (PrimState m) (Plucker a))

basicLength :: Vector (Plucker a) -> Int

basicUnsafeSlice :: Int -> Int -> Vector (Plucker a) -> Vector (Plucker a)

basicUnsafeIndexM :: Monad m => Vector (Plucker a) -> Int -> m (Plucker a)

basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) (Plucker a) -> Vector (Plucker a) -> m ()

elemseq :: Vector (Plucker a) -> Plucker a -> b -> b

Unbox a => MVector MVector (Plucker a)

Defined in Linear.Plucker

Methods

basicLength :: MVector s (Plucker a) -> Int

basicUnsafeSlice :: Int -> Int -> MVector s (Plucker a) -> MVector s (Plucker a)

basicOverlaps :: MVector s (Plucker a) -> MVector s (Plucker a) -> Bool

basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) (Plucker a))

basicInitialize :: PrimMonad m => MVector (PrimState m) (Plucker a) -> m ()

basicUnsafeReplicate :: PrimMonad m => Int -> Plucker a -> m (MVector (PrimState m) (Plucker a))

basicUnsafeRead :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> m (Plucker a)

basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> Plucker a -> m ()

basicClear :: PrimMonad m => MVector (PrimState m) (Plucker a) -> m ()

basicSet :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Plucker a -> m ()

basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) (Plucker a) -> MVector (PrimState m) (Plucker a) -> m ()

basicUnsafeMove :: PrimMonad m => MVector (PrimState m) (Plucker a) -> MVector (PrimState m) (Plucker a) -> m ()

basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> m (MVector (PrimState m) (Plucker a))

Eq a => Eq (Plucker a)

Defined in Linear.Plucker

Methods

(==) :: Plucker a -> Plucker a -> Bool

(/=) :: Plucker a -> Plucker a -> Bool

Floating a => Floating (Plucker a)

Defined in Linear.Plucker

Methods

pi :: Plucker a

exp :: Plucker a -> Plucker a

log :: Plucker a -> Plucker a

sqrt :: Plucker a -> Plucker a

(**) :: Plucker a -> Plucker a -> Plucker a

logBase :: Plucker a -> Plucker a -> Plucker a

sin :: Plucker a -> Plucker a

cos :: Plucker a -> Plucker a

tan :: Plucker a -> Plucker a

asin :: Plucker a -> Plucker a

acos :: Plucker a -> Plucker a

atan :: Plucker a -> Plucker a

sinh :: Plucker a -> Plucker a

cosh :: Plucker a -> Plucker a

tanh :: Plucker a -> Plucker a

asinh :: Plucker a -> Plucker a

acosh :: Plucker a -> Plucker a

atanh :: Plucker a -> Plucker a

log1p :: Plucker a -> Plucker a

expm1 :: Plucker a -> Plucker a

log1pexp :: Plucker a -> Plucker a

log1mexp :: Plucker a -> Plucker a

Fractional a => Fractional (Plucker a)

Defined in Linear.Plucker

Methods

(/) :: Plucker a -> Plucker a -> Plucker a

recip :: Plucker a -> Plucker a

fromRational :: Rational -> Plucker a

Num a => Num (Plucker a)

Defined in Linear.Plucker

Methods

(+) :: Plucker a -> Plucker a -> Plucker a

(-) :: Plucker a -> Plucker a -> Plucker a

(*) :: Plucker a -> Plucker a -> Plucker a

negate :: Plucker a -> Plucker a

abs :: Plucker a -> Plucker a

signum :: Plucker a -> Plucker a

fromInteger :: Integer -> Plucker a

Ord a => Ord (Plucker a)

Defined in Linear.Plucker

Methods

compare :: Plucker a -> Plucker a -> Ordering

(<) :: Plucker a -> Plucker a -> Bool

(<=) :: Plucker a -> Plucker a -> Bool

(>) :: Plucker a -> Plucker a -> Bool

(>=) :: Plucker a -> Plucker a -> Bool

max :: Plucker a -> Plucker a -> Plucker a

min :: Plucker a -> Plucker a -> Plucker a

Read a => Read (Plucker a)

Defined in Linear.Plucker

Methods

readsPrec :: Int -> ReadS (Plucker a)

readList :: ReadS [Plucker a]

readPrec :: ReadPrec (Plucker a)

readListPrec :: ReadPrec [Plucker a]

