```{-# LANGUAGE CPP #-}
#define Flt Float
#define VECT_Float

{-# LANGUAGE MultiParamTypeClasses, FunctionalDependencies, GeneralizedNewtypeDeriving #-}

module Data.Vect.Flt.Base
( AbelianGroup(..) , vecSum
, MultSemiGroup(..) , Ring , semigroupProduct
, LeftModule(..) , RightModule(..)
, Vector(..) , DotProd(..) , CrossProd(..)
, normalize , distance , angle , angle'
, UnitVector(..)
, Pointwise(..)
, Extend(..) , HasCoordinates(..) , Dimension(..)
, Matrix(..) , Tensor(..) , Diagonal (..) , Determinant(..)
, Orthogonal(..) , Projective(..) , MatrixNorms(..)
, Vec2(..) , Vec3(..) , Vec4(..)
, Mat2(..) , Mat3(..) , Mat4(..)
, Ortho2 , Ortho3 , Ortho4
, Normal2 , Normal3 , Normal4
, Proj3 , Proj4
, mkVec2 , mkVec3 , mkVec4
, project , project' , projectUnsafe , flipNormal
, householder, householderOrtho
)
where

import System.Random
import Foreign

--------------------------------------------------------------------------------
-- class declarations

class AbelianGroup g where
(&+) :: g -> g -> g
(&-) :: g -> g -> g
neg  :: g -> g
zero :: g

infixl 6 &+
infixl 6 &-

vecSum :: AbelianGroup g => [g] -> g
vecSum l = foldl (&+) zero l

class MultSemiGroup r where
(.*.) :: r -> r -> r
one   :: r

class (AbelianGroup r, MultSemiGroup r) => Ring r

infixl 7 .*.

-- was: ringProduct :: Ring r => [r] -> r
semigroupProduct :: MultSemiGroup r => [r] -> r
semigroupProduct l = foldl (.*.) one l

class LeftModule r m where
lmul :: r -> m -> m
(*.) :: r -> m -> m
(*.) = lmul

class RightModule m r where
rmul :: m -> r -> m
(.*) :: m -> r -> m
(.*) = rmul

{- RULES
"matrix multiplication left"   forall m n x.  (n .*. m) *. x = n *. (m *. x)
"matrix multiplication right"  forall m n x.  x .* (m .*. n) = (x .* m) .* n
-}

infixr 7 *.
infixl 7 .*

class AbelianGroup v => Vector v where
mapVec    :: (Flt -> Flt) -> v -> v
scalarMul :: Flt -> v -> v
(*&) ::      Flt -> v -> v
(&*) ::      v -> Flt -> v
(*&) s v = scalarMul s v
(&*) v s = scalarMul s v

infixr 7 *&
infixl 7 &*

{-# RULES
"scalar multiplication left"   forall s t x.  t *& (s *& x) = (t*s) *& x
"scalar multiplication right"  forall s t x.  (x &* s) &* t = x &* (s*t)
#-}

class DotProd v where
(&.) :: v -> v -> Flt
norm    :: v -> Flt
normsqr :: v -> Flt
len     :: v -> Flt
lensqr  :: v -> Flt
len = norm
lensqr = normsqr
dotprod :: v -> v -> Flt
normsqr v = (v &. v)
norm = sqrt . lensqr
dotprod = (&.)

infix 7 &.

{-# RULES
"len/square 1"   forall x.  (len x)*(len x) = lensqr x
"len/square 2"   forall x.  (len x)^2 = lensqr x
"norm/square 1"  forall x.  (norm x)*(norm x) = normsqr x
"norm/square 2"  forall x.  (norm x)^2 = normsqr x
#-}

normalize :: (Vector v, DotProd v) => v -> v
normalize v = scalarMul (1.0/(len v)) v

distance :: (Vector v, DotProd v) => v -> v -> Flt
distance x y = norm (x &- y)

-- | the angle between two vectors
angle :: (Vector v, DotProd v) => v -> v -> Flt
angle x y = acos \$ (x &. y) / (norm x * norm y)

-- | the angle between two unit vectors
angle' {- ' CPP is sensitive to primes -} :: (Vector v, UnitVector v u, DotProd v) => u -> u -> Flt
angle' x y = acos (fromNormal x &. fromNormal y)

{-# RULES
"normalize is idempotent"  forall x. normalize (normalize x) = normalize x
#-}

class (Vector v, DotProd v) => UnitVector v u | v->u, u->v  where
mkNormal         :: v -> u       -- ^ normalizes the input
toNormalUnsafe   :: v -> u       -- ^ does not normalize the input!
