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

Methods

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

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

return :: a -> Plucker a

fail :: String -> Plucker a

Functor Plucker

Methods

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

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

MonadFix Plucker

Methods

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

Applicative Plucker

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

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

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

Associated Types

type Rep1 (Plucker :: * -> *) :: * -> *

Methods

from1 :: Plucker a -> Rep1 Plucker a

to1 :: Rep1 Plucker a -> Plucker a

Distributive 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

Associated Types

type Rep (Plucker :: * -> *) :: *

Methods

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

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

MonadZip 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

Methods

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

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

Traversable1 Plucker

Methods

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

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

Apply Plucker

Methods

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

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

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

Bind Plucker

Methods

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

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

Foldable1 Plucker

Methods

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

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

Eq1 Plucker

Methods

eq1 :: Eq a => Plucker a -> Plucker a -> Bool

Ord1 Plucker

Methods

compare1 :: Ord a => Plucker a -> Plucker a -> Ordering

Read1 Plucker

Methods

readsPrec1 :: Read a => Int -> ReadS (Plucker a)

Show1 Plucker

Methods

showsPrec1 :: Show a => Int -> Plucker a -> ShowS

Additive 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

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

Trace Plucker

Methods

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

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

Affine Plucker

Associated Types

type Diff (Plucker :: * -> *) :: * -> *

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)

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)

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)

Methods

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

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

Floating a => Floating (Plucker a)

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

Fractional a => Fractional (Plucker a)

Methods

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

recip :: Plucker a -> Plucker a

fromRational :: Rational -> Plucker a

Num a => Num (Plucker a)

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)

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)

Methods

readsPrec :: Int -> ReadS (Plucker a)

readList :: ReadS [Plucker a]

readPrec :: ReadPrec (Plucker a)

readListPrec :: ReadPrec [Plucker a]

Show a => Show (Plucker a)

Methods

showsPrec :: Int -> Plucker a -> ShowS

show :: Plucker a -> String

showList :: [Plucker a] -> ShowS

Ix a => Ix (Plucker a)

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)

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)

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)

Methods

put :: Plucker a -> Put

get :: Get (Plucker a)

Serial a => Serial (Plucker a)

Methods

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

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

Serialize a => Serialize (Plucker a)

Methods

put :: Putter (Plucker a)

get :: Get (Plucker a)

NFData a => NFData (Plucker a)

Methods

rnf :: Plucker a -> ()

Hashable a => Hashable (Plucker a)

Methods

hashWithSalt :: Int -> Plucker a -> Int

hash :: Plucker a -> Int

Unbox a => Unbox (Plucker a)

Ixed (Plucker a)

Methods

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

Epsilon a => Epsilon (Plucker a)

Methods

nearZero :: Plucker a -> Bool

FunctorWithIndex (E Plucker) Plucker

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

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

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

Methods

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

type Rep1 Plucker

type Rep Plucker = E Plucker

type Diff Plucker = Plucker

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

type Rep (Plucker a)

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

type Index (Plucker a) = E Plucker

type IxValue (Plucker a) = a

squaredError :: (Eq a, 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

Methods

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

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

Show LinePass

Methods

showsPrec :: Int -> LinePass -> ShowS

show :: LinePass -> String

showList :: [LinePass] -> ShowS

Generic LinePass

Associated Types

type Rep LinePass :: * -> *

Methods

from :: LinePass -> Rep LinePass x

to :: Rep LinePass x -> LinePass

type Rep LinePass

passes :: (Epsilon a, Num 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