```{-# OPTIONS -fno-warn-missing-methods #-}
{-# LANGUAGE FlexibleInstances #-}

-- | Geometric functions concerning vectors.
module Graphics.Gloss.Geometry.Vector
( magV
, argV
, dotV
, detV
, mulSV
, rotateV
, angleVV
, normaliseV
, unitVectorAtAngle )
where
import Graphics.Gloss.Picture		(Vector)
import Graphics.Gloss.Geometry.Angle

-- | Pretend a vector is a number.
--	Vectors aren't real numbes according to Haskell, because they don't
--	support the multiply and divide field operators. We can pretend they
--	are though, and use the (+) and (-) operators as component-wise
--
instance Num (Float, Float) where
(+) (x1, y1) (x2, y2)	= (x1 + x2, y1 + y2)
(-) (x1, y1) (x2, y2)	= (x1 - x2, y1 - y2)
negate (x, y)		= (negate x, negate y)

-- | The magnitude of a vector.
magV :: Vector -> Float
{-# INLINE magV #-}
magV (x, y)
= sqrt (x * x + y * y)

-- | The angle of this vector, relative to the +ve x-axis.
argV :: Vector -> Float
{-# INLINE argV #-}
argV (x, y)
= normaliseAngle \$ atan2 y x

-- | The dot product of two vectors.
dotV :: Vector -> Vector -> Float
{-# INLINE dotV #-}
dotV (x1, x2) (y1, y2)
= x1 * y1 + x2 * y2

-- | The determinant of two vectors.
detV :: Vector -> Vector -> Float
{-# INLINE detV #-}
detV (x1, y1) (x2, y2)
= x1 * y2 - y1 * x2

-- | Multiply a vector by a scalar.
mulSV :: Float -> Vector -> Vector
{-# INLINE mulSV #-}
mulSV s (x, y)
= (s * x, s * y)

-- | Rotate a vector by an angle (in radians). +ve angle is counter-clockwise.
rotateV :: Float -> Vector -> Vector
{-# INLINE rotateV #-}
rotateV r (x, y)
= 	(  x * cos r - y * sin r
,  x * sin r + y * cos r)

-- | Compute the inner angle (in radians) between two vectors.
angleVV :: Vector -> Vector -> Float
{-# INLINE angleVV #-}
angleVV p1@(x1, y1) p2@(x2, y2)
= let 	m1	= magV p1
m2	= magV p2
d	= p1 `dotV` p2
aDiff	= acos \$ d / (m1 * m2)

-- | Normalise a vector, so it has a magnitude of 1.
normaliseV :: Vector -> Vector
{-# INLINE normaliseV #-}
normaliseV v	= mulSV (1 / magV v) v

-- | Produce a unit vector at a given angle relative to the +ve x-axis.
--	The provided angle is in radians.
unitVectorAtAngle :: Float -> Vector
{-# INLINE unitVectorAtAngle #-}
unitVectorAtAngle r
= (cos r, sin r)

```