fromNormal       :: u -> v
fromNormalRadius :: Flt -> u -> v
fromNormalRadius t n = t *& fromNormal n

-- | Projects the first vector down to the hyperplane orthogonal to the second (unit) vector
project' :: (Vector v, UnitVector v u, DotProd v) => v -> u -> v
project' what dir = projectUnsafe what (fromNormal dir)

-- | Direction (second argument) is assumed to be a /unit/ vector!
projectUnsafe :: (Vector v, DotProd v) => v -> v -> v
projectUnsafe what dir = what &- dir &* (what &. dir)

project :: (Vector v, DotProd v) => v -> v -> v
project what dir = what &- dir &* ((what &. dir) / (dir &. dir))

-- | Since unit vectors are not a group, we need a separate function.
flipNormal :: UnitVector v n => n -> n
flipNormal = toNormalUnsafe . neg . fromNormal

-- | Cross product
class CrossProd v where
crossprod :: v -> v -> v
(&^)      :: v -> v -> v
(&^) = crossprod

-- | Pointwise multiplication
class Pointwise v where
pointwise :: v -> v -> v
(&!)      :: v -> v -> v
(&!) = pointwise

infix 7 &^
infix 7 &!

class HasCoordinates v x | v->x where
_1 :: v -> x
_2 :: v -> x
_3 :: v -> x
_4 :: v -> x

-- | conversion between vectors (and matrices) of different dimensions
class Extend u v where
extendZero :: u -> v          -- ^ example: @extendZero (Vec2 5 6) = Vec4 5 6 0 0@
extendWith :: Flt -> u -> v   -- ^ example: @extendWith 1 (Vec2 5 6) = Vec4 5 6 1 1@
trim :: v -> u                -- ^ example: @trim (Vec4 5 6 7 8) = Vec2 5 6@

-- | makes a diagonal matrix from a vector
class Diagonal s t | t->s where
diag :: s -> t

class Matrix m where
transpose :: m -> m
inverse :: m -> m
idmtx :: m

{-# RULES
"transpose is an involution"  forall m. transpose (transpose m) = m
"inverse is an involution"    forall m. inverse (inverse m) = m
#-}

class Matrix m => Orthogonal m o | m->o, o->m where
fromOrtho     :: o -> m
toOrthoUnsafe :: m -> o

class (AbelianGroup m, Matrix m) => MatrixNorms m where
frobeniusNorm  :: m -> Flt       -- ^ the frobenius norm (= euclidean norm in the space of matrices)
matrixDistance :: m -> m -> Flt  -- ^ euclidean distance in the space of matrices
operatorNorm   :: m -> Flt       -- ^ (euclidean) operator norm (not implemented yet)
matrixDistance m n = frobeniusNorm (n &- m)
operatorNorm = error "operatorNorm: not implemented yet"

-- | Outer product (could be unified with Diagonal?)
class Tensor t v | t->v where
outer :: v -> v -> t

class Determinant m where
det :: m -> Flt

class Dimension a where
dim :: a -> Int

-- | Householder matrix, see <http://en.wikipedia.org/wiki/Householder_transformation>.
-- In plain words, it is the reflection to the hyperplane orthogonal to the input vector.
householder :: (Vector v, UnitVector v u, Matrix m, Vector m, Tensor m v) => u -> m
householder u = idmtx &- (2 *& outer v v)
where v = fromNormal u

householderOrtho :: (Vector v, UnitVector v u, Matrix m, Vector m, Tensor m v, Orthogonal m o) => u -> o
householderOrtho = toOrthoUnsafe . householder

-- | \"Projective\" matrices have the following form: the top left corner
-- is an any matrix, the bottom right corner is 1, and the top-right
-- column is zero. These describe the affine orthogonal transformation of
-- the space one dimension less.
class (Vector v, Orthogonal n o, Diagonal v n) => Projective v n o m p
| m->p, p->m, p->o, o->p, p->n, n->p, p->v, v->p, n->o, n->v, v->n where
fromProjective     :: p -> m
toProjectiveUnsafe :: m -> p
orthogonal         :: o -> p
linear             :: n -> p
translation        :: v -> p
scaling            :: v -> p

--------------------------------------------------------------------------------
-- Vec / Mat datatypes

data Vec2 = Vec2 {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt
data Vec3 = Vec3 {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt
data Vec4 = Vec4 {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt {-# UNPACK #-} !Flt

-- | The components are /row/ vectors
data Mat2 = Mat2 !Vec2 !Vec2              deriving (Read,Show)
data Mat3 = Mat3 !Vec3 !Vec3 !Vec3        deriving (Read,Show)
data Mat4 = Mat4 !Vec4 !Vec4 !Vec4 !Vec4  deriving (Read,Show)

-- | The assumption when dealing with these is always that they are of unit length.
