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
Description
Plücker coordinates for lines in 3d homogeneous space.
Synopsis
- data Plucker a = Plucker !a !a !a !a !a !a
- squaredError :: Num a => Plucker a -> a
- isotropic :: Epsilon a => Plucker a -> Bool
- (><) :: Num a => Plucker a -> Plucker a -> a
- plucker :: Num a => V4 a -> V4 a -> Plucker a
- plucker3D :: Num a => V3 a -> V3 a -> Plucker a
- parallel :: Epsilon a => Plucker a -> Plucker a -> Bool
- intersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool
- data LinePass
- passes :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> LinePass
- quadranceToOrigin :: Fractional a => Plucker a -> a
- closestToOrigin :: Fractional a => Plucker a -> V3 a
- isLine :: Epsilon a => Plucker a -> Bool
- coincides :: (Epsilon a, Fractional a) => Plucker a -> Plucker a -> Bool
- coincides' :: (Epsilon a, Fractional a, Ord a) => Plucker a -> Plucker a -> Bool
- p01 :: Lens' (Plucker a) a
- p02 :: Lens' (Plucker a) a
- p03 :: Lens' (Plucker a) a
- p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p12 :: Lens' (Plucker a) a
- p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p23 :: Lens' (Plucker a) a
- p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- p31 :: Lens' (Plucker a) a
- p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
- e01 :: E Plucker
- e02 :: E Plucker
- e03 :: E Plucker
- e12 :: E Plucker
- e31 :: E Plucker
- e23 :: E Plucker
Documentation
Plücker coordinates for lines in a 3-dimensional space.
Constructors
Plucker !a !a !a !a !a !a |
Instances
Monad Plucker Source # | |
Functor Plucker Source # | |
MonadFix Plucker Source # | |
Defined in Linear.Plucker | |
Applicative Plucker Source # | |
Foldable Plucker Source # | |
Defined in Linear.Plucker Methods fold :: Monoid m => Plucker m -> m foldMap :: Monoid m => (a -> m) -> Plucker a -> 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 elem :: Eq a => a -> Plucker a -> Bool maximum :: Ord a => Plucker a -> a minimum :: Ord a => Plucker a -> a | |
Traversable Plucker Source # | |
Distributive Plucker Source # | |
Defined in Linear.Plucker | |
Representable Plucker Source # | |
Show1 Plucker Source # | |
Defined in Linear.Plucker Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Plucker a -> ShowS liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Plucker a] -> ShowS | |
Read1 Plucker Source # | |
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] | |
Ord1 Plucker Source # | |
Defined in Linear.Plucker Methods liftCompare :: (a -> b -> Ordering) -> Plucker a -> Plucker b -> Ordering | |
Eq1 Plucker Source # | |
Defined in Linear.Plucker | |
MonadZip Plucker Source # | |
Serial1 Plucker Source # | |
Defined in Linear.Plucker Methods serializeWith :: MonadPut m => (a -> m ()) -> Plucker a -> m () Source # deserializeWith :: MonadGet m => m a -> m (Plucker a) Source # | |
Apply Plucker Source # | |
Traversable1 Plucker Source # | |
Foldable1 Plucker Source # | |
Bind Plucker Source # | |
Additive Plucker Source # | |
Defined in Linear.Plucker 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 # | |
Finite Plucker Source # | |
Trace Plucker Source # | |
Affine Plucker Source # | |
Unbox a => Vector Vector (Plucker a) Source # | |
Defined in Linear.Plucker Methods basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) (Plucker a) -> m (Vector (Plucker a)) Source # basicUnsafeThaw :: PrimMonad m => Vector (Plucker a) -> m (Mutable Vector (PrimState m) (Plucker a)) Source # basicLength :: Vector (Plucker a) -> Int Source # basicUnsafeSlice :: Int -> Int -> Vector (Plucker a) -> Vector (Plucker a) Source # basicUnsafeIndexM :: Monad m => Vector (Plucker a) -> Int -> m (Plucker a) Source # basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) (Plucker a) -> Vector (Plucker a) -> m () Source # elemseq :: Vector (Plucker a) -> Plucker a -> b -> b Source # | |
Unbox a => MVector MVector (Plucker a) Source # | |
Defined in Linear.Plucker Methods basicLength :: MVector s (Plucker a) -> Int Source # basicUnsafeSlice :: Int -> Int -> MVector s (Plucker a) -> MVector s (Plucker a) Source # basicOverlaps :: MVector s (Plucker a) -> MVector s (Plucker a) -> Bool Source # basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) (Plucker a)) Source # basicInitialize :: PrimMonad m => MVector (PrimState m) (Plucker a) -> m () Source # basicUnsafeReplicate :: PrimMonad m => Int -> Plucker a -> m (MVector (PrimState m) (Plucker a)) Source # basicUnsafeRead :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> m (Plucker a) Source # basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> Plucker a -> m () Source # basicClear :: PrimMonad m => MVector (PrimState m) (Plucker a) -> m () Source # basicSet :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Plucker a -> m () Source # basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) (Plucker a) -> MVector (PrimState m) (Plucker a) -> m () Source # basicUnsafeMove :: PrimMonad m => MVector (PrimState m) (Plucker a) -> MVector (PrimState m) (Plucker a) -> m () Source # basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) (Plucker a) -> Int -> m (MVector (PrimState m) (Plucker a)) Source # | |
