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

Linear.Affine

Description

Operations on affine spaces.

Synopsis

Documentation

class Additive (Diff p) => Affine p where #

An affine space is roughly a vector space in which we have forgotten or at least pretend to have forgotten the origin.

a .+^ (b .-. a)  =  b@
(a .+^ u) .+^ v  =  a .+^ (u ^+^ v)@
(a .-. b) ^+^ v  =  (a .+^ v) .-. q@

Minimal complete definition

(.-.), (.+^)

Associated Types

type Diff p :: * -> * #

Methods

(.-.) :: Num a => p a -> p a -> Diff p a infixl 6 #

Get the difference between two points as a vector offset.

(.+^) :: Num a => p a -> Diff p a -> p a infixl 6 #

Add a vector offset to a point.

(.-^) :: Num a => p a -> Diff p a -> p a infixl 6 #

Subtract a vector offset from a point.

Instances

Affine [] #
Associated Types type Diff ([] :: * -> ) :: -> * # Methods (.-.) :: Num a => [a] -> [a] -> Diff [] a # (.+^) :: Num a => [a] -> Diff [] a -> [a] # (.-^) :: Num a => [a] -> Diff [] a -> [a] #
Affine Maybe #
Associated Types type Diff (Maybe :: * -> ) :: -> * # Methods (.-.) :: Num a => Maybe a -> Maybe a -> Diff Maybe a # (.+^) :: Num a => Maybe a -> Diff Maybe a -> Maybe a # (.-^) :: Num a => Maybe a -> Diff Maybe a -> Maybe a #
Affine Identity #
Associated Types type Diff (Identity :: * -> ) :: -> * # Methods (.-.) :: Num a => Identity a -> Identity a -> Diff Identity a # (.+^) :: Num a => Identity a -> Diff Identity a -> Identity a # (.-^) :: Num a => Identity a -> Diff Identity a -> Identity a #
Affine Complex #
Associated Types type Diff (Complex :: * -> ) :: -> * # Methods (.-.) :: Num a => Complex a -> Complex a -> Diff Complex a # (.+^) :: Num a => Complex a -> Diff Complex a -> Complex a # (.-^) :: Num a => Complex a -> Diff Complex a -> Complex a #
Affine ZipList #
Associated Types type Diff (ZipList :: * -> ) :: -> * # Methods (.-.) :: Num a => ZipList a -> ZipList a -> Diff ZipList a # (.+^) :: Num a => ZipList a -> Diff ZipList a -> ZipList a # (.-^) :: Num a => ZipList a -> Diff ZipList a -> ZipList a #
Affine IntMap #
Associated Types type Diff (IntMap :: * -> ) :: -> * # Methods (.-.) :: Num a => IntMap a -> IntMap a -> Diff IntMap a # (.+^) :: Num a => IntMap a -> Diff IntMap a -> IntMap a # (.-^) :: Num a => IntMap a -> Diff IntMap a -> IntMap a #
Affine Vector #
Associated Types type Diff (Vector :: * -> ) :: -> * # Methods (.-.) :: Num a => Vector a -> Vector a -> Diff Vector a # (.+^) :: Num a => Vector a -> Diff Vector a -> Vector a # (.-^) :: Num a => Vector a -> Diff Vector a -> Vector a #
Affine V0 #
Associated Types type Diff (V0 :: * -> ) :: -> * # Methods (.-.) :: Num a => V0 a -> V0 a -> Diff V0 a # (.+^) :: Num a => V0 a -> Diff V0 a -> V0 a # (.-^) :: Num a => V0 a -> Diff V0 a -> V0 a #
Affine V1 #
