module Fractal.RUFF.Types.Complex
( Complex((:+)), cis, mkPolar
, realPart, imagPart, conjugate
, magnitude, phase, polar
) where
import Data.Data (Data)
import Data.Typeable (Typeable)
data Complex r = !r :+ !r
deriving (Read, Show, Eq, Ord, Data, Typeable)
instance Num r => Num (Complex r) where
(x :+ y) + (u :+ v) = (x + u) :+ (y + v)
(x :+ y) (u :+ v) = (x u) :+ (y v)
(x :+ y) * (u :+ v) = (x * u y * v) :+ (x * v + y * u)
negate (x :+ y) = negate x :+ negate y
abs = error "Fractal.Types.Complex.Num.abs"
signum = error "Fractal.Types.Complex.Num.signum"
fromInteger n = fromInteger n :+ 0
instance Fractional r => Fractional (Complex r) where
(x :+ y) / (u :+ v) = ((x * u + y * v) / d) :+ ((y * u x * v) / d) where d = u * u + v * v
fromRational r = fromRational r :+ 0
realPart :: Complex r -> r
realPart (r :+ _) = r
imagPart :: Complex r -> r
imagPart (_ :+ i) = i
conjugate :: Num r => Complex r -> Complex r
conjugate (r :+ i) = r :+ negate i
phase :: (Ord r, Floating r) => Complex r -> r
phase (r :+ i)
| r > 0 && i > 0 = atan ( i / r)
| r > 0 && i < 0 = atan (abs i / r)
| r < 0 && i > 0 = pi atan ( i / abs r)
| r < 0 && i < 0 = atan (abs i / abs r) pi
| i > 0 = pi / 2
| i < 0 = pi / 2
| r < 0 = pi
| otherwise = 0
magnitude :: Floating r => Complex r -> r
magnitude (r :+ i) = sqrt $ r * r + i * i
mkPolar :: Floating r => r -> r -> Complex r
mkPolar r t = (r * cos t) :+ (r * sin t)
cis :: Floating r => r -> Complex r
cis t = cos t :+ sin t
polar :: (Ord r, Floating r) => Complex r -> (r, r)
polar z = (magnitude z, phase z)