Copyright	(C) 2012-2015 Edward Kmett
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable
Safe Haskell	Trustworthy
Language	Haskell98

Linear.Plucker

Contents

Operations on lines
Basis elements

Description

Plücker coordinates for lines in 3d homogeneous space.

Synopsis

Documentation

data Plucker a Source #

Plücker coordinates for lines in a 3-dimensional space.

Constructors

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

Instances

Monad Plucker Source #
Methods (>>=) :: Plucker a -> (a -> Plucker b) -> Plucker b # (>>) :: Plucker a -> Plucker b -> Plucker b # return :: a -> Plucker a # fail :: String -> Plucker a #
Functor Plucker Source #
Methods fmap :: (a -> b) -> Plucker a -> Plucker b # (<$) :: a -> Plucker b -> Plucker a #
MonadFix Plucker Source #
Methods mfix :: (a -> Plucker a) -> Plucker a #
Applicative Plucker Source #
Methods pure :: a -> Plucker a # (<>) :: Plucker (a -> b) -> Plucker a -> Plucker b # (>) :: Plucker a -> Plucker b -> Plucker b # (<*) :: Plucker a -> Plucker b -> Plucker a #
Foldable Plucker Source #
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 Source #
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) #
Generic1 Plucker Source #
Associated Types type Rep1 (Plucker :: * -> ) :: -> * # Methods from1 :: Plucker a -> Rep1 Plucker a # to1 :: Rep1 Plucker a -> Plucker a #
Distributive Plucker Source #
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 Source #
Associated Types type Rep (Plucker :: * -> ) :: # Methods tabulate :: (Rep Plucker -> a) -> Plucker a # index :: Plucker a -> Rep Plucker -> a #
Eq1 Plucker Source #
Methods liftEq :: (a -> b -> Bool) -> Plucker a -> Plucker b -> Bool #
Ord1 Plucker Source #
Methods liftCompare :: (a -> b -> Ordering) -> Plucker a -> Plucker b -> Ordering #
Read1 Plucker Source #
Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Plucker a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Plucker a] #
Show1 Plucker Source #
Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Plucker a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Plucker a] -> ShowS #
MonadZip Plucker Source #
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 Source #
Methods serializeWith :: MonadPut m => (a -> m ()) -> Plucker a -> m () # deserializeWith :: MonadGet m => m a -> m (Plucker a) #
Traversable1 Plucker Source #
Methods traverse1 :: Apply f => (a -> f b) -> Plucker a -> f (Plucker b) # sequence1 :: Apply f => Plucker (f b) -> f (Plucker b) #
Apply Plucker Source #
Methods (<.>) :: Plucker (a -> b) -> Plucker a -> Plucker b # (.>) :: Plucker a -> Plucker b -> Plucker b # (<.) :: Plucker a -> Plucker b -> Plucker a #
Bind Plucker Source #
Methods (>>-) :: Plucker a -> (a -> Plucker b) -> Plucker b # join :: Plucker (Plucker a) -> Plucker a #
Foldable1 Plucker Source #
Methods fold1 :: Semigroup m => Plucker m -> m # foldMap1 :: Semigroup m => (a -> m) -> Plucker a -> m #
Additive Plucker Source #
Methods zero :: Num a => Plucker a Source # (^+^) :: Num a => Plucker a -> Plucker a -> Plucker a Source # (^-^) :: Num a => Plucker a -> Plucker a -> Plucker a Source # lerp :: Num a => a -> Plucker a -> Plucker a -> Plucker a Source # liftU2 :: (a -> a -> a) -> Plucker a -> Plucker a -> Plucker a Source # liftI2 :: (a -> b -> c) -> Plucker a -> Plucker b -> Plucker c Source #
Metric Plucker Source #
Methods dot :: Num a => Plucker a -> Plucker a -> a Source # quadrance :: Num a => Plucker a -> a Source # qd :: Num a => Plucker a -> Plucker a -> a Source # distance :: Floating a => Plucker a -> Plucker a -> a Source # norm :: Floating a => Plucker a -> a Source # signorm :: Floating a => Plucker a -> Plucker a Source #
Trace Plucker Source #
Methods trace :: Num a => Plucker (Plucker a) -> a Source # diagonal :: Plucker (Plucker a) -> Plucker a Source #
Affine Plucker Source #
Associated Types type Diff (Plucker :: * -> ) :: -> * Source # Methods (.-.) :: Num a => Plucker a -> Plucker a -> Diff Plucker a Source # (.+^) :: Num a => Plucker a -> Diff Plucker a -> Plucker a Source # (.-^) :: Num a => Plucker a -> Diff Plucker a -> Plucker a Source #
Unbox a => Vector Vector (Plucker a) Source #
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) Source #
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) Source #
Methods (==) :: Plucker a -> Plucker a -> Bool # (/=) :: Plucker a -> Plucker a -> Bool #
Floating a => Floating (Plucker a) Source #
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) Source #
Methods (/) :: Plucker a -> Plucker a -> Plucker a # recip :: Plucker a -> Plucker a # fromRational :: Rational -> Plucker a #
Num a => Num (Plucker a) Source #
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) Source #
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) Source #
Methods readsPrec :: Int -> ReadS (Plucker a) # readList :: ReadS [Plucker a] # readPrec :: ReadPrec (Plucker a) # readListPrec :: ReadPrec [Plucker a] #
Show a => Show (Plucker a) Source #
Methods showsPrec :: Int -> Plucker a -> ShowS # show :: Plucker a -> String # showList :: [Plucker a] -> ShowS #
Ix a => Ix (Plucker a) Source #
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) Source #
Associated Types type Rep (Plucker a) :: * -> * # Methods from :: Plucker a -> Rep (Plucker a) x # to :: Rep (Plucker a) x -> Plucker a #
Storable a => Storable (Plucker a) Source #
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) Source #
Methods put :: Plucker a -> Put # get :: Get (Plucker a) # putList :: [Plucker a] -> Put #
Serial a => Serial (Plucker a) Source #
Methods serialize :: MonadPut m => Plucker a -> m () # deserialize :: MonadGet m => m (Plucker a) #
Serialize a => Serialize (Plucker a) Source #
Methods put :: Putter (Plucker a) # get :: Get (Plucker a) #
NFData a => NFData (Plucker a) Source #
Methods rnf :: Plucker a -> () #
Hashable a => Hashable (Plucker a) Source #
Methods hashWithSalt :: Int -> Plucker a -> Int # hash :: Plucker a -> Int #
Unbox a => Unbox (Plucker a) Source #

