```{-# LANGUAGE DeriveDataTypeable, PatternGuards, ScopedTypeVariables #-}
module Linear.Quaternion
( Quaternion(..)
, Complicated(..)
, Hamiltonian(..)
, slerp
, asinq
, acosq
, atanq
, asinhq
, acoshq
, atanhq
, absi
, pow
, rotate
, axisAngle
) where
import Control.Applicative
import Control.Lens
import Data.Complex (Complex((:+)))
import Data.Data
import Data.Distributive
import Data.Foldable
import qualified Data.Foldable as F
import Data.Monoid
import Foreign.Ptr (castPtr, plusPtr)
import Foreign.Storable (Storable(..))
import Linear.Epsilon
import Linear.Conjugate
import Linear.Metric
import Linear.V3
import Linear.Vector
import Prelude hiding (any)

data Quaternion a = Quaternion a {-# UNPACK #-}!(V3 a)

instance Functor Quaternion where
fmap f (Quaternion e v) = Quaternion (f e) (fmap f v)
a <\$ _ = Quaternion a (V3 a a a)

instance Applicative Quaternion where
pure a = Quaternion a (pure a)
Quaternion f fv <*> Quaternion a v = Quaternion (f a) (fv <*> v)

instance Monad Quaternion where
return = pure
(>>=) = bindRep -- the diagonal of a sedenion is super useful!

instance Representable Quaternion where
rep f = Quaternion (f _e) (V3 (f _i) (f _j) (f _k))

instance Foldable Quaternion where
foldMap f (Quaternion e v) = f e `mappend` foldMap f v
foldr f z (Quaternion e v) = f e (F.foldr f z v)

instance Traversable Quaternion where
traverse f (Quaternion e v) = Quaternion <\$> f e <*> traverse f v

instance forall a. Storable a => Storable (Quaternion a) where
sizeOf _ = 4 * sizeOf (undefined::a)
alignment _ = alignment (undefined::a)
poke ptr (Quaternion e v) = poke (castPtr ptr) e >>
poke (castPtr (ptr `plusPtr` sz)) v
where sz = sizeOf (undefined::a)
peek ptr = Quaternion <\$> peek (castPtr ptr)
<*> peek (castPtr (ptr `plusPtr` sz))
where sz = sizeOf (undefined::a)

instance RealFloat a => Num (Quaternion a) where
{-# SPECIALIZE instance Num (Quaternion Float) #-}
{-# SPECIALIZE instance Num (Quaternion Double) #-}
(+) = liftA2 (+)
(-) = liftA2 (-)
negate = fmap negate
Quaternion s1 v1 * Quaternion s2 v2 = Quaternion (s1*s2 - (v1 `dot` v2)) \$
(v1 `cross` v2) + s1*^v2 + s2*^v1
fromInteger x = Quaternion (fromInteger x) 0
abs z = Quaternion (norm z) 0
signum q@(Quaternion e (V3 i j k))
| m == 0.0 = q
| not (isInfinite m || isNaN m) = q ^/ sqrt m
| any isNaN q = qNaN
| not (ii || ij || ik) = Quaternion 1 (V3 0 0 0)
| not (ie || ij || ik) = Quaternion 0 (V3 1 0 0)
| not (ie || ii || ik) = Quaternion 0 (V3 0 1 0)
| not (ie || ii || ij) = Quaternion 0 (V3 0 0 1)
| otherwise = qNaN
where
m = quadrance q
ie = isInfinite e
ii = isInfinite i
ij = isInfinite j
ik = isInfinite k

-- abs    = error "Quaternion.abs: use norm"
-- signum = error "Quaternion.signum: use signorm"

qNaN :: RealFloat a => Quaternion a
qNaN = Quaternion fNaN (V3 fNaN fNaN fNaN) where fNaN = 0/0

-- {-# 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.