-- Also, interpolation works differently.
newtype Normal2 = Normal2 Vec2 deriving (Read,Show,Storable,DotProd,Dimension)
newtype Normal3 = Normal3 Vec3 deriving (Read,Show,Storable,DotProd,Dimension)
newtype Normal4 = Normal4 Vec4 deriving (Read,Show,Storable,DotProd,Dimension)

mkVec2 :: (Flt,Flt) -> Vec2
mkVec3 :: (Flt,Flt,Flt) -> Vec3
mkVec4 :: (Flt,Flt,Flt,Flt) -> Vec4

mkVec2 (x,y)     = Vec2 x y
mkVec3 (x,y,z)   = Vec3 x y z
mkVec4 (x,y,z,w) = Vec4 x y z w

-- | Orthogonal matrices.
--
-- Note: the "Random" instances generates orthogonal matrices with determinant 1
-- (that is, orientation-preserving orthogonal transformations)!
newtype Ortho2 = Ortho2 Mat2 deriving (Read,Show,Storable,MultSemiGroup,Determinant,Dimension)
newtype Ortho3 = Ortho3 Mat3 deriving (Read,Show,Storable,MultSemiGroup,Determinant,Dimension)
newtype Ortho4 = Ortho4 Mat4 deriving (Read,Show,Storable,MultSemiGroup,Determinant,Dimension)

-- | Projective matrices, encoding affine transformations in dimension one less.
newtype Proj3 = Proj3 Mat3 deriving (Read,Show,Storable,MultSemiGroup)
newtype Proj4 = Proj4 Mat4 deriving (Read,Show,Storable,MultSemiGroup)

--------------------------------------------------------------------------------
-- Unit vectors

instance UnitVector Vec2 Normal2 where
mkNormal v = Normal2 (normalize v)
fromNormal (Normal2 v) = v
toNormalUnsafe = Normal2

instance UnitVector Vec3 Normal3 where
mkNormal v = Normal3 (normalize v)
fromNormal (Normal3 v) = v
toNormalUnsafe = Normal3

instance UnitVector Vec4 Normal4 where
mkNormal v = Normal4 (normalize v)
fromNormal (Normal4 v) = v
toNormalUnsafe = Normal4

_rndUnit :: (RandomGen g, Random v, Vector v, DotProd v) => g -> (v,g)
_rndUnit g =
if d > 0.01
then ( v &* (1.0/d) , h )
else _rndUnit h
where
(v,h) = random g
d = norm v

instance Random Normal2 where
random g = let (v,h) = _rndUnit g in (Normal2 v, h)
randomR _ = random

instance Random Normal3 where
random g = let (v,h) = _rndUnit g in (Normal3 v, h)
randomR _ = random

instance Random Normal4 where
random g = let (v,h) = _rndUnit g in (Normal4 v, h)
randomR _ = random

instance CrossProd Normal3 where
crossprod (Normal3 v) (Normal3 w) = mkNormal (crossprod v w)

--------------------------------------------------------------------------------
-- Orthogonal matrices

instance Orthogonal Mat2 Ortho2 where
fromOrtho (Ortho2 o) = o
toOrthoUnsafe = Ortho2

instance Orthogonal Mat3 Ortho3 where
fromOrtho (Ortho3 o) = o
toOrthoUnsafe = Ortho3

instance Orthogonal Mat4 Ortho4 where
fromOrtho (Ortho4 o) = o
toOrthoUnsafe = Ortho4

------

instance Matrix Ortho2 where
transpose (Ortho2 o) = Ortho2 (transpose o)
idmtx = Ortho2 idmtx
inverse = transpose

instance Matrix Ortho3 where
transpose (Ortho3 o) = Ortho3 (transpose o)
idmtx = Ortho3 idmtx
inverse = transpose

instance Matrix Ortho4 where
transpose (Ortho4 o) = Ortho4 (transpose o)
idmtx = Ortho4 idmtx
inverse = transpose

------

instance Random Ortho2 where
random g = let (o,h) = _rndOrtho2 g in (toOrthoUnsafe (_flip1stRow2 o), h)
randomR _ = random

instance Random Ortho3 where
random g = let (o,h) = _rndOrtho3 g in (toOrthoUnsafe (             o), h)
randomR _ = random

instance Random Ortho4 where
random g = let (o,h) = _rndOrtho4 g in (toOrthoUnsafe (_flip1stRow4 o), h)
randomR _ = random

------

-- determinant will be -1
_rndOrtho2 :: RandomGen g => g -> (Mat2, g)
_rndOrtho2 g = (h2, g1) where
h2 = householder u2 :: Mat2
(u2,g1) = random g

-- generates a uniformly random orthogonal 3x3 matrix
-- /with determinant +1/, with respect to the Haar measure of SO3.