Associated Types type Diff (V1 :: * -> ) :: -> * # Methods (.-.) :: Num a => V1 a -> V1 a -> Diff V1 a # (.+^) :: Num a => V1 a -> Diff V1 a -> V1 a # (.-^) :: Num a => V1 a -> Diff V1 a -> V1 a #
Affine V2 #
Associated Types type Diff (V2 :: * -> ) :: -> * # Methods (.-.) :: Num a => V2 a -> V2 a -> Diff V2 a # (.+^) :: Num a => V2 a -> Diff V2 a -> V2 a # (.-^) :: Num a => V2 a -> Diff V2 a -> V2 a #
Affine V3 #
Associated Types type Diff (V3 :: * -> ) :: -> * # Methods (.-.) :: Num a => V3 a -> V3 a -> Diff V3 a # (.+^) :: Num a => V3 a -> Diff V3 a -> V3 a # (.-^) :: Num a => V3 a -> Diff V3 a -> V3 a #
Affine V4 #
Associated Types type Diff (V4 :: * -> ) :: -> * # Methods (.-.) :: Num a => V4 a -> V4 a -> Diff V4 a # (.+^) :: Num a => V4 a -> Diff V4 a -> V4 a # (.-^) :: Num a => V4 a -> Diff V4 a -> V4 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 #
Affine Quaternion #
Associated Types type Diff (Quaternion :: * -> ) :: -> * # Methods (.-.) :: Num a => Quaternion a -> Quaternion a -> Diff Quaternion a # (.+^) :: Num a => Quaternion a -> Diff Quaternion a -> Quaternion a # (.-^) :: Num a => Quaternion a -> Diff Quaternion a -> Quaternion a #
Affine ((->) b) #
Associated Types type Diff ((->) b :: * -> ) :: -> * # Methods (.-.) :: Num a => (b -> a) -> (b -> a) -> Diff ((->) b) a # (.+^) :: Num a => (b -> a) -> Diff ((->) b) a -> b -> a # (.-^) :: Num a => (b -> a) -> Diff ((->) b) a -> b -> a #
Ord k => Affine (Map k) #
Associated Types type Diff (Map k :: * -> ) :: -> * # Methods (.-.) :: Num a => Map k a -> Map k a -> Diff (Map k) a # (.+^) :: Num a => Map k a -> Diff (Map k) a -> Map k a # (.-^) :: Num a => Map k a -> Diff (Map k) a -> Map k a #
(Eq k, Hashable k) => Affine (HashMap k) #
Associated Types type Diff (HashMap k :: * -> ) :: -> * # Methods (.-.) :: Num a => HashMap k a -> HashMap k a -> Diff (HashMap k) a # (.+^) :: Num a => HashMap k a -> Diff (HashMap k) a -> HashMap k a # (.-^) :: Num a => HashMap k a -> Diff (HashMap k) a -> HashMap k a #
Additive f => Affine (Point f) #
Associated Types type Diff (Point f :: * -> ) :: -> * # Methods (.-.) :: Num a => Point f a -> Point f a -> Diff (Point f) a # (.+^) :: Num a => Point f a -> Diff (Point f) a -> Point f a # (.-^) :: Num a => Point f a -> Diff (Point f) a -> Point f a #
Dim * n => Affine (V * n) #
Associated Types type Diff (V * n :: * -> ) :: -> * # Methods (.-.) :: Num a => V * n a -> V * n a -> Diff (V * n) a # (.+^) :: Num a => V * n a -> Diff (V * n) a -> V * n a # (.-^) :: Num a => V * n a -> Diff (V * n) a -> V * n a #

qdA :: (Affine p, Foldable (Diff p), Num a) => p a -> p a -> a #

Compute the quadrance of the difference (the square of the distance)

distanceA :: (Floating a, Foldable (Diff p), Affine p) => p a -> p a -> a #

Distance between two points in an affine space

newtype Point f a #

A handy wrapper to help distinguish points from vectors at the type level

Constructors

P (f a)