Show a => Show (Plucker a)

Defined in Linear.Plucker

Methods

showsPrec :: Int -> Plucker a -> ShowS

show :: Plucker a -> String

showList :: [Plucker a] -> ShowS

Ix a => Ix (Plucker a)

Defined in Linear.Plucker

Methods

range :: (Plucker a, Plucker a) -> [Plucker a]

index :: (Plucker a, Plucker a) -> Plucker a -> Int

unsafeIndex :: (Plucker a, Plucker a) -> Plucker a -> Int

inRange :: (Plucker a, Plucker a) -> Plucker a -> Bool

rangeSize :: (Plucker a, Plucker a) -> Int

unsafeRangeSize :: (Plucker a, Plucker a) -> Int

Generic (Plucker a)

Defined in Linear.Plucker

Associated Types

type Rep (Plucker a) :: Type -> Type

Methods

from :: Plucker a -> Rep (Plucker a) x

to :: Rep (Plucker a) x -> Plucker a

Lift a => Lift (Plucker a)

Defined in Linear.Plucker

Methods

lift :: Plucker a -> Q Exp

Storable a => Storable (Plucker a)

Defined in Linear.Plucker

Methods

sizeOf :: Plucker a -> Int

alignment :: Plucker a -> Int

peekElemOff :: Ptr (Plucker a) -> Int -> IO (Plucker a)

pokeElemOff :: Ptr (Plucker a) -> Int -> Plucker a -> IO ()

peekByteOff :: Ptr b -> Int -> IO (Plucker a)

pokeByteOff :: Ptr b -> Int -> Plucker a -> IO ()

peek :: Ptr (Plucker a) -> IO (Plucker a)

poke :: Ptr (Plucker a) -> Plucker a -> IO ()

Binary a => Binary (Plucker a)

Defined in Linear.Plucker

Methods

put :: Plucker a -> Put

get :: Get (Plucker a)

putList :: [Plucker a] -> Put

Serial a => Serial (Plucker a)

Defined in Linear.Plucker

Methods

serialize :: MonadPut m => Plucker a -> m ()

deserialize :: MonadGet m => m (Plucker a)

Serialize a => Serialize (Plucker a)

Defined in Linear.Plucker

Methods

put :: Putter (Plucker a)

get :: Get (Plucker a)

NFData a => NFData (Plucker a)

Defined in Linear.Plucker

Methods

rnf :: Plucker a -> ()

Hashable a => Hashable (Plucker a)

Defined in Linear.Plucker

Methods

hashWithSalt :: Int -> Plucker a -> Int

hash :: Plucker a -> Int

Unbox a => Unbox (Plucker a)

Defined in Linear.Plucker

Ixed (Plucker a)

Defined in Linear.Plucker

Methods

ix :: Index (Plucker a) -> Traversal' (Plucker a) (IxValue (Plucker a))

Epsilon a => Epsilon (Plucker a)

Defined in Linear.Plucker

Methods

nearZero :: Plucker a -> Bool

Generic1 Plucker

Defined in Linear.Plucker

Associated Types

type Rep1 Plucker :: k -> Type

Methods

from1 :: Plucker a -> Rep1 Plucker a

to1 :: Rep1 Plucker a -> Plucker a

FunctorWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

imap :: (E Plucker -> a -> b) -> Plucker a -> Plucker b

imapped :: IndexedSetter (E Plucker) (Plucker a) (Plucker b) a b

FoldableWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

ifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m

ifolded :: IndexedFold (E Plucker) (Plucker a) a

ifoldr :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b

ifoldl :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b

ifoldr' :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b

ifoldl' :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b

TraversableWithIndex (E Plucker) Plucker

Defined in Linear.Plucker

Methods

itraverse :: Applicative f => (E Plucker -> a -> f b) -> Plucker a -> f (Plucker b)

itraversed :: IndexedTraversal (E Plucker) (Plucker a) (Plucker b) a b

Each (Plucker a) (Plucker b) a b

Defined in Linear.Plucker

Methods

each :: Traversal (Plucker a) (Plucker b) a b

Field1 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_1 :: Lens (Plucker a) (Plucker a) a a

