{-# LANGUAGE CPP #-}
{-# LANGUAGE DeriveDataTypeable #-}
{-# LANGUAGE DeriveGeneric #-}
{-# LANGUAGE DeriveTraversable #-}
{-# LANGUAGE ScopedTypeVariables #-}
{-# LANGUAGE TypeFamilies #-}

module Nonlinear.Quaternion
  ( Quaternion (..),
    Complicated (..),
    Hamiltonian (..),
    slerp,
    asinq,
    acosq,
    atanq,
    asinhq,
    acoshq,
    atanhq,
    absi,
    pow,
    rotate,
    axisAngle,
  )
where

import Control.Applicative
import Control.Monad.Fix
import Control.Monad.Zip
import Data.Complex (Complex ((:+)))
import Data.Data
import Data.Foldable
import Data.Functor ((<&>))
import Data.Functor.Classes
import Foreign.Ptr (castPtr, plusPtr)
import Foreign.Storable (Storable (..))
import GHC.Generics (Generic, Generic1)
import Nonlinear.Internal
import Nonlinear.V1
import Nonlinear.V2
import Nonlinear.V3
import Nonlinear.V4
import Nonlinear.Vector
import Prelude hiding (any)

#if MIN_VERSION_base(4,14,0)
import GHC.Ix (Ix (..))
#else
import Data.Ix (Ix (..))
#endif

-- | Quaternions
data Quaternion a = Quaternion !a {-# UNPACK #-} !(V3 a)
  deriving (Quaternion a -> Quaternion a -> Bool
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$cmapM :: forall (m :: * -> *) a b.
Monad m =>
(a -> m b) -> Quaternion a -> m (Quaternion b)
sequenceA :: Quaternion (f a) -> f (Quaternion a)
$csequenceA :: forall (f :: * -> *) a.
Applicative f =>
Quaternion (f a) -> f (Quaternion a)
traverse :: (a -> f b) -> Quaternion a -> f (Quaternion b)
$ctraverse :: forall (f :: * -> *) a b.
Applicative f =>
(a -> f b) -> Quaternion a -> f (Quaternion b)
$cp2Traversable :: Foldable Quaternion
$cp1Traversable :: Functor Quaternion
Traversable)

instance Applicative Quaternion where
  pure :: a -> Quaternion a
pure a
a = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (a -> V3 a
forall (f :: * -> *) a. Applicative f => a -> f a
pure a
a)
  {-# INLINE pure #-}
  Quaternion a -> b
f V3 (a -> b)
fv <*> :: Quaternion (a -> b) -> Quaternion a -> Quaternion b
<*> Quaternion a
a V3 a
v = b -> V3 b -> Quaternion b
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> b
f a
a) (V3 (a -> b)
fv V3 (a -> b) -> V3 a -> V3 b
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> V3 a
v)
  {-# INLINE (<*>) #-}

instance Monad Quaternion where
  return :: a -> Quaternion a
return = a -> Quaternion a
forall (f :: * -> *) a. Applicative f => a -> f a
pure
  {-# INLINE return #-}

  -- the diagonal of a sedenion is super useful!
  Quaternion a
a (V3 a
b a
c a
d) >>= :: Quaternion a -> (a -> Quaternion b) -> Quaternion b
>>= a -> Quaternion b
f = b -> V3 b -> Quaternion b
forall a. a -> V3 a -> Quaternion a
Quaternion b
a' (b -> b -> b -> V3 b
forall a. a -> a -> a -> V3 a
V3 b
b' b
c' b
d')
    where
      Quaternion b
a' V3 b
_ = a -> Quaternion b
f a
a
      Quaternion b
_ (V3 b
b' b
_ b
_) = a -> Quaternion b
f a
b
      Quaternion b
_ (V3 b
_ b
c' b
_) = a -> Quaternion b
f a
c
      Quaternion b
_ (V3 b
_ b
_ b
d') = a -> Quaternion b
f a
d
  {-# INLINE (>>=) #-}

instance Vec Quaternion where
  construct :: ((forall b. Lens' (Quaternion b) b) -> a) -> Quaternion a
construct (forall b. Lens' (Quaternion b) b) -> a
f = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion ((forall b. Lens' (Quaternion b) b) -> a
f forall b. Lens' (Quaternion b) b
forall (t :: * -> *) a. R4 t => Lens' (t a) a
_w) (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 ((forall b. Lens' (Quaternion b) b) -> a
f forall b. Lens' (Quaternion b) b
forall (t :: * -> *) a. R1 t => Lens' (t a) a
_x) ((forall b. Lens' (Quaternion b) b) -> a
f forall b. Lens' (Quaternion b) b
forall (t :: * -> *) a. R2 t => Lens' (t a) a
_y) ((forall b. Lens' (Quaternion b) b) -> a
f forall b. Lens' (Quaternion b) b
forall (t :: * -> *) a. R3 t => Lens' (t a) a
_z))

instance Ix a => Ix (Quaternion a) where
  {-# SPECIALIZE instance Ix (Quaternion Int) #-}

  range :: (Quaternion a, Quaternion a) -> [Quaternion a]
range (Quaternion a
l1 V3 a
l2, Quaternion a
u1 V3 a
u2) =
    [a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
i1 V3 a
i2 | a
i1 <- (a, a) -> [a]
forall a. Ix a => (a, a) -> [a]
range (a
l1, a
u1), V3 a
i2 <- (V3 a, V3 a) -> [V3 a]
forall a. Ix a => (a, a) -> [a]
range (V3 a
l2, V3 a
u2)]
  {-# INLINE range #-}

  inRange :: (Quaternion a, Quaternion a) -> Quaternion a -> Bool
inRange (Quaternion a
l1 V3 a
l2, Quaternion a
u1 V3 a
u2) (Quaternion a
i1 V3 a
i2) =
    (a, a) -> a -> Bool
forall a. Ix a => (a, a) -> a -> Bool
inRange (a
l1, a
u1) a
i1 Bool -> Bool -> Bool
&& (V3 a, V3 a) -> V3 a -> Bool
forall a. Ix a => (a, a) -> a -> Bool
inRange (V3 a
l2, V3 a
u2) V3 a
i2
  {-# INLINE inRange #-}

