6w3

import Prelude

data Quaternion = Quat { a :: Double,
                         b :: Double,
                         c :: Double,
                         d :: Double }
  deriving Eq

-- Take the real part of a quaternion (used for testing)
-- realPart (a + b*i + c*j + d*k) == a
realPart :: Quaternion -> Double
realPart q = (a q)

i, j, k, zeroQuat :: Quaternion
i = Quat 0 1 0 0
j = Quat 0 0 1 0
k = Quat 0 0 0 1
zeroQuat = Quat 0 0 0 0

fromDouble :: Double -> Quaternion
fromDouble x = Quat x 0 0 0

instance Show Quaternion where
  show (Quat a b c d) = show a ++ " + " ++ show b ++ "i + " ++ show c ++ "j + " ++ show d ++ "k"


instance Num Quaternion where
  q1 + q2 = Quat (a q1 + a q2) (b q1 + b q2) (c q1 + c q2) (d q1 + d q2)
  q1 * q2 = Quat a_term b_term c_term d_term
    where
      a_term = a q1 * a q2 - b q1 * b q2 - c q1 * c q2 - d q1 * d q2
      b_term = a q1 * b q2 + b q1 * a q2 + c q1 * d q2 - d q1 * c q2
      c_term = a q1 * c q2 - b q1 * d q2 + c q1 * a q2 + d q1 * b q2
      d_term = a q1 * d q2 + b q1 * c q2 - c q1 * b q2 + d q1 * a q2

  abs q = fromDouble (sqrt (sum_prod))
    where
      sum_prod = a q * a q + b q * b q + c q * c q + d q * d q

  -- Divide each Double component by the (abs x)
  signum q =
    if q /= zeroQuat then
      Quat (a q / absVal) (b q / absVal) (c q / absVal) (d q / absVal)
    else
      zeroQuat
    where
      absVal = realPart (abs q)

  negate q = Quat (- a q) (- b q) (- c q) (-d q)
  fromInteger n = Quat (fromIntegral n) 0 0 0