Field2 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_2 :: Lens (Plucker a) (Plucker a) a a

Field3 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_3 :: Lens (Plucker a) (Plucker a) a a

Field4 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_4 :: Lens (Plucker a) (Plucker a) a a

Field5 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_5 :: Lens (Plucker a) (Plucker a) a a

Field6 (Plucker a) (Plucker a) a a

Defined in Linear.Plucker

Methods

_6 :: Lens (Plucker a) (Plucker a) a a

type Rep Plucker

Defined in Linear.Plucker

type Rep Plucker = E Plucker

type Size Plucker

Defined in Linear.Plucker

type Size Plucker = 6

type Diff Plucker

Defined in Linear.Affine

type Diff Plucker = Plucker

data MVector s (Plucker a)

Defined in Linear.Plucker

data MVector s (Plucker a) = MV_Plucker !Int (MVector s a)

type Rep (Plucker a)

Defined in Linear.Plucker

type Rep (Plucker a) = D1 (MetaData "Plucker" "Linear.Plucker" "linear-1.20.9-2NLGh2yhzqjHRUiWebmTMl" False) (C1 (MetaCons "Plucker" PrefixI False) ((S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a))) :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)))))

data Vector (Plucker a)

Defined in Linear.Plucker

data Vector (Plucker a) = V_Plucker !Int (Vector a)

type Index (Plucker a)

Defined in Linear.Plucker

type Index (Plucker a) = E Plucker

type IxValue (Plucker a)

Defined in Linear.Plucker

type IxValue (Plucker a) = a

type Rep1 Plucker

Defined in Linear.Plucker

type Rep1 Plucker = D1 (MetaData "Plucker" "Linear.Plucker" "linear-1.20.9-2NLGh2yhzqjHRUiWebmTMl" False) (C1 (MetaCons "Plucker" PrefixI False) ((S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1)) :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1))))

squaredError :: 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.

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

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

(><) :: Num a => Plucker a -> Plucker a -> a infixl 5

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

plucker :: 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.

plucker3D :: 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

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

Checks if two lines are parallel.

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

Checks if two lines intersect (or nearly intersect).

data 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).

Instances

Eq LinePass

Defined in Linear.Plucker

Methods

(==) :: LinePass -> LinePass -> Bool

(/=) :: LinePass -> LinePass -> Bool

Show LinePass

Defined in Linear.Plucker

Methods

showsPrec :: Int -> LinePass -> ShowS

show :: LinePass -> String

showList :: [LinePass] -> ShowS

Generic LinePass

Defined in Linear.Plucker

Associated Types

type Rep LinePass :: Type -> Type

Methods

from :: LinePass -> Rep LinePass x

to :: Rep LinePass x -> LinePass

type Rep LinePass

Defined in Linear.Plucker

type Rep LinePass = D1 (MetaData "LinePass" "Linear.Plucker" "linear-1.20.9-2NLGh2yhzqjHRUiWebmTMl" False) (C1 (MetaCons "Coplanar" PrefixI False) (U1 :: Type -> Type) :+: (C1 (MetaCons "Clockwise" PrefixI False) (U1 :: Type -> Type) :+: C1 (MetaCons "Counterclockwise" PrefixI False) (U1 :: Type -> Type)))

passes :: (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.

quadranceToOrigin :: Fractional a => Plucker a -> a

The minimum squared distance of a line from the origin.

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

The point where a line is closest to the origin.

isLine :: 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.

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

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

coincides' :: (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

p01 :: Lens' (Plucker a) 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

p02 :: Lens' (Plucker a) 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

p03 :: Lens' (Plucker a) 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

p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

p12 :: Lens' (Plucker a) 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

p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

p23 :: Lens' (Plucker a) 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

p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

p31 :: Lens' (Plucker a) 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

p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)

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

p10 :: Num a => Lens' (Plucker a) a
p20 :: Num a => Lens' (Plucker a) a
p30 :: Num a => Lens' (Plucker a) a
p32 :: Num a => Lens' (Plucker a) a
p13 :: Num a => Lens' (Plucker a) a
p21 :: Num a => Lens' (Plucker a) a

e01 :: E Plucker

e02 :: E Plucker

e03 :: E Plucker

e12 :: E Plucker

e31 :: E Plucker

e23 :: E Plucker