#if MIN_VERSION_base(4,14,0)
  unsafeIndex :: (Quaternion a, Quaternion a) -> Quaternion a -> Int
unsafeIndex (Quaternion a
l1 V3 a
l2, Quaternion a
u1 V3 a
u2) (Quaternion a
i1 V3 a
i2) =
    (a, a) -> a -> Int
forall a. Ix a => (a, a) -> a -> Int
unsafeIndex (a
l1, a
u1) a
i1 Int -> Int -> Int
forall a. Num a => a -> a -> a
* (V3 a, V3 a) -> Int
forall a. Ix a => (a, a) -> Int
unsafeRangeSize (V3 a
l2, V3 a
u2) Int -> Int -> Int
forall a. Num a => a -> a -> a
+ (V3 a, V3 a) -> V3 a -> Int
forall a. Ix a => (a, a) -> a -> Int
unsafeIndex (V3 a
l2, V3 a
u2) V3 a
i2
  {-# INLINE unsafeIndex #-}
#else
  index (Quaternion l1 l2, Quaternion u1 u2) (Quaternion i1 i2) =
    index (l1, u1) i1 * rangeSize (l2, u2) + index (l2, u2) i2
  {-# INLINE index #-}
#endif

instance Storable a => Storable (Quaternion a) where
  sizeOf :: Quaternion a -> Int
sizeOf Quaternion a
_ = Int
4 Int -> Int -> Int
forall a. Num a => a -> a -> a
* a -> Int
forall a. Storable a => a -> Int
sizeOf (a
forall a. HasCallStack => a
undefined :: a)
  {-# INLINE sizeOf #-}
  alignment :: Quaternion a -> Int
alignment Quaternion a
_ = a -> Int
forall a. Storable a => a -> Int
alignment (a
forall a. HasCallStack => a
undefined :: a)
  {-# INLINE alignment #-}
  poke :: Ptr (Quaternion a) -> Quaternion a -> IO ()
poke Ptr (Quaternion a)
ptr (Quaternion a
e V3 a
v) =
    Ptr a -> a -> IO ()
forall a. Storable a => Ptr a -> a -> IO ()
poke (Ptr (Quaternion a) -> Ptr a
forall a b. Ptr a -> Ptr b
castPtr Ptr (Quaternion a)
ptr) a
e
      IO () -> IO () -> IO ()
forall (m :: * -> *) a b. Monad m => m a -> m b -> m b
>> Ptr (V3 a) -> V3 a -> IO ()
forall a. Storable a => Ptr a -> a -> IO ()
poke (Ptr Any -> Ptr (V3 a)
forall a b. Ptr a -> Ptr b
castPtr (Ptr (Quaternion a)
ptr Ptr (Quaternion a) -> Int -> Ptr Any
forall a b. Ptr a -> Int -> Ptr b
`plusPtr` Int
sz)) V3 a
v
    where
      sz :: Int
sz = a -> Int
forall a. Storable a => a -> Int
sizeOf (a
forall a. HasCallStack => a
undefined :: a)
  {-# INLINE poke #-}
  peek :: Ptr (Quaternion a) -> IO (Quaternion a)
peek Ptr (Quaternion a)
ptr =
    a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> V3 a -> Quaternion a) -> IO a -> IO (V3 a -> Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> Ptr a -> IO a
forall a. Storable a => Ptr a -> IO a
peek (Ptr (Quaternion a) -> Ptr a
forall a b. Ptr a -> Ptr b
castPtr Ptr (Quaternion a)
ptr)
      IO (V3 a -> Quaternion a) -> IO (V3 a) -> IO (Quaternion a)
forall (f :: * -> *) a b. Applicative f => f (a -> b) -> f a -> f b
<*> Ptr (V3 a) -> IO (V3 a)
forall a. Storable a => Ptr a -> IO a
peek (Ptr Any -> Ptr (V3 a)
forall a b. Ptr a -> Ptr b
castPtr (Ptr (Quaternion a)
ptr Ptr (Quaternion a) -> Int -> Ptr Any
forall a b. Ptr a -> Int -> Ptr b
`plusPtr` Int
sz))
    where
      sz :: Int
sz = a -> Int
forall a. Storable a => a -> Int
sizeOf (a
forall a. HasCallStack => a
undefined :: a)
  {-# INLINE peek #-}