Eq a => Eq (Plucker a) Source # | |
Floating a => Floating (Plucker a) Source # | |
Defined in Linear.Plucker Methods sqrt :: Plucker a -> Plucker a (**) :: Plucker a -> Plucker a -> Plucker a logBase :: Plucker a -> 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 | |
Fractional a => Fractional (Plucker a) Source # | |
Defined in Linear.Plucker | |
Num a => Num (Plucker a) Source # | |
Ord a => Ord (Plucker a) Source # | |
Defined in Linear.Plucker | |
Read a => Read (Plucker a) Source # | |
Defined in Linear.Plucker | |
Show a => Show (Plucker a) Source # | |
Ix a => Ix (Plucker a) Source # | |
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) Source # | |
Lift a => Lift (Plucker a) Source # | |
Defined in Linear.Plucker | |
Storable a => Storable (Plucker a) Source # | |
Defined in Linear.Plucker Methods 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 () | |
Serial a => Serial (Plucker a) Source # | |
Serialize a => Serialize (Plucker a) Source # | |
Hashable a => Hashable (Plucker a) Source # | |
Defined in Linear.Plucker | |
Unbox a => Unbox (Plucker a) Source # | |
Defined in Linear.Plucker | |
Ixed (Plucker a) Source # | |
Defined in Linear.Plucker | |
Binary a => Binary (Plucker a) Source # | |
NFData a => NFData (Plucker a) Source # | |
Defined in Linear.Plucker | |
Epsilon a => Epsilon (Plucker a) Source # | |
Defined in Linear.Plucker | |
Generic1 Plucker Source # | |
FunctorWithIndex (E Plucker) Plucker Source # | |
FoldableWithIndex (E Plucker) Plucker Source # | |
Defined in Linear.Plucker Methods ifoldMap :: Monoid m => (E Plucker -> a -> m) -> Plucker a -> m Source # ifolded :: IndexedFold (E Plucker) (Plucker a) a Source # ifoldr :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b Source # ifoldl :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b Source # ifoldr' :: (E Plucker -> a -> b -> b) -> b -> Plucker a -> b Source # ifoldl' :: (E Plucker -> b -> a -> b) -> b -> Plucker a -> b Source # | |
TraversableWithIndex (E Plucker) Plucker Source # | |
Each (Plucker a) (Plucker b) a b Source # | |
Field1 (Plucker a) (Plucker a) a a Source # | |
Field2 (Plucker a) (Plucker a) a a Source # | |
Field3 (Plucker a) (Plucker a) a a Source # | |
Field4 (Plucker a) (Plucker a) a a Source # | |
Field5 (Plucker a) (Plucker a) a a Source # | |
Field6 (Plucker a) (Plucker a) a a Source # | |
type Rep Plucker Source # | |
Defined in Linear.Plucker | |
type Size Plucker Source # | |
Defined in Linear.Plucker | |
type Diff Plucker Source # | |
Defined in Linear.Affine | |
data MVector s (Plucker a) Source # | |
Defined in Linear.Plucker | |
type Rep (Plucker a) Source # | |
Defined in Linear.Plucker type Rep (Plucker a) = D1 ('MetaData "Plucker" "Linear.Plucker" "linear-1.20.9-RPUBINYq384VPzdgkhYOj" '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) Source # | |
Defined in Linear.Plucker | |
type Index (Plucker a) Source # | |
Defined in Linear.Plucker | |
type IxValue (Plucker a) Source # | |
Defined in Linear.Plucker | |
type Rep1 Plucker Source # | |
Defined in Linear.Plucker type Rep1 Plucker = D1 ('MetaData "Plucker" "Linear.Plucker" "linear-1.20.9-RPUBINYq384VPzdgkhYOj" '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 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
intersects :: (Epsilon a, Ord a) => Plucker a -> Plucker a -> Bool Source #
Checks if two lines intersect (or nearly intersect).
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 # | |
Show LinePass Source # | |
Generic LinePass Source # | |
type Rep LinePass Source # | |
Defined in Linear.Plucker type Rep LinePass = D1 ('MetaData "LinePass" "Linear.Plucker" "linear-1.20.9-RPUBINYq384VPzdgkhYOj" '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 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
p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a
p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a
p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a
p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a
p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a
p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a) Source #
These elements form an alternate basis for the Plücker space, or the Grassmanian manifold Gr(2,V4)
.
p10
::Num
a =>Lens'
(Plucker
a) ap20
::Num
a =>Lens'
(Plucker
a) ap30
::Num
a =>Lens'
(Plucker
a) ap32
::Num
a =>Lens'
(Plucker
a) ap13
::Num
a =>Lens'
(Plucker
a) ap21
::Num
a =>Lens'
(Plucker
a) a