--
-- see Theorem 4 in:
-- Francesco Mezzadri: How to Generate Random Matrices from the Classical Compact Groups
-- Notices of the AMS, May 2007 issue
_rndOrtho3 :: RandomGen g => g -> (Mat3, g)
_rndOrtho3 g = ( (h3 .*. m3), g2) where
m3 = (extendWith :: Flt -> Mat2 -> Mat3) 1 o2
h3 = householder u3 :: Mat3
(u3,g1) = random g
(o2,g2) = _rndOrtho2 g1

-- determinant will be -1
_rndOrtho4 :: RandomGen g => g -> (Mat4, g)
_rndOrtho4 g = ( (h4 .*. m4), g2) where
m4 = (extendWith :: Flt -> Mat3 -> Mat4) 1 o3
h4 = householder u4 :: Mat4
(u4,g1) = random g
(o3,g2) = _rndOrtho3 g1

------

_flip1stRow2 :: Mat2 -> Mat2
_flip1stRow2 (Mat2 a b) = Mat2 (neg a) b

_flip1stRow3 :: Mat3 -> Mat3
_flip1stRow3 (Mat3 a b c) = Mat3 (neg a) b c

_flip1stRow4 :: Mat4 -> Mat4
_flip1stRow4 (Mat4 a b c d) = Mat4 (neg a) b c d

--------------------------------------------------------------------------------
-- projective matrices

instance Projective Vec2 Mat2 Ortho2 Mat3 Proj3 where
fromProjective (Proj3 m) = m
toProjectiveUnsafe = Proj3
orthogonal = Proj3 . extendWith 1 . fromOrtho
linear     = Proj3 . extendWith 1
translation v = Proj3 \$ Mat3 (Vec3 1 0 0) (Vec3 0 1 0) (extendWith 1 v)
scaling     v = Proj3 \$ diag (extendWith 1 v)

instance Projective Vec3 Mat3 Ortho3 Mat4 Proj4 where
fromProjective (Proj4 m) = m
toProjectiveUnsafe = Proj4
orthogonal = Proj4 . extendWith 1 . fromOrtho
linear     = Proj4 . extendWith 1
translation v = Proj4 \$ Mat4 (Vec4 1 0 0 0) (Vec4 0 1 0 0) (Vec4 0 0 1 0) (extendWith 1 v)
scaling     v = Proj4 \$ diag (extendWith 1 v)

instance Matrix Proj3 where
idmtx = Proj3 idmtx
transpose (Proj3 m) = Proj3 (transpose m)
inverse = _invertProj3

instance Matrix Proj4 where
idmtx = Proj4 idmtx
transpose (Proj4 m) = Proj4 (transpose m)
inverse = _invertProj4

_invertProj3 :: Proj3 -> Proj3
_invertProj3 (Proj3 mat@(Mat3 _ _ t)) =
Proj3 \$ Mat3 (extendZero a) (extendZero b) (extendWith 1 t')
where
t' = neg \$ (trim t :: Vec2) .* invm2
invm2@(Mat2 a b) = inverse \$ (trim mat :: Mat2)

-- Inverts a projective 4x4 matrix. But you can simply use "inverse" instead.
-- We assume that the bottom-right corner is 1.