Ixed (Plucker a) Source #
Methods ix :: Index (Plucker a) -> Traversal' (Plucker a) (IxValue (Plucker a)) #
Epsilon a => Epsilon (Plucker a) Source #
Methods nearZero :: Plucker a -> Bool Source #
FunctorWithIndex (E Plucker) Plucker Source #
Methods imap :: (E Plucker -> a -> b) -> Plucker a -> Plucker b # imapped :: (Indexable (E Plucker) p, Settable f) => p a (f b) -> Plucker a -> f (Plucker b) #
FoldableWithIndex (E Plucker) Plucker Source #
Methods ifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m # ifolded :: (Indexable (E Plucker) p, Contravariant f, Applicative f) => p a (f a) -> Plucker a -> f (Plucker 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 Source #
Methods itraverse :: Applicative f => (E Plucker -> a -> f b) -> Plucker a -> f (Plucker b) # itraversed :: (Indexable (E Plucker) p, Applicative f) => p a (f b) -> Plucker a -> f (Plucker b) #
Each (Plucker a) (Plucker b) a b Source #
Methods each :: Traversal (Plucker a) (Plucker b) a b #
type Rep1 Plucker Source #
type Rep1 Plucker = D1 (MetaData "Plucker" "Linear.Plucker" "linear-1.20.5-ALqHTYew01KD1PdDd6TYmz" False) (C1 (MetaCons "Plucker" PrefixI False) ((::) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1))) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) ((:*:) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1)))))
type Rep Plucker Source #
type Rep Plucker = E Plucker
type Diff Plucker Source #
type Diff Plucker = Plucker
data MVector s (Plucker a) Source #
data MVector s (Plucker a) = MV_Plucker !Int (MVector s a)
type Rep (Plucker a) Source #
type Rep (Plucker a) = D1 (MetaData "Plucker" "Linear.Plucker" "linear-1.20.5-ALqHTYew01KD1PdDd6TYmz" False) (C1 (MetaCons "Plucker" PrefixI False) ((::) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)))) ((::) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)) ((:*:) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a)) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a))))))
data Vector (Plucker a) Source #
data Vector (Plucker a) = V_Plucker !Int (Vector a)
type Index (Plucker a) Source #
type Index (Plucker a) = E Plucker
type IxValue (Plucker a) Source #
type IxValue (Plucker a) = a

squaredError :: Num a => Plucker a -> a Source #

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 Source #

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 Source #

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 Source #

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 Source #

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 Source #

Checks if two lines are parallel.

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

Checks if two lines intersect (or nearly intersect).

data LinePass Source #

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 Source #
Methods (==) :: LinePass -> LinePass -> Bool # (/=) :: LinePass -> LinePass -> Bool #
Show LinePass Source #
Methods showsPrec :: Int -> LinePass -> ShowS # show :: LinePass -> String # showList :: [LinePass] -> ShowS #
Generic LinePass Source #
Associated Types type Rep LinePass :: * -> * # Methods from :: LinePass -> Rep LinePass x # to :: Rep LinePass x -> LinePass #
type Rep LinePass Source #
type Rep LinePass = D1 (MetaData "LinePass" "Linear.Plucker" "linear-1.20.5-ALqHTYew01KD1PdDd6TYmz" False) ((:+:) (C1 (MetaCons "Coplanar" PrefixI False) U1) ((:+:) (C1 (MetaCons "Clockwise" PrefixI False) U1) (C1 (MetaCons "Counterclockwise" PrefixI False) U1)))

passes :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> LinePass Source #

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

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

The minimum squared distance of a line from the origin.

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

The point where a line is closest to the origin.

isLine :: Epsilon a => Plucker a -> Bool Source #

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 Source #

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

coincides' :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Plucker a -> Bool Source #

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

Basis elements

p01 :: Lens' (Plucker a) a Source #

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 Source #

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 Source #

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) Source #