instance RealFloat a => Num (Quaternion a) where
  {-# SPECIALIZE instance Num (Quaternion Float) #-}
  {-# SPECIALIZE instance Num (Quaternion Double) #-}
  + :: Quaternion a -> Quaternion a -> Quaternion a
(+) = (a -> a -> a) -> Quaternion a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 a -> a -> a
forall a. Num a => a -> a -> a
(+)
  {-# INLINE (+) #-}
  (-) = (a -> a -> a) -> Quaternion a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 (-)
  {-# INLINE (-) #-}
  negate :: Quaternion a -> Quaternion a
negate = (a -> a) -> Quaternion a -> Quaternion a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
fmap a -> a
forall a. Num a => a -> a
negate
  {-# INLINE negate #-}
  Quaternion a
s1 V3 a
v1 * :: Quaternion a -> Quaternion a -> Quaternion a
* Quaternion a
s2 V3 a
v2 =
    a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a
s1 a -> a -> a
forall a. Num a => a -> a -> a
* a
s2 a -> a -> a
forall a. Num a => a -> a -> a
- (V3 a
v1 V3 a -> V3 a -> a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> f a -> a
`dot` V3 a
v2)) (V3 a -> Quaternion a) -> V3 a -> Quaternion a
forall a b. (a -> b) -> a -> b
$
      (V3 a
v1 V3 a -> V3 a -> V3 a
forall a. Num a => V3 a -> V3 a -> V3 a
`cross` V3 a
v2) V3 a -> V3 a -> V3 a
forall a. Num a => a -> a -> a
+ a
s1 a -> V3 a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ V3 a
v2 V3 a -> V3 a -> V3 a
forall a. Num a => a -> a -> a
+ a
s2 a -> V3 a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ V3 a
v1
  {-# INLINE (*) #-}
  fromInteger :: Integer -> Quaternion a
fromInteger Integer
x = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (Integer -> a
forall a. Num a => Integer -> a
fromInteger Integer
x) V3 a
0
  {-# INLINE fromInteger #-}
  abs :: Quaternion a -> Quaternion a
abs Quaternion a
z = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (Quaternion a -> a
forall (f :: * -> *) a. (Vec f, Floating a) => f a -> a
norm Quaternion a
z) V3 a
0
  {-# INLINE abs #-}
  signum :: Quaternion a -> Quaternion a
signum q :: Quaternion a
q@(Quaternion a
e (V3 a
i a
j a
k))
    | a
m a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0.0 = Quaternion a
q
    | Bool -> Bool
not (a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
m Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
m) = Quaternion a
q Quaternion a -> a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Fractional a) => f a -> a -> f a
^/ a -> a
forall a. Floating a => a -> a
sqrt a
m
    | (a -> Bool) -> Quaternion a -> Bool
forall (t :: * -> *) a. Foldable t => (a -> Bool) -> t a -> Bool
any a -> Bool
forall a. RealFloat a => a -> Bool
isNaN Quaternion a
q = Quaternion a
forall a. RealFloat a => Quaternion a
qNaN
    | Bool -> Bool
not (Bool
ii Bool -> Bool -> Bool
|| Bool
ij Bool -> Bool -> Bool
|| Bool
ik) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
1 (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
0 a
0 a
0)
    | Bool -> Bool
not (Bool
ie Bool -> Bool -> Bool
|| Bool
ij Bool -> Bool -> Bool
|| Bool
ik) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
0 (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
1 a
0 a
0)
    | Bool -> Bool
not (Bool
ie Bool -> Bool -> Bool
|| Bool
ii Bool -> Bool -> Bool
|| Bool
ik) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
0 (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
0 a
1 a
0)
    | Bool -> Bool
not (Bool
ie Bool -> Bool -> Bool
|| Bool
ii Bool -> Bool -> Bool
|| Bool
ij) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
0 (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
0 a
0 a
1)
    | Bool
otherwise = Quaternion a
forall a. RealFloat a => Quaternion a
qNaN
    where
      m :: a
m = Quaternion a -> a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a
quadrance Quaternion a
q
      ie :: Bool
ie = a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
e
      ii :: Bool
ii = a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
i
      ij :: Bool
ij = a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
j
      ik :: Bool
ik = a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
k
  {-# INLINE signum #-}

-- {-# RULES "abs/norm" abs x = Quaternion (norm x) 0 #-}
-- {-# RULES "signum/signorm" signum = signorm #-}

-- this will attempt to rewrite calls to abs to use norm intead when it is available.

qNaN :: RealFloat a => Quaternion a
qNaN :: Quaternion a
qNaN = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
fNaN (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
fNaN a
fNaN a
fNaN) where fNaN :: a
fNaN = a
0 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
0
{-# INLINE qNaN #-}

instance RealFloat a => Fractional (Quaternion a) where
  {-# SPECIALIZE instance Fractional (Quaternion Float) #-}
  {-# SPECIALIZE instance Fractional (Quaternion Double) #-}
  Quaternion a
q0 (V3 a
q1 a
q2 a
q3) / :: Quaternion a -> Quaternion a -> Quaternion a
/ Quaternion a
r0 (V3 a
r1 a
r2 a
r3) =
    a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion
      (a
r0 a -> a -> a
forall a. Num a => a -> a -> a
* a
q0 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r1 a -> a -> a
forall a. Num a => a -> a -> a
* a
q1 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r2 a -> a -> a
forall a. Num a => a -> a -> a
* a
q2 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r3 a -> a -> a
forall a. Num a => a -> a -> a
* a
q3)
      ( a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3
          (a
r0 a -> a -> a
forall a. Num a => a -> a -> a
* a
q1 a -> a -> a
forall a. Num a => a -> a -> a
- a
r1 a -> a -> a
forall a. Num a => a -> a -> a
* a
q0 a -> a -> a
forall a. Num a => a -> a -> a
- a
r2 a -> a -> a
forall a. Num a => a -> a -> a
* a
q3 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r3 a -> a -> a
forall a. Num a => a -> a -> a
* a
q2)
          (a
r0 a -> a -> a
forall a. Num a => a -> a -> a
* a
q2 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r1 a -> a -> a
forall a. Num a => a -> a -> a
* a
q3 a -> a -> a
forall a. Num a => a -> a -> a
- a
r2 a -> a -> a
forall a. Num a => a -> a -> a
* a
q0 a -> a -> a
forall a. Num a => a -> a -> a
- a
r3 a -> a -> a
forall a. Num a => a -> a -> a
* a
q1)
          (a
r0 a -> a -> a
forall a. Num a => a -> a -> a
* a
q3 a -> a -> a
forall a. Num a => a -> a -> a
- a
r1 a -> a -> a
forall a. Num a => a -> a -> a
* a
q2 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r2 a -> a -> a
forall a. Num a => a -> a -> a
* a
q1 a -> a -> a
forall a. Num a => a -> a -> a
- a
r3 a -> a -> a
forall a. Num a => a -> a -> a
* a
q0)
      )
      Quaternion a -> a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Fractional a) => f a -> a -> f a
^/ (a
r0 a -> a -> a
forall a. Num a => a -> a -> a
* a
r0 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r1 a -> a -> a
forall a. Num a => a -> a -> a
* a
r1 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r2 a -> a -> a
forall a. Num a => a -> a -> a
* a
r2 a -> a -> a
forall a. Num a => a -> a -> a
+ a
r3 a -> a -> a
forall a. Num a => a -> a -> a
* a
r3)
  {-# INLINE (/) #-}
  recip :: Quaternion a -> Quaternion a
recip q :: Quaternion a
q@(Quaternion a
e V3 a
v) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
e (V3 a -> V3 a
forall a. Num a => a -> a
negate V3 a
v) Quaternion a -> a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Fractional a) => f a -> a -> f a
^/ Quaternion a -> a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a
quadrance Quaternion a
q
  {-# INLINE recip #-}
  fromRational :: Rational -> Quaternion a
fromRational Rational
x = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (Rational -> a
forall a. Fractional a => Rational -> a
fromRational Rational
x) V3 a
0
  {-# INLINE fromRational #-}