_invertProj4 :: Proj4 -> Proj4
_invertProj4 (Proj4 mat@(Mat4 _ _ _ t)) =
Proj4 \$ Mat4 (extendZero a) (extendZero b) (extendZero c) (extendWith 1 t')
where
t' = neg \$ (trim t :: Vec3) .* invm3
invm3@(Mat3 a b c) = inverse \$ (trim mat :: Mat3)

--------------------------------------------------------------------------------
-- Vec2 instances

instance HasCoordinates Vec2 Flt where
_1 (Vec2 x _) = x
_2 (Vec2 _ y) = y
_3 _ = error "has only 2 coordinates"
_4 _ = error "has only 2 coordinates"

instance AbelianGroup Vec2 where
(&+) (Vec2 x1 y1) (Vec2 x2 y2) = Vec2 (x1+x2) (y1+y2)
(&-) (Vec2 x1 y1) (Vec2 x2 y2) = Vec2 (x1-x2) (y1-y2)
neg  (Vec2 x y)                = Vec2 (-x) (-y)
zero = Vec2 0 0

instance Vector Vec2 where
scalarMul s (Vec2 x y) = Vec2 (s*x) (s*y)
mapVec    f (Vec2 x y) = Vec2 (f x) (f y)

instance DotProd Vec2 where
(&.) (Vec2 x1 y1) (Vec2 x2 y2) = x1*x2 + y1*y2

instance Pointwise Vec2 where
pointwise (Vec2 x1 y1) (Vec2 x2 y2) = Vec2 (x1*x2) (y1*y2)

instance Determinant (Vec2,Vec2) where
det (Vec2 x1 y1 , Vec2 x2 y2) = x1*y2 - x2*y1

{-
instance Show Vec2 where
show (Vec2 x y) = "( " ++ show x ++ " , " ++ show y ++ " )"
-}

instance Random Vec2 where
random = randomR (Vec2 (-1) (-1),Vec2 1 1)
randomR (Vec2 a b, Vec2 c d) gen =
let (x,gen1) = randomR (a,c) gen
(y,gen2) = randomR (b,d) gen1
in (Vec2 x y, gen2)

instance Storable Vec2 where
sizeOf    _ = 2 * sizeOf (undefined::Flt)
alignment _ = sizeOf (undefined::Flt)

peek q = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
x <- peek        p
y <- peekByteOff p k
return (Vec2 x y)

poke q (Vec2 x y) = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
poke        p   x
pokeByteOff p k y

instance Dimension Vec2 where dim _ = 2

--------------------------------------------------------------------------------
-- Mat2 instances

instance HasCoordinates Mat2 Vec2 where
_1 (Mat2 x _) = x
_2 (Mat2 _ y) = y
_3 _ = error "has only 2 coordinates"
_4 _ = error "has only 2 coordinates"

instance Matrix Mat2 where
transpose (Mat2 row1 row2) =
Mat2 (Vec2 (_1 row1) (_1 row2))
(Vec2 (_2 row1) (_2 row2))
idmtx = Mat2 (Vec2 1 0) (Vec2 0 1)
inverse (Mat2 (Vec2 a b) (Vec2 c d)) =
Mat2 (Vec2 (d*r) (-b*r)) (Vec2 (-c*r) (a*r))
where r = 1.0 / (a*d - b*c)

instance AbelianGroup Mat2 where
(&+) (Mat2 r1 r2) (Mat2 s1 s2) = Mat2 (r1 &+ s1) (r2 &+ s2)
(&-) (Mat2 r1 r2) (Mat2 s1 s2) = Mat2 (r1 &- s1) (r2 &- s2)
neg  (Mat2 r1 r2)              = Mat2 (neg r1) (neg r2)
zero = Mat2 zero zero

instance Vector Mat2 where
scalarMul s (Mat2 r1 r2) = Mat2 (g r1) (g r2) where g = scalarMul s
mapVec    f (Mat2 r1 r2) = Mat2 (g r1) (g r2) where g = mapVec f

instance MultSemiGroup Mat2 where
(.*.) (Mat2 r1 r2) n =
let (Mat2 c1 c2) = transpose n
in Mat2 (Vec2 (r1 &. c1) (r1 &. c2))
(Vec2 (r2 &. c1) (r2 &. c2))
one = idmtx

instance Ring Mat2

instance LeftModule Mat2 Vec2 where
lmul (Mat2 row1 row2) v = Vec2 (row1 &. v) (row2 &. v)