Instances

Unbox (f a) => Vector Vector (Point f a) #
Methods basicUnsafeFreeze :: PrimMonad m => Mutable Vector (PrimState m) (Point f a) -> m (Vector (Point f a)) # basicUnsafeThaw :: PrimMonad m => Vector (Point f a) -> m (Mutable Vector (PrimState m) (Point f a)) # basicLength :: Vector (Point f a) -> Int # basicUnsafeSlice :: Int -> Int -> Vector (Point f a) -> Vector (Point f a) # basicUnsafeIndexM :: Monad m => Vector (Point f a) -> Int -> m (Point f a) # basicUnsafeCopy :: PrimMonad m => Mutable Vector (PrimState m) (Point f a) -> Vector (Point f a) -> m () # elemseq :: Vector (Point f a) -> Point f a -> b -> b #
Unbox (f a) => MVector MVector (Point f a) #
Methods basicLength :: MVector s (Point f a) -> Int # basicUnsafeSlice :: Int -> Int -> MVector s (Point f a) -> MVector s (Point f a) # basicOverlaps :: MVector s (Point f a) -> MVector s (Point f a) -> Bool # basicUnsafeNew :: PrimMonad m => Int -> m (MVector (PrimState m) (Point f a)) # basicInitialize :: PrimMonad m => MVector (PrimState m) (Point f a) -> m () # basicUnsafeReplicate :: PrimMonad m => Int -> Point f a -> m (MVector (PrimState m) (Point f a)) # basicUnsafeRead :: PrimMonad m => MVector (PrimState m) (Point f a) -> Int -> m (Point f a) # basicUnsafeWrite :: PrimMonad m => MVector (PrimState m) (Point f a) -> Int -> Point f a -> m () # basicClear :: PrimMonad m => MVector (PrimState m) (Point f a) -> m () # basicSet :: PrimMonad m => MVector (PrimState m) (Point f a) -> Point f a -> m () # basicUnsafeCopy :: PrimMonad m => MVector (PrimState m) (Point f a) -> MVector (PrimState m) (Point f a) -> m () # basicUnsafeMove :: PrimMonad m => MVector (PrimState m) (Point f a) -> MVector (PrimState m) (Point f a) -> m () # basicUnsafeGrow :: PrimMonad m => MVector (PrimState m) (Point f a) -> Int -> m (MVector (PrimState m) (Point f a)) #
Monad f => Monad (Point f) #
Methods (>>=) :: Point f a -> (a -> Point f b) -> Point f b # (>>) :: Point f a -> Point f b -> Point f b # return :: a -> Point f a # fail :: String -> Point f a #
Functor f => Functor (Point f) #
Methods fmap :: (a -> b) -> Point f a -> Point f b # (<$) :: a -> Point f b -> Point f a #
Applicative f => Applicative (Point f) #
Methods pure :: a -> Point f a # (<>) :: Point f (a -> b) -> Point f a -> Point f b # (>) :: Point f a -> Point f b -> Point f b # (<*) :: Point f a -> Point f b -> Point f a #
Foldable f => Foldable (Point f) #
Methods fold :: Monoid m => Point f m -> m # foldMap :: Monoid m => (a -> m) -> Point f a -> m # foldr :: (a -> b -> b) -> b -> Point f a -> b # foldr' :: (a -> b -> b) -> b -> Point f a -> b # foldl :: (b -> a -> b) -> b -> Point f a -> b # foldl' :: (b -> a -> b) -> b -> Point f a -> b # foldr1 :: (a -> a -> a) -> Point f a -> a # foldl1 :: (a -> a -> a) -> Point f a -> a # toList :: Point f a -> [a] # null :: Point f a -> Bool # length :: Point f a -> Int # elem :: Eq a => a -> Point f a -> Bool # maximum :: Ord a => Point f a -> a # minimum :: Ord a => Point f a -> a # sum :: Num a => Point f a -> a # product :: Num a => Point f a -> a #
Traversable f => Traversable (Point f) #
Methods traverse :: Applicative f => (a -> f b) -> Point f a -> f (Point f b) # sequenceA :: Applicative f => Point f (f a) -> f (Point f a) # mapM :: Monad m => (a -> m b) -> Point f a -> m (Point f b) # sequence :: Monad m => Point f (m a) -> m (Point f a) #
Generic1 (Point f) #
Associated Types type Rep1 (Point f :: * -> ) :: -> * # Methods from1 :: Point f a -> Rep1 (Point f) a # to1 :: Rep1 (Point f) a -> Point f a #
Distributive f => Distributive (Point f) #