-- | A vector space that includes the basis elements '_e' and '_i'
class Complicated t where
  _e, _i :: Lens' (t a) a

instance Complicated Complex where
  _e :: (a -> m a) -> Complex a -> m (Complex a)
_e a -> m a
f (a
a :+ a
b) = (a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a
b) (a -> Complex a) -> m a -> m (Complex a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> m a
f a
a
  {-# INLINE _e #-}
  _i :: (a -> m a) -> Complex a -> m (Complex a)
_i a -> m a
f (a
a :+ a
b) = (a
a a -> a -> Complex a
forall a. a -> a -> Complex a
:+) (a -> Complex a) -> m a -> m (Complex a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> m a
f a
b
  {-# INLINE _i #-}

instance Complicated Quaternion where
  _e :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_e a -> m a
f (Quaternion a
a V3 a
v) = (a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
`Quaternion` V3 a
v) (a -> Quaternion a) -> m a -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> a -> m a
f a
a
  {-# INLINE _e #-}
  _i :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_i a -> m a
f (Quaternion a
a V3 a
v) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (V3 a -> Quaternion a) -> m (V3 a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (a -> m a) -> V3 a -> m (V3 a)
forall (t :: * -> *) a. R1 t => Lens' (t a) a
_x a -> m a
f V3 a
v
  {-# INLINE _i #-}

-- | A vector space that includes the basis elements '_e', '_i', '_j' and '_k'
class Complicated t => Hamiltonian t where
  _j, _k :: Lens' (t a) a
  _ijk :: Lens' (t a) (V3 a)

instance Hamiltonian Quaternion where
  _j :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_j a -> m a
f (Quaternion a
a V3 a
v) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (V3 a -> Quaternion a) -> m (V3 a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (a -> m a) -> V3 a -> m (V3 a)
forall (t :: * -> *) a. R2 t => Lens' (t a) a
_y a -> m a
f V3 a
v
  {-# INLINE _j #-}
  _k :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_k a -> m a
f (Quaternion a
a V3 a
v) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (V3 a -> Quaternion a) -> m (V3 a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> (a -> m a) -> V3 a -> m (V3 a)
forall (t :: * -> *) a. R3 t => Lens' (t a) a
_z a -> m a
f V3 a
v
  {-# INLINE _k #-}
  _ijk :: (V3 a -> m (V3 a)) -> Quaternion a -> m (Quaternion a)
_ijk V3 a -> m (V3 a)
f (Quaternion a
a V3 a
v) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (V3 a -> Quaternion a) -> m (V3 a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> V3 a -> m (V3 a)
f V3 a
v
  {-# INLINE _ijk #-}

reimagine :: RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine :: a -> a -> Quaternion a -> Quaternion a
reimagine a
r a
s (Quaternion a
_ V3 a
v)
  | a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
s Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
s =
    let aux :: a -> a
aux a
0 = a
0
        aux a
x = a
s a -> a -> a
forall a. Num a => a -> a -> a
* a
x
     in a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
r (a -> a
aux (a -> a) -> V3 a -> V3 a
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> V3 a
v)
  | Bool
otherwise = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
r (V3 a
v V3 a -> a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a -> f a
^* a
s)
{-# INLINE reimagine #-}

-- | quadrance of the imaginary component
qi :: Num a => Quaternion a -> a
qi :: Quaternion a -> a
qi (Quaternion a
_ V3 a
v) = V3 a -> a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a
quadrance V3 a
v
{-# INLINE qi #-}

-- | norm of the imaginary component
absi :: Floating a => Quaternion a -> a
absi :: Quaternion a -> a
absi = a -> a
forall a. Floating a => a -> a
sqrt (a -> a) -> (Quaternion a -> a) -> Quaternion a -> a
forall b c a. (b -> c) -> (a -> b) -> a -> c
. Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi
{-# INLINE absi #-}

-- | raise a 'Quaternion' to a scalar power
pow :: RealFloat a => Quaternion a -> a -> Quaternion a
pow :: Quaternion a -> a -> Quaternion a
pow Quaternion a
q a
t = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
exp (a
t a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ Quaternion a -> Quaternion a
forall a. Floating a => a -> a
log Quaternion a
q)
{-# INLINE pow #-}

sqrte2pqiq :: (Floating a, Ord a) => a -> a -> a
sqrte2pqiq :: a -> a -> a
sqrte2pqiq a
e a
qiq -- = sqrt (e*e + qiq)
  | a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< -a
1.5097698010472593e153 = -(a
qiq a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
e) a -> a -> a
forall a. Num a => a -> a -> a
- a
e
  | a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
5.582399551122541e57 = a -> a
forall a. Floating a => a -> a
sqrt (a
e a -> a -> a
forall a. Num a => a -> a -> a
* a
e a -> a -> a
forall a. Num a => a -> a -> a
+ a
qiq) -- direct definition
  | Bool
otherwise = (a
qiq a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
e) a -> a -> a
forall a. Num a => a -> a -> a
+ a
e