instance RightModule Vec2 Mat2 where
rmul v mt = lmul (transpose mt) v

instance Diagonal Vec2 Mat2 where
diag (Vec2 x y) = Mat2 (Vec2 x 0) (Vec2 0 y)

instance Tensor Mat2 Vec2 where
outer (Vec2 a b) (Vec2 x y) = Mat2
(Vec2 (a*x) (a*y))
(Vec2 (b*x) (b*y))

instance Determinant Mat2 where
det (Mat2 (Vec2 a b) (Vec2 c d)) = a*d - b*c

{-
instance Show Mat2 where
show (Mat2 r1 r2) = show r1 ++ "\n" ++ show r2
-}

instance Storable Mat2 where
sizeOf    _ = 2 * sizeOf (undefined::Vec2)
alignment _ = alignment  (undefined::Vec2)

peek q = do
let p = castPtr q :: Ptr Vec2
k = sizeOf (undefined::Vec2)
r1 <- peek        p
r2 <- peekByteOff p k
return (Mat2 r1 r2)

poke q (Mat2 r1 r2) = do
let p = castPtr q :: Ptr Vec2
k = sizeOf (undefined::Vec2)
poke        p   r1
pokeByteOff p k r2

instance Random Mat2 where
random = randomR (Mat2 v1 v1 , Mat2 v2 v2) where
v1 = Vec2 (-1) (-1)
v2 = Vec2   1    1
randomR (Mat2 a b, Mat2 c d) gen =
let (x,gen1) = randomR (a,c) gen
(y,gen2) = randomR (b,d) gen1
in (Mat2 x y, gen2)

instance Dimension Mat2 where dim _ = 2

instance MatrixNorms Mat2 where
frobeniusNorm (Mat2 r1 r2) =
sqrt \$
normsqr r1 +
normsqr r2

instance Pointwise Mat2 where
pointwise (Mat2 x1 y1) (Mat2 x2 y2) = Mat2 (x1 &! x2) (y1 &! y2)

--------------------------------------------------------------------------------
-- Vec3 instances

instance HasCoordinates Vec3 Flt where
_1 (Vec3 x _ _) = x
_2 (Vec3 _ y _) = y
_3 (Vec3 _ _ z) = z
_4 _ = error "has only 3 coordinates"

instance AbelianGroup Vec3 where
(&+) (Vec3 x1 y1 z1) (Vec3 x2 y2 z2) = Vec3 (x1+x2) (y1+y2) (z1+z2)
(&-) (Vec3 x1 y1 z1) (Vec3 x2 y2 z2) = Vec3 (x1-x2) (y1-y2) (z1-z2)
neg  (Vec3 x y z)                    = Vec3 (-x) (-y) (-z)
zero = Vec3 0 0 0

instance Vector Vec3 where
scalarMul s (Vec3 x y z) = Vec3 (s*x) (s*y) (s*z)
mapVec    f (Vec3 x y z) = Vec3 (f x) (f y) (f z)

instance DotProd Vec3 where
(&.) (Vec3 x1 y1 z1) (Vec3 x2 y2 z2) = x1*x2 + y1*y2 + z1*z2

instance Pointwise Vec3 where
pointwise (Vec3 x1 y1 z1) (Vec3 x2 y2 z2) = Vec3 (x1*x2) (y1*y2) (z1*z2)

{-
instance Show Vec3 where
show (Vec3 x y z) = "( " ++ show x ++ " , " ++ show y ++ " , " ++ show z ++ " )"
-}

instance Random Vec3 where
random = randomR (Vec3 (-1) (-1) (-1),Vec3 1 1 1)
randomR (Vec3 a b c, Vec3 d e f) gen =
let (x,gen1) = randomR (a,d) gen
(y,gen2) = randomR (b,e) gen1
(z,gen3) = randomR (c,f) gen2
in (Vec3 x y z, gen3)

instance CrossProd Vec3 where
crossprod (Vec3 x1 y1 z1) (Vec3 x2 y2 z2) = Vec3 (y1*z2-y2*z1) (z1*x2-z2*x1) (x1*y2-x2*y1)

instance Determinant (Vec3,Vec3,Vec3) where
det (u,v,w) = u &. (v &^ w)

instance Storable Vec3 where
sizeOf    _ = 3 * sizeOf (undefined::Flt)
alignment _ = sizeOf (undefined::Flt)

peek q = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
x <- peek        p
y <- peekByteOff p (k  )
z <- peekByteOff p (k+k)
return (Vec3 x y z)

poke q (Vec3 x y z) = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