Methods distribute :: Functor f => f (Point f a) -> Point f (f a) # collect :: Functor f => (a -> Point f b) -> f a -> Point f (f b) # distributeM :: Monad m => m (Point f a) -> Point f (m a) # collectM :: Monad m => (a -> Point f b) -> m a -> Point f (m b) #
Representable f => Representable (Point f) #
Associated Types type Rep (Point f :: * -> ) :: # Methods tabulate :: (Rep (Point f) -> a) -> Point f a # index :: Point f a -> Rep (Point f) -> a #
Eq1 f => Eq1 (Point f) #
Methods liftEq :: (a -> b -> Bool) -> Point f a -> Point f b -> Bool #
Ord1 f => Ord1 (Point f) #
Methods liftCompare :: (a -> b -> Ordering) -> Point f a -> Point f b -> Ordering #
Read1 f => Read1 (Point f) #
Methods liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Point f a) # liftReadList :: (Int -> ReadS a) -> ReadS [a] -> ReadS [Point f a] #
Show1 f => Show1 (Point f) #
Methods liftShowsPrec :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> Point f a -> ShowS # liftShowList :: (Int -> a -> ShowS) -> ([a] -> ShowS) -> [Point f a] -> ShowS #
Serial1 f => Serial1 (Point f) #
Methods serializeWith :: MonadPut m => (a -> m ()) -> Point f a -> m () # deserializeWith :: MonadGet m => m a -> m (Point f a) #
Apply f => Apply (Point f) #
Methods (<.>) :: Point f (a -> b) -> Point f a -> Point f b # (.>) :: Point f a -> Point f b -> Point f b # (<.) :: Point f a -> Point f b -> Point f a #
Bind f => Bind (Point f) #
Methods (>>-) :: Point f a -> (a -> Point f b) -> Point f b # join :: Point f (Point f a) -> Point f a #
Additive f => Additive (Point f) #
Methods zero :: Num a => Point f a # (^+^) :: Num a => Point f a -> Point f a -> Point f a # (^-^) :: Num a => Point f a -> Point f a -> Point f a # lerp :: Num a => a -> Point f a -> Point f a -> Point f a # liftU2 :: (a -> a -> a) -> Point f a -> Point f a -> Point f a # liftI2 :: (a -> b -> c) -> Point f a -> Point f b -> Point f c #
Metric f => Metric (Point f) #
Methods dot :: Num a => Point f a -> Point f a -> a # quadrance :: Num a => Point f a -> a # qd :: Num a => Point f a -> Point f a -> a # distance :: Floating a => Point f a -> Point f a -> a # norm :: Floating a => Point f a -> a # signorm :: Floating a => Point f a -> Point f a #
Finite f => Finite (Point f) #
Associated Types type Size (Point f :: * -> *) :: Nat # Methods toV :: Point f a -> V Nat (Size (Point f)) a # fromV :: V Nat (Size (Point f)) a -> Point f a #
R1 f => R1 (Point f) #
Methods _x :: Functor f => (a -> f a) -> Point f a -> f (Point f a) #
R2 f => R2 (Point f) #
Methods _y :: Functor f => (a -> f a) -> Point f a -> f (Point f a) # _xy :: Functor f => (V2 a -> f (V2 a)) -> Point f a -> f (Point f a) #
R3 f => R3 (Point f) #
Methods _z :: Functor f => (a -> f a) -> Point f a -> f (Point f a) # _xyz :: Functor f => (V3 a -> f (V3 a)) -> Point f a -> f (Point f a) #
R4 f => R4 (Point f) #
Methods _w :: Functor f => (a -> f a) -> Point f a -> f (Point f a) # _xyzw :: Functor f => (V4 a -> f (V4 a)) -> Point f a -> f (Point f a) #
Additive f => Affine (Point f) #
Associated Types type Diff (Point f :: * -> ) :: -> * # Methods (.-.) :: Num a => Point f a -> Point f a -> Diff (Point f) a # (.+^) :: Num a => Point f a -> Diff (Point f) a -> Point f a # (.-^) :: Num a => Point f a -> Diff (Point f) a -> Point f a #
Eq (f a) => Eq (Point f a) #
Methods (==) :: Point f a -> Point f a -> Bool # (/=) :: Point f a -> Point f a -> Bool #
Fractional (f a) => Fractional (Point f a) #
Methods (/) :: Point f a -> Point f a -> Point f a # recip :: Point f a -> Point f a # fromRational :: Rational -> Point f a #