-- {-# SPECIALIZE sqrte2pqiq :: Double -> Double -> Double #-}
-- {-# SPECIALIZE sqrte2pqiq :: Float -> Float -> Float #-}

tanrhs :: (Floating a, Ord a) => a -> a -> a -> a
tanrhs :: a -> a -> a -> a
tanrhs a
sai a
ai a
d -- = cosh ai * (sai / ai) / d -- improved from 6.04 bits of error to 0.19 bits
  | a
sai a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< -a
4.618902267687042e-52 = (a
sai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cosh a
ai
  | a
sai a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
1.038530535935153e-39 = (a -> a
forall a. Floating a => a -> a
cosh a
ai a -> a -> a
forall a. Num a => a -> a -> a
* a
sai) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d
  | Bool
otherwise = (a
sai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cosh a
ai

-- {-# SPECIALIZE tanrhs :: Double -> Double -> Double -> Double #-}
-- {-# SPECIALIZE tanrhs :: Float -> Float -> Float -> Float #-}

-- ehh..
instance RealFloat a => Floating (Quaternion a) where
  {-# SPECIALIZE instance Floating (Quaternion Float) #-}
  {-# SPECIALIZE instance Floating (Quaternion Double) #-}
  pi :: Quaternion a
pi = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
forall a. Floating a => a
pi V3 a
0
  {-# INLINE pi #-}
  exp :: Quaternion a -> Quaternion a
exp q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
exp a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq, a
exe <- a -> a
forall a. Floating a => a -> a
exp a
e = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a
exe a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cos a
ai) (a
exe a -> a -> a
forall a. Num a => a -> a -> a
* (a -> a
forall a. Floating a => a -> a
sin a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai)) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE exp #-}
  log :: Quaternion a -> Quaternion a
log q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 =
      if a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= a
0
        then a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
log a
e) V3 a
v -- Using v rather than 0 preserves negative zeros
        else a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Num a => a -> a
negate (a -> a
forall a. Floating a => a -> a
log (a -> a
forall a. Num a => a -> a
negate a
e))) V3 a
v -- negative scalar: negate quaternion, take log, negate again, preserves negative zeros
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
log a
m) (a -> a
forall a. Floating a => a -> a
acos (a
e a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
m) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
      m :: a
m = a -> a -> a
forall a. (Floating a, Ord a) => a -> a -> a
sqrte2pqiq a
e a
qiq
  {-# INLINE log #-}

  Quaternion a
x ** :: Quaternion a -> Quaternion a -> Quaternion a
** Quaternion a
y = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
exp (Quaternion a
y Quaternion a -> Quaternion a -> Quaternion a
forall a. Num a => a -> a -> a
* Quaternion a -> Quaternion a
forall a. Floating a => a -> a
log Quaternion a
x)
  {-# INLINE (**) #-}

  sqrt :: Quaternion a -> Quaternion a
sqrt q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
m a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = Quaternion a
q
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 =
      if a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> a
0
        then a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
sqrt a
e) V3 a
0
        else a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
0 (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 (a -> a
forall a. Floating a => a -> a
sqrt (a -> a
forall a. Num a => a -> a
negate a
e)) a
0 a
0)
    | a
im <- a -> a
forall a. Floating a => a -> a
sqrt (a
0.5 a -> a -> a
forall a. Num a => a -> a -> a
* (a
m a -> a -> a
forall a. Num a => a -> a -> a
- a
e)) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a
0.5 a -> a -> a
forall a. Num a => a -> a -> a
* (a
m a -> a -> a
forall a. Num a => a -> a -> a
+ a
e)) (V3 a
v V3 a -> a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a -> f a
^* a
im)
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
      m :: a
m = a -> a -> a
forall a. (Floating a, Ord a) => a -> a -> a
sqrte2pqiq a
e a
qiq
  {-# INLINE sqrt #-}

  cos :: Quaternion a -> Quaternion a
cos q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
cos a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
cos a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cosh a
ai) (-a -> a
forall a. Floating a => a -> a
sin a
e a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sinh a
ai) Quaternion a
q -- 0.15 bits error
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE cos #-}

  sin :: Quaternion a -> Quaternion a
sin q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
sin a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
sin a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cosh a
ai) (a -> a
forall a. Floating a => a -> a
cos a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
sinh a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE sin #-}

  tan :: Quaternion a -> Quaternion a
tan q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
tan a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq,
      a
ce <- a -> a
forall a. Floating a => a -> a
cos a
e,
      a
sai <- a -> a
forall a. Floating a => a -> a
sinh a
ai,
      a
d <- a
ce a -> a -> a
forall a. Num a => a -> a -> a
* a
ce a -> a -> a
forall a. Num a => a -> a -> a
+ a
sai a -> a -> a
forall a. Num a => a -> a -> a
* a
sai =
      a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a
ce a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
sin a
e a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d) (a -> a -> a -> a
forall a. (Floating a, Ord a) => a -> a -> a -> a
tanrhs a
sai a
ai a
d) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE tan #-}

  sinh :: Quaternion a -> Quaternion a
sinh q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
sinh a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
sinh a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cos a
ai) (a -> a
forall a. Floating a => a -> a
cosh a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
sin a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE sinh #-}

  cosh :: Quaternion a -> Quaternion a
cosh q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
cosh a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
cosh a
e a -> a -> a
forall a. Num a => a -> a -> a
* a -> a
forall a. Floating a => a -> a
cos a
ai) (a -> a
forall a. Floating a => a -> a
sin a
ai a -> a -> a
forall a. Num a => a -> a -> a
* (a -> a
forall a. Floating a => a -> a
sinh a
e a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai)) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE cosh #-}