instance RealFloat a => Fractional (Quaternion a) where
{-# SPECIALIZE instance Fractional (Quaternion Float) #-}
{-# SPECIALIZE instance Fractional (Quaternion Double) #-}
Quaternion q0 (V3 q1 q2 q3) / Quaternion r0 (V3 r1 r2 r3) =
Quaternion (r0*q0+r1*q1+r2*q2+r3*q3)
(V3 (r0*q1-r1*q0-r2*q3+r3*q2)
(r0*q2+r1*q3-r2*q0-r3*q1)
(r0*q3-r1*q2+r2*q1-r3*q0))
^/ (r0*r0 + r1*r1 + r2*r2 + r3*r3)
recip q = q ^/ quadrance q
fromRational x = Quaternion (fromRational x) 0

instance Metric Quaternion where
Quaternion e v `dot` Quaternion e' v' = e*e' + (v `dot` v')

class Complicated t where
_e :: Functor f => (a -> f a) -> t a -> f (t a)
_i :: Functor f => (a -> f a) -> t a -> f (t a)

instance Complicated Complex where
_e f (a :+ b) = (:+ b) <\$> f a
_i f (a :+ b) = (a :+) <\$> f b

instance Complicated Quaternion where
_e f (Quaternion a v) = (\a' -> Quaternion a' v) <\$> f a
_i f (Quaternion a v) = Quaternion a <\$> traverseOf _x f v
--_i f (Quaternion a (V3 b c d)) = (\b' -> Quaternion a (V3 b' c d)) <\$> f b

class Complicated t => Hamiltonian t where
_j :: Functor f => (a -> f a) -> t a -> f (t a)
_k :: Functor f => (a -> f a) -> t a -> f (t a)
_ijk :: Functor f => (V3 a -> f (V3 a)) -> t a -> f (t a)

instance Hamiltonian Quaternion where
_j f (Quaternion a v) = Quaternion a <\$> traverseOf _y f v
_k f (Quaternion a v) = Quaternion a <\$> traverseOf _z f v
-- _j f (Quaternion a (V3 b c d)) = (\c' -> Quaternion a (V3 b c' d)) <\$> f c
-- _k f (Quaternion a (V3 b c d)) = Quaternion a . V3 b c <\$> f d

_ijk f (Quaternion a v) = Quaternion a <\$> f v

instance Distributive Quaternion where
distribute = distributeRep

instance (Conjugate a, RealFloat a) => Conjugate (Quaternion a) where
conjugate (Quaternion e v) = Quaternion (conjugate e) (negate v)

reimagine :: RealFloat a => a -> a -> Quaternion a -> Quaternion a
reimagine r s (Quaternion _ v)
| isNaN s || isInfinite s = let aux 0 = 0
aux x = s * x
in Quaternion r (aux <\$> v)
| otherwise = Quaternion r (v^*s)

-- | quadrance of the imaginary component
qi :: Num a => Quaternion a -> a
qi (Quaternion _ v) = quadrance v

-- | norm of the imaginary component
absi :: Floating a => Quaternion a -> a
absi = sqrt . qi

-- | raise a 'Quaternion' to a scalar power
pow :: RealFloat a => Quaternion a -> a -> Quaternion a
pow q t = exp (t *^ log q)

-- ehh..
instance RealFloat a => Floating (Quaternion a) where
{-# SPECIALIZE instance Floating (Quaternion Float) #-}
{-# SPECIALIZE instance Floating (Quaternion Double) #-}
pi = Quaternion pi 0
exp q@(Quaternion e v)
| qiq == 0 = Quaternion (exp e) v
| ai <- sqrt qiq, ee <- exp e = reimagine (ee * cos ai) (ee * (sin ai / ai)) q
where qiq = qi q
log q@(Quaternion e v@(V3 _i j k))
| qiq == 0 = if e >= 0
then Quaternion (log e) v
else Quaternion (log (negate e)) (V3 pi j k) -- mmm, pi
| ai <- sqrt qiq, m <- sqrt (e*e + qiq) = reimagine (log m) (atan2 m e / ai) q
where qiq = qi q
x ** y = exp (y * log x)
sqrt q@(Quaternion e v)
| m   == 0 = q
| qiq == 0 = if e > 0
then Quaternion (sqrt e) 0
else Quaternion 0 (V3 (sqrt (negate e)) 0 0)
| im <- sqrt (0.5*(m-e)) / sqrt qiq = Quaternion (0.5*(m+e)) (v^*im)
where qiq = qi q
m = sqrt (e*e + qiq)
cos q@(Quaternion e v)
| qiq == 0 = Quaternion (cos e) v
| ai <- sqrt qiq = reimagine (cos e * cosh ai) (- sin e * (sinh ai / ai)) q
where qiq = qi q
sin q@(Quaternion e v)
| qiq == 0 = Quaternion (sin e) v
| ai <- sqrt qiq = reimagine (sin e * cosh ai) (cos e * (sinh ai / ai)) q
where qiq = qi q
tan q@(Quaternion e v)
| qiq == 0 = Quaternion (tan e) v
| ai <- sqrt qiq, ce <- cos e, sai <- sinh ai, d <- ce*ce + sai*sai =
reimagine (ce * sin e / d) (cosh ai * (sai / ai) / d) q
where qiq = qi q
sinh q@(Quaternion e v)
| qiq == 0 = Quaternion (sinh e) v
| ai <- sqrt qiq = reimagine (sinh e * cos ai) (cosh e * (sin ai / ai)) q
where qiq = qi q
cosh q@(Quaternion e v)
| qiq == 0 = Quaternion (cosh e) v
| ai <- sqrt qiq = reimagine (cosh e * cos ai) ((sinh e * sin ai) / ai) q
where qiq = qi q
tanh q@(Quaternion e v)
| qiq == 0 = Quaternion (tanh e) v
| ai <- sqrt qiq, se <- sinh e, cai <- cos ai, d <- se*se + cai*cai =
reimagine ((cosh e * se) / d) ((cai * (sin ai / ai)) / d) q
where qiq = qi q

asin q = cut asin q
acos q = cut acos q
atan q = cut atan q

asinh q = cut asinh q
acosh q = cut acosh q
atanh q = cut atanh q

-- | Helper for calculating with specific branch cuts
cut :: RealFloat a => (Complex a -> Complex a) -> Quaternion a -> Quaternion a
cut f q@(Quaternion e v)
| qiq == 0 = Quaternion a (_x.~b\$v)
| otherwise = reimagine a (b / ai) q
where qiq = qi q
ai = sqrt qiq
a :+ b = f (e :+ ai)

-- | Helper for calculating with specific branch cuts
cutWith :: RealFloat a => Complex a -> Quaternion a -> Quaternion a
cutWith (r :+ im) q@(Quaternion e v)
| e /= 0 || qiq == 0 || isNaN qiq || isInfinite qiq = error "bad cut"
| s <- im / sqrt qiq = Quaternion r (v^*s)
where qiq = qi q

-- | 'asin' with a specified branch cut.