(Data (f a), Typeable * a, Typeable (* -> *) f) => Data (Point f a) #
Methods gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Point f a -> c (Point f a) # gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Point f a) # toConstr :: Point f a -> Constr # dataTypeOf :: Point f a -> DataType # dataCast1 :: Typeable (* -> ) t => (forall d. Data d => c (t d)) -> Maybe (c (Point f a)) # dataCast2 :: Typeable ( -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Point f a)) # gmapT :: (forall b. Data b => b -> b) -> Point f a -> Point f a # gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Point f a -> r # gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Point f a -> r # gmapQ :: (forall d. Data d => d -> u) -> Point f a -> [u] # gmapQi :: Int -> (forall d. Data d => d -> u) -> Point f a -> u # gmapM :: Monad m => (forall d. Data d => d -> m d) -> Point f a -> m (Point f a) # gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Point f a -> m (Point f a) # gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Point f a -> m (Point f a) #
Num (f a) => Num (Point f a) #
Methods (+) :: Point f a -> Point f a -> Point f a # (-) :: Point f a -> Point f a -> Point f a # (*) :: Point f a -> Point f a -> Point f a # negate :: Point f a -> Point f a # abs :: Point f a -> Point f a # signum :: Point f a -> Point f a # fromInteger :: Integer -> Point f a #
Ord (f a) => Ord (Point f a) #
Methods compare :: Point f a -> Point f a -> Ordering # (<) :: Point f a -> Point f a -> Bool # (<=) :: Point f a -> Point f a -> Bool # (>) :: Point f a -> Point f a -> Bool # (>=) :: Point f a -> Point f a -> Bool # max :: Point f a -> Point f a -> Point f a # min :: Point f a -> Point f a -> Point f a #
Read (f a) => Read (Point f a) #
Methods readsPrec :: Int -> ReadS (Point f a) # readList :: ReadS [Point f a] # readPrec :: ReadPrec (Point f a) # readListPrec :: ReadPrec [Point f a] #
Show (f a) => Show (Point f a) #
Methods showsPrec :: Int -> Point f a -> ShowS # show :: Point f a -> String # showList :: [Point f a] -> ShowS #
Ix (f a) => Ix (Point f a) #
Methods range :: (Point f a, Point f a) -> [Point f a] # index :: (Point f a, Point f a) -> Point f a -> Int # unsafeIndex :: (Point f a, Point f a) -> Point f a -> Int inRange :: (Point f a, Point f a) -> Point f a -> Bool # rangeSize :: (Point f a, Point f a) -> Int # unsafeRangeSize :: (Point f a, Point f a) -> Int
Generic (Point f a) #
Associated Types type Rep (Point f a) :: * -> * # Methods from :: Point f a -> Rep (Point f a) x # to :: Rep (Point f a) x -> Point f a #
Storable (f a) => Storable (Point f a) #
Methods sizeOf :: Point f a -> Int # alignment :: Point f a -> Int # peekElemOff :: Ptr (Point f a) -> Int -> IO (Point f a) # pokeElemOff :: Ptr (Point f a) -> Int -> Point f a -> IO () # peekByteOff :: Ptr b -> Int -> IO (Point f a) # pokeByteOff :: Ptr b -> Int -> Point f a -> IO () # peek :: Ptr (Point f a) -> IO (Point f a) # poke :: Ptr (Point f a) -> Point f a -> IO () #
Binary (f a) => Binary (Point f a) #
Methods put :: Point f a -> Put # get :: Get (Point f a) # putList :: [Point f a] -> Put #
Serial (f a) => Serial (Point f a) #
Methods serialize :: MonadPut m => Point f a -> m () # deserialize :: MonadGet m => m (Point f a) #
Serialize (f a) => Serialize (Point f a) #
Methods put :: Putter (Point f a) # get :: Get (Point f a) #
NFData (f a) => NFData (Point f a) #
Methods rnf :: Point f a -> () #
Hashable (f a) => Hashable (Point f a) #
Methods hashWithSalt :: Int -> Point f a -> Int # hash :: Point f a -> Int #
Unbox (f a) => Unbox (Point f a) #