  tanh :: Quaternion a -> Quaternion a
tanh q :: Quaternion a
q@(Quaternion a
e V3 a
v)
    | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
tanh a
e) V3 a
v
    | a
ai <- a -> a
forall a. Floating a => a -> a
sqrt a
qiq,
      a
se <- a -> a
forall a. Floating a => a -> a
sinh a
e,
      a
cai <- a -> a
forall a. Floating a => a -> a
cos a
ai,
      a
d <- a
se a -> a -> a
forall a. Num a => a -> a -> a
* a
se a -> a -> a
forall a. Num a => a -> a -> a
+ a
cai a -> a -> a
forall a. Num a => a -> a -> a
* a
cai =
      a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine (a -> a
forall a. Floating a => a -> a
cosh a
e a -> a -> a
forall a. Num a => a -> a -> a
* a
se a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d) (a -> a -> a -> a
forall a. (Floating a, Ord a) => a -> a -> a -> a
tanhrhs a
cai a
ai a
d) Quaternion a
q
    where
      qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
  {-# INLINE tanh #-}

  asin :: Quaternion a -> Quaternion a
asin = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
asin
  {-# INLINE asin #-}
  acos :: Quaternion a -> Quaternion a
acos = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
acos
  {-# INLINE acos #-}
  atan :: Quaternion a -> Quaternion a
atan = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
atan
  {-# INLINE atan #-}

  asinh :: Quaternion a -> Quaternion a
asinh = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
asinh
  {-# INLINE asinh #-}
  acosh :: Quaternion a -> Quaternion a
acosh = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
acosh
  {-# INLINE acosh #-}
  atanh :: Quaternion a -> Quaternion a
atanh = (Complex a -> Complex a) -> Quaternion a -> Quaternion a
forall a.
RealFloat a =>
(Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
forall a. Floating a => a -> a
atanh
  {-# INLINE atanh #-}

tanhrhs :: (Floating a, Ord a) => a -> a -> a -> a
tanhrhs :: a -> a -> a -> a
tanhrhs a
cai a
ai a
d -- = cai * (sin ai / ai) / d
  | a
d a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= -a
4.2173720203427147e-29 Bool -> Bool -> Bool
&& a
d a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
4.446702369113811e64 = a
cai a -> a -> a
forall a. Fractional a => a -> a -> a
/ (a
d a -> a -> a
forall a. Num a => a -> a -> a
* (a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sin a
ai))
  | Bool
otherwise = a
cai a -> a -> a
forall a. Num a => a -> a -> a
* (a
1 a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sin a
ai) a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
d

-- {-# SPECIALIZE tanhrhs :: Double -> Double -> Double -> Double #-}
-- {-# SPECIALIZE tanhrhs :: Float -> Float -> Float -> Float #-}

-- | Helper for calculating with specific branch cuts
cut :: RealFloat a => (Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut :: (Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut Complex a -> Complex a
f q :: Quaternion a
q@(Quaternion a
e (V3 a
_ a
y a
z))
  | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
a (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
b a
y a
z)
  | Bool
otherwise = a -> a -> Quaternion a -> Quaternion a
forall a. RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine a
a (a
b a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
ai) Quaternion a
q
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
    ai :: a
ai = a -> a
forall a. Floating a => a -> a
sqrt a
qiq
    a
a :+ a
b = Complex a -> Complex a
f (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a
ai)
{-# INLINE cut #-}

-- | Helper for calculating with specific branch cuts
cutWith :: RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith :: Complex a -> Quaternion a -> Quaternion a
cutWith (a
r :+ a
im) q :: Quaternion a
q@(Quaternion a
e V3 a
v)
  | a
e a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0 Bool -> Bool -> Bool
|| a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
== a
0 Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isNaN a
qiq Bool -> Bool -> Bool
|| a -> Bool
forall a. RealFloat a => a -> Bool
isInfinite a
qiq = String -> Quaternion a
forall a. HasCallStack => String -> a
error String
"bad cut"
  | a
s <- a
im a -> a -> a
forall a. Fractional a => a -> a -> a
/ a -> a
forall a. Floating a => a -> a
sqrt a
qiq = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
r (V3 a
v V3 a -> a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> a -> f a
^* a
s)
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE cutWith #-}

-- | 'asin' with a specified branch cut.
asinq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
asinq :: Quaternion a -> Quaternion a -> Quaternion a
asinq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= -a
1 Bool -> Bool -> Bool
&& a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
asin Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
asin (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE asinq #-}

-- | 'acos' with a specified branch cut.
acosq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
acosq :: Quaternion a -> Quaternion a -> Quaternion a
acosq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= -a
1 Bool -> Bool -> Bool
&& a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
acos Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
acos (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE acosq #-}

-- | 'atan' with a specified branch cut.
atanq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
atanq :: Quaternion a -> Quaternion a -> Quaternion a
atanq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
e a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
qiq a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= -a
1 Bool -> Bool -> Bool
&& a
qiq a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
atan Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
atan (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE atanq #-}

-- | 'asinh' with a specified branch cut.
asinhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
asinhq :: Quaternion a -> Quaternion a -> Quaternion a
asinhq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
e a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
qiq a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= -a
1 Bool -> Bool -> Bool
&& a
qiq a -> a -> Bool
forall a. Ord a => a -> a -> Bool
<= a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
asinh Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
asinh (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE asinhq #-}

-- | 'acosh' with a specified branch cut.
acoshq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
acoshq :: Quaternion a -> Quaternion a -> Quaternion a
acoshq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
>= a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
asinh Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
acosh (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE acoshq #-}

-- | 'atanh' with a specified branch cut.
atanhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
atanhq :: Quaternion a -> Quaternion a -> Quaternion a
atanhq q :: Quaternion a
q@(Quaternion a
e V3 a
_) Quaternion a
u
  | a
qiq a -> a -> Bool
forall a. Eq a => a -> a -> Bool
/= a
0.0 Bool -> Bool -> Bool
|| a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
> -a
1 Bool -> Bool -> Bool
&& a
e a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
1 = Quaternion a -> Quaternion a
forall a. Floating a => a -> a
atanh Quaternion a
q
  | Bool
otherwise = Complex a -> Quaternion a -> Quaternion a
forall a. RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (Complex a -> Complex a
forall a. Floating a => a -> a
atanh (a
e a -> a -> Complex a
forall a. a -> a -> Complex a
:+ a -> a
forall a. Floating a => a -> a
sqrt a
qiq)) Quaternion a
u
  where
    qiq :: a
qiq = Quaternion a -> a
forall a. Num a => Quaternion a -> a
qi Quaternion a
q
{-# INLINE atanhq #-}