poke        p       x
pokeByteOff p (k  ) y
pokeByteOff p (k+k) z

instance Dimension Vec3 where dim _ = 3

--------------------------------------------------------------------------------
-- Mat3 instances

instance HasCoordinates Mat3 Vec3 where
_1 (Mat3 x _ _) = x
_2 (Mat3 _ y _) = y
_3 (Mat3 _ _ z) = z
_4 _ = error "has only 3 coordinates"

instance Matrix Mat3 where

transpose (Mat3 row1 row2 row3) =
Mat3 (Vec3 (_1 row1) (_1 row2) (_1 row3))
(Vec3 (_2 row1) (_2 row2) (_2 row3))
(Vec3 (_3 row1) (_3 row2) (_3 row3))

idmtx = Mat3 (Vec3 1 0 0) (Vec3 0 1 0) (Vec3 0 0 1)

inverse (Mat3 (Vec3 a b c) (Vec3 e f g) (Vec3 i j k)) =
Mat3 (Vec3 (d11*r) (d21*r) (d31*r))
(Vec3 (d12*r) (d22*r) (d32*r))
(Vec3 (d13*r) (d23*r) (d33*r))
where
r = 1.0 / ( a*d11 + b*d12 + c*d13 )

d11 = f*k - g*j
d12 = g*i - e*k
d13 = e*j - f*i

d31 = b*g - c*f
d32 = c*e - a*g
d33 = a*f - b*e

d21 = c*j - b*k
d22 = a*k - c*i
d23 = b*i - a*j

instance AbelianGroup Mat3 where
(&+) (Mat3 r1 r2 r3) (Mat3 s1 s2 s3) = Mat3 (r1 &+ s1) (r2 &+ s2) (r3 &+ s3)
(&-) (Mat3 r1 r2 r3) (Mat3 s1 s2 s3) = Mat3 (r1 &- s1) (r2 &- s2) (r3 &- s3)
neg  (Mat3 r1 r2 r3)                 = Mat3 (neg r1) (neg r2) (neg r3)
zero = Mat3 zero zero zero

instance Vector Mat3 where
scalarMul s (Mat3 r1 r2 r3) = Mat3 (g r1) (g r2) (g r3) where g = scalarMul s
mapVec    f (Mat3 r1 r2 r3) = Mat3 (g r1) (g r2) (g r3) where g = mapVec f

instance MultSemiGroup Mat3 where
(.*.) (Mat3 r1 r2 r3) n =
let (Mat3 c1 c2 c3) = transpose n
in Mat3 (Vec3 (r1 &. c1) (r1 &. c2) (r1 &. c3))
(Vec3 (r2 &. c1) (r2 &. c2) (r2 &. c3))
(Vec3 (r3 &. c1) (r3 &. c2) (r3 &. c3))
one = idmtx

instance Ring Mat3

instance LeftModule Mat3 Vec3 where
lmul (Mat3 row1 row2 row3) v = Vec3 (row1 &. v) (row2 &. v) (row3 &. v)

instance RightModule Vec3 Mat3 where
rmul v mt = lmul (transpose mt) v

instance Diagonal Vec3 Mat3 where
diag (Vec3 x y z) = Mat3 (Vec3 x 0 0) (Vec3 0 y 0) (Vec3 0 0 z)

instance Tensor Mat3 Vec3 where
outer (Vec3 a b c) (Vec3 x y z) = Mat3
(Vec3 (a*x) (a*y) (a*z))
(Vec3 (b*x) (b*y) (b*z))
(Vec3 (c*x) (c*y) (c*z))

instance Determinant Mat3 where
det (Mat3 r1 r2 r3) = det (r1,r2,r3)

{-
instance Show Mat3 where
show (Mat3 r1 r2 r3) = show r1 ++ "\n" ++ show r2 ++ "\n" ++ show r3
-}

instance Storable Mat3 where
sizeOf    _ = 3 * sizeOf (undefined::Vec3)
alignment _ = alignment  (undefined::Vec3)

peek q = do
let p = castPtr q :: Ptr Vec3
k = sizeOf (undefined::Vec3)
r1 <- peek        p
r2 <- peekByteOff p (k  )
r3 <- peekByteOff p (k+k)
return (Mat3 r1 r2 r3)

poke q (Mat3 r1 r2 r3) = do
let p = castPtr q :: Ptr Vec3
k = sizeOf (undefined::Vec3)
poke        p       r1
pokeByteOff p (k  ) r2
pokeByteOff p (k+k) r3

instance Random Mat3 where
random = randomR (Mat3 v1 v1 v1 , Mat3 v2 v2 v2) where
v1 = Vec3 (-1) (-1) (-1)
v2 = Vec3   1    1    1
randomR (Mat3 a b c, Mat3 d e f) gen =
let (x,gen1) = randomR (a,d) gen