asinq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
asinq q@(Quaternion e _) u
| qiq /= 0.0 || e >= -1 && e <= 1 = asin q
| otherwise = cutWith (asin (e :+ sqrt qiq)) u
where qiq = qi q

-- | 'acos' with a specified branch cut.
acosq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
acosq q@(Quaternion e _) u
| qiq /= 0.0 || e >= -1 && e <= 1 = acos q
| otherwise = cutWith (acos (e :+ sqrt qiq)) u
where qiq = qi q

-- | 'atan' with a specified branch cut.
atanq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
atanq q@(Quaternion e _) u
| e /= 0.0 || qiq >= -1 && qiq <= 1 = atan q
| otherwise = cutWith (atan (e :+ sqrt qiq)) u
where qiq = qi q

-- | 'asinh' with a specified branch cut.
asinhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
asinhq q@(Quaternion e _) u
| e /= 0.0 || qiq >= -1 && qiq <= 1 = asinh q
| otherwise = cutWith (asinh (e :+ sqrt qiq)) u
where qiq = qi q

-- | 'acosh' with a specified branch cut.
acoshq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
acoshq q@(Quaternion e _) u
| qiq /= 0.0 || e >= 1 = asinh q
| otherwise = cutWith (acosh (e :+ sqrt qiq)) u
where qiq = qi q

-- | 'atanh' with a specified branch cut.
atanhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
atanhq q@(Quaternion e _) u
| qiq /= 0.0 || e > -1 && e < 1 = atanh q
| otherwise = cutWith (atanh (e :+ sqrt qiq)) u
where qiq = qi q

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

--slerp :: RealFloat a => Quaternion a -> Quaternion a -> a -> Quaternion a
--slerp q0 q1 = let q10 = q1 / q0 in \t -> pow q10 t * q0

-- | Apply a rotation to a vector.
rotate :: (Conjugate a, RealFloat a) => Quaternion a -> V3 a -> V3 a
rotate q v = (q * Quaternion 0 v * conjugate q)^._ijk

{-
rotate :: Num a => Quaternion a -> V3 a -> V3 a
rotate (Quaternion a' b c d) (V3 x y z) = V3
(2*((t8+t10)*x+(t6- t4)*y+(t3+t7)*z)+x)
(2*((t4+ t6)*y+(t5+t10)*y+(t9-t2)*z)+y)
(2*((t7- t3)*z+(t2+ t9)*z+(t5+t8)*z)+z)
where
a = -a'
t2 = a*b
t3 = a*c
t4 = a*d
t5 = -b*b
t6 = b*c
t7 = b*d
t8 = -c*c
t9 = c*d
t10 = -d*d
-}
{-# SPECIALIZE rotate :: Quaternion Float -> V3 Float -> V3 Float #-}
{-# SPECIALIZE rotate :: Quaternion Double -> V3 Double -> V3 Double #-}

instance (RealFloat a, Epsilon a) => Epsilon (Quaternion a) where
nearZero = nearZero . quadrance

-- | @'axisAngle' axis theta@ builds a 'Quaternion' representing a
-- rotation of @theta@ radians about @axis@.
axisAngle :: (Epsilon a, Floating a) => V3 a -> a -> Quaternion a
axisAngle axis theta = normalize \$ Quaternion (cos half) \$ (sin half) *^ axis
where half = theta / 2
```