-- | Spherical linear interpolation between two quaternions.
slerp :: RealFloat a => Quaternion a -> Quaternion a -> a -> Quaternion a
slerp :: Quaternion a -> Quaternion a -> a -> Quaternion a
slerp Quaternion a
q Quaternion a
p a
t
  | a
1.0 a -> a -> a
forall a. Num a => a -> a -> a
- a
cosphi a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
1e-8 = Quaternion a
q
  | Bool
otherwise = ((a -> a
forall a. Floating a => a -> a
sin ((a
1 a -> a -> a
forall a. Num a => a -> a -> a
- a
t) a -> a -> a
forall a. Num a => a -> a -> a
* a
phi) a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ Quaternion a
q) Quaternion a -> Quaternion a -> Quaternion a
forall a. Num a => a -> a -> a
+ a -> a
forall a. Floating a => a -> a
sin (a
t a -> a -> a
forall a. Num a => a -> a -> a
* a
phi) a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ Quaternion a -> Quaternion a
f Quaternion a
p) Quaternion a -> a -> Quaternion a
forall (f :: * -> *) a. (Vec f, Fractional a) => f a -> a -> f a
^/ a -> a
forall a. Floating a => a -> a
sin a
phi
  where
    dqp :: a
dqp = Quaternion a -> Quaternion a -> a
forall (f :: * -> *) a. (Vec f, Num a) => f a -> f a -> a
dot Quaternion a
q Quaternion a
p
    (a
cosphi, Quaternion a -> Quaternion a
f) = if a
dqp a -> a -> Bool
forall a. Ord a => a -> a -> Bool
< a
0 then (-a
dqp, Quaternion a -> Quaternion a
forall a. Num a => a -> a
negate) else (a
dqp, Quaternion a -> Quaternion a
forall a. a -> a
id)
    phi :: a
phi = a -> a
forall a. Floating a => a -> a
acos a
cosphi
{-# SPECIALIZE slerp :: Quaternion Float -> Quaternion Float -> Float -> Quaternion Float #-}
{-# SPECIALIZE slerp :: Quaternion Double -> Quaternion Double -> Double -> Quaternion Double #-}

-- | Apply a rotation to a vector.
rotate :: (RealFloat a) => Quaternion a -> V3 a -> V3 a
rotate :: Quaternion a -> V3 a -> V3 a
rotate Quaternion a
q V3 a
v = V3 a
ijk
  where
    Quaternion a
_ V3 a
ijk = Quaternion a
q Quaternion a -> Quaternion a -> Quaternion a
forall a. Num a => a -> a -> a
* a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
0 V3 a
v Quaternion a -> Quaternion a -> Quaternion a
forall a. Num a => a -> a -> a
* Quaternion a
q
{-# SPECIALIZE rotate :: Quaternion Float -> V3 Float -> V3 Float #-}
{-# SPECIALIZE rotate :: Quaternion Double -> V3 Double -> V3 Double #-}

-- | @'axisAngle' axis theta@ builds a 'Quaternion' representing a
-- rotation of @theta@ radians about @axis@.
axisAngle :: Floating a => V3 a -> a -> Quaternion a
axisAngle :: V3 a -> a -> Quaternion a
axisAngle V3 a
axis a
theta = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion (a -> a
forall a. Floating a => a -> a
cos a
half) (a -> a
forall a. Floating a => a -> a
sin a
half a -> V3 a -> V3 a
forall (f :: * -> *) a. (Vec f, Num a) => a -> f a -> f a
*^ V3 a -> V3 a
forall (f :: * -> *) a. (Vec f, Floating a) => f a -> f a
normalize V3 a
axis)
  where
    half :: a
half = a
theta a -> a -> a
forall a. Fractional a => a -> a -> a
/ a
2
{-# INLINE axisAngle #-}

instance MonadZip Quaternion where
  mzipWith :: (a -> b -> c) -> Quaternion a -> Quaternion b -> Quaternion c
mzipWith = (a -> b -> c) -> Quaternion a -> Quaternion b -> Quaternion c
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2

instance MonadFix Quaternion where
  mfix :: (a -> Quaternion a) -> Quaternion a
mfix a -> Quaternion a
f =
    a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion
      (let Quaternion a
a V3 a
_ = a -> Quaternion a
f a
a in a
a)
      ( a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3
          (let Quaternion a
_ (V3 a
a a
_ a
_) = a -> Quaternion a
f a
a in a
a)
          (let Quaternion a
_ (V3 a
_ a
a a
_) = a -> Quaternion a
f a
a in a
a)
          (let Quaternion a
_ (V3 a
_ a
_ a
a) = a -> Quaternion a
f a
a in a
a)
      )

instance Eq1 Quaternion where
  liftEq :: (a -> b -> Bool) -> Quaternion a -> Quaternion b -> Bool
liftEq a -> b -> Bool
f (Quaternion a
a V3 a
b) (Quaternion b
c V3 b
d) = a -> b -> Bool
f a
a b
c Bool -> Bool -> Bool
&& (a -> b -> Bool) -> V3 a -> V3 b -> Bool
forall (f :: * -> *) a b.
Eq1 f =>
(a -> b -> Bool) -> f a -> f b -> Bool
liftEq a -> b -> Bool
f V3 a
b V3 b
d