Ixed (f a) => Ixed (Point f a) #
Methods ix :: Index (Point f a) -> Traversal' (Point f a) (IxValue (Point f a)) #
Wrapped (Point f a) #
Associated Types type Unwrapped (Point f a) :: * # Methods _Wrapped' :: Iso' (Point f a) (Unwrapped (Point f a)) #
Epsilon (f a) => Epsilon (Point f a) #
Methods nearZero :: Point f a -> Bool #
(~) * t (Point g b) => Rewrapped (Point f a) t #

Traversable f => Each (Point f a) (Point f b) a b #
Methods each :: Traversal (Point f a) (Point f b) a b #
data MVector s (Point f a) #
data MVector s (Point f a) = MV_P (MVector s (f a))
type Rep1 (Point f) #
type Rep1 (Point f) = D1 (MetaData "Point" "Linear.Affine" "linear-1.20.6-6UAb21vhoOd3sQpfz3EoRJ" True) (C1 (MetaCons "P" PrefixI False) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec1 f)))
type Rep (Point f) #
type Rep (Point f) = Rep f
type Size (Point f) #
type Size (Point f) = Size f
type Diff (Point f) #
type Diff (Point f) = f
type Rep (Point f a) #
type Rep (Point f a) = D1 (MetaData "Point" "Linear.Affine" "linear-1.20.6-6UAb21vhoOd3sQpfz3EoRJ" True) (C1 (MetaCons "P" PrefixI False) (S1 (MetaSel (Nothing Symbol) NoSourceUnpackedness NoSourceStrictness DecidedLazy) (Rec0 (f a))))
data Vector (Point f a) #
data Vector (Point f a) = V_P (Vector (f a))
type Index (Point f a) #
type Index (Point f a) = Index (f a)
type IxValue (Point f a) #
type IxValue (Point f a) = IxValue (f a)
type Unwrapped (Point f a) #
type Unwrapped (Point f a) = f a

lensP :: Lens' (Point g a) (g a) #

_Point :: Iso' (Point f a) (f a) #

(.#) :: Coercible b a => (b -> c) -> (a -> b) -> a -> c #

(#.) :: Coercible c b => (b -> c) -> (a -> b) -> a -> c #

unP :: Point f a -> f a #

origin :: (Additive f, Num a) => Point f a #

Vector spaces have origins.

relative :: (Additive f, Num a) => Point f a -> Iso' (Point f a) (f a) #

An isomorphism between points and vectors, given a reference point.