instance Ord1 Quaternion where
  liftCompare :: (a -> b -> Ordering) -> Quaternion a -> Quaternion b -> Ordering
liftCompare a -> b -> Ordering
f (Quaternion a
a V3 a
b) (Quaternion b
c V3 b
d) = a -> b -> Ordering
f a
a b
c Ordering -> Ordering -> Ordering
forall a. Monoid a => a -> a -> a
`mappend` (a -> b -> Ordering) -> V3 a -> V3 b -> Ordering
forall (f :: * -> *) a b.
Ord1 f =>
(a -> b -> Ordering) -> f a -> f b -> Ordering
liftCompare a -> b -> Ordering
f V3 a
b V3 b
d

instance Show1 Quaternion where
  liftShowsPrec :: (Int -> a -> ShowS)
-> ([a] -> ShowS) -> Int -> Quaternion a -> ShowS
liftShowsPrec Int -> a -> ShowS
f [a] -> ShowS
g Int
d (Quaternion a
a V3 a
b) = (Int -> a -> ShowS)
-> (Int -> V3 a -> ShowS) -> String -> Int -> a -> V3 a -> ShowS
forall a b.
(Int -> a -> ShowS)
-> (Int -> b -> ShowS) -> String -> Int -> a -> b -> ShowS
showsBinaryWith Int -> a -> ShowS
f ((Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> V3 a -> ShowS
forall (f :: * -> *) a.
Show1 f =>
(Int -> a -> ShowS) -> ([a] -> ShowS) -> Int -> f a -> ShowS
liftShowsPrec Int -> a -> ShowS
f [a] -> ShowS
g) String
"Quaternion" Int
d a
a V3 a
b

instance Read1 Quaternion where
  liftReadsPrec :: (Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (Quaternion a)
liftReadsPrec Int -> ReadS a
f ReadS [a]
g = (String -> ReadS (Quaternion a)) -> Int -> ReadS (Quaternion a)
forall a. (String -> ReadS a) -> Int -> ReadS a
readsData ((String -> ReadS (Quaternion a)) -> Int -> ReadS (Quaternion a))
-> (String -> ReadS (Quaternion a)) -> Int -> ReadS (Quaternion a)
forall a b. (a -> b) -> a -> b
$ (Int -> ReadS a)
-> (Int -> ReadS (V3 a))
-> String
-> (a -> V3 a -> Quaternion a)
-> String
-> ReadS (Quaternion a)
forall a b t.
(Int -> ReadS a)
-> (Int -> ReadS b) -> String -> (a -> b -> t) -> String -> ReadS t
readsBinaryWith Int -> ReadS a
f ((Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (V3 a)
forall (f :: * -> *) a.
Read1 f =>
(Int -> ReadS a) -> ReadS [a] -> Int -> ReadS (f a)
liftReadsPrec Int -> ReadS a
f ReadS [a]
g) String
"Quaternion" a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion

instance Semigroup a => Semigroup (Quaternion a) where
  <> :: Quaternion a -> Quaternion a -> Quaternion a
(<>) = (a -> a -> a) -> Quaternion a -> Quaternion a -> Quaternion a
forall (f :: * -> *) a b c.
Applicative f =>
(a -> b -> c) -> f a -> f b -> f c
liftA2 a -> a -> a
forall a. Semigroup a => a -> a -> a
(<>)

instance Monoid a => Monoid (Quaternion a) where
  mempty :: Quaternion a
mempty = a -> Quaternion a
forall (f :: * -> *) a. Applicative f => a -> f a
pure a
forall a. Monoid a => a
mempty

instance R1 Quaternion where
  _x :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_x a -> m a
f (Quaternion a
w (V3 a
x a
y a
z)) = a -> m a
f a
x m a -> (a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \a
x' -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
x' a
y a
z)

instance R2 Quaternion where
  _y :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_y a -> m a
f (Quaternion a
w (V3 a
x a
y a
z)) = a -> m a
f a
y m a -> (a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \a
y' -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
x a
y' a
z)
  _xy :: (V2 a -> m (V2 a)) -> Quaternion a -> m (Quaternion a)
_xy V2 a -> m (V2 a)
f (Quaternion a
w (V3 a
x a
y a
z)) = V2 a -> m (V2 a)
f (a -> a -> V2 a
forall a. a -> a -> V2 a
V2 a
x a
y) m (V2 a) -> (V2 a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \(V2 a
x' a
y') -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
x' a
y' a
z)

instance R3 Quaternion where
  _z :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_z a -> m a
f (Quaternion a
w (V3 a
x a
y a
z)) = a -> m a
f a
z m a -> (a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \a
z' -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
x a
y a
z')
  _xyz :: (V3 a -> m (V3 a)) -> Quaternion a -> m (Quaternion a)
_xyz V3 a -> m (V3 a)
f (Quaternion a
w V3 a
xyz) = a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w (V3 a -> Quaternion a) -> m (V3 a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => (a -> b) -> f a -> f b
<$> V3 a -> m (V3 a)
f V3 a
xyz

instance R4 Quaternion where
  _w :: (a -> m a) -> Quaternion a -> m (Quaternion a)
_w a -> m a
f (Quaternion a
w V3 a
xyz) = a -> m a
f a
w m a -> (a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \a
w' -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w' V3 a
xyz
  _xyzw :: (V4 a -> m (V4 a)) -> Quaternion a -> m (Quaternion a)
_xyzw V4 a -> m (V4 a)
f (Quaternion a
w (V3 a
x a
y a
z)) = V4 a -> m (V4 a)
f (a -> a -> a -> a -> V4 a
forall a. a -> a -> a -> a -> V4 a
V4 a
x a
y a
z a
w) m (V4 a) -> (V4 a -> Quaternion a) -> m (Quaternion a)
forall (f :: * -> *) a b. Functor f => f a -> (a -> b) -> f b
<&> \(V4 a
x' a
y' a
z' a
w') -> a -> V3 a -> Quaternion a
forall a. a -> V3 a -> Quaternion a
Quaternion a
w' (a -> a -> a -> V3 a
forall a. a -> a -> a -> V3 a
V3 a
x' a
y' a
z')