(y,gen2) = randomR (b,e) gen1
(z,gen3) = randomR (c,f) gen2
in (Mat3 x y z, gen3)

instance Dimension Mat3 where dim _ = 3

instance MatrixNorms Mat3 where
frobeniusNorm (Mat3 r1 r2 r3)  =
sqrt \$
normsqr r1 +
normsqr r2 +
normsqr r3

instance Pointwise Mat3 where
pointwise (Mat3 x1 y1 z1) (Mat3 x2 y2 z2) = Mat3 (x1 &! x2) (y1 &! y2) (z1 &! z2)

--------------------------------------------------------------------------------
-- Vec4 instances

instance HasCoordinates Vec4 Flt where
_1 (Vec4 x _ _ _) = x
_2 (Vec4 _ y _ _) = y
_3 (Vec4 _ _ z _) = z
_4 (Vec4 _ _ _ w) = w

instance AbelianGroup Vec4 where
(&+) (Vec4 x1 y1 z1 w1) (Vec4 x2 y2 z2 w2) = Vec4 (x1+x2) (y1+y2) (z1+z2) (w1+w2)
(&-) (Vec4 x1 y1 z1 w1) (Vec4 x2 y2 z2 w2) = Vec4 (x1-x2) (y1-y2) (z1-z2) (w1-w2)
neg  (Vec4 x y z w)                        = Vec4 (-x) (-y) (-z) (-w)
zero = Vec4 0 0 0 0

instance Vector Vec4 where
scalarMul s (Vec4 x y z w) = Vec4 (s*x) (s*y) (s*z) (s*w)
mapVec    f (Vec4 x y z w) = Vec4 (f x) (f y) (f z) (f w)

instance DotProd Vec4 where
(&.) (Vec4 x1 y1 z1 w1) (Vec4 x2 y2 z2 w2) = x1*x2 + y1*y2 + z1*z2 + w1*w2

instance Pointwise Vec4 where
pointwise (Vec4 x1 y1 z1 w1) (Vec4 x2 y2 z2 w2) = Vec4 (x1*x2) (y1*y2) (z1*z2) (w1*w2)

{-
instance Show Vec4 where
show (Vec4 x y z w) = "( " ++ show x ++ " , " ++ show y ++ " , " ++ show z ++ " , " ++ show w ++ " )"
-}

instance Random Vec4 where
random = randomR (Vec4 (-1) (-1) (-1) (-1),Vec4 1 1 1 1)
randomR (Vec4 a b c d, Vec4 e f g h) gen =
let (x,gen1) = randomR (a,e) gen
(y,gen2) = randomR (b,f) gen1
(z,gen3) = randomR (c,g) gen2
(w,gen4) = randomR (d,h) gen3
in (Vec4 x y z w, gen4)

instance Storable Vec4 where
sizeOf    _ = 4 * sizeOf (undefined::Flt)
alignment _ = sizeOf (undefined::Flt)

peek q = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
x <- peek        p
y <- peekByteOff p (k  )
z <- peekByteOff p (k+k)
w <- peekByteOff p (3*k)
return (Vec4 x y z w)

poke q (Vec4 x y z w) = do
let p = castPtr q :: Ptr Flt
k = sizeOf (undefined::Flt)
poke        p       x
pokeByteOff p (k  ) y
pokeByteOff p (k+k) z
pokeByteOff p (3*k) w

instance Dimension Vec4 where dim _ = 4

--------------------------------------------------------------------------------
-- Mat4 instances

instance HasCoordinates Mat4 Vec4 where
_1 (Mat4 x _ _ _) = x
_2 (Mat4 _ y _ _) = y
_3 (Mat4 _ _ z _) = z
_4 (Mat4 _ _ _ w) = w

instance Matrix Mat4 where
transpose (Mat4 row1 row2 row3 row4) =
Mat4 (Vec4 (_1 row1) (_1 row2) (_1 row3) (_1 row4))
(Vec4 (_2 row1) (_2 row2) (_2 row3) (_2 row4))
(Vec4 (_3 row1) (_3 row2) (_3 row3) (_3 row4))
(Vec4 (_4 row1) (_4 row2) (_4 row3) (_4 row4))
idmtx = Mat4 (Vec4 1 0 0 0) (Vec4 0 1 0 0) (Vec4 0 0 1 0) (Vec4 0 0 0 1)
inverse = error "inverse/Mat4: not implemented yet"

instance AbelianGroup Mat4 where
(&+) (Mat4 r1 r2 r3 r4) (Mat4 s1 s2 s3 s4) = Mat4 (r1 &+ s1) (r2 &+ s2) (r3 &+ s3) (r4 &+ s4)
(&-) (Mat4 r1 r2 r3 r4) (Mat4 s1 s2 s3 s4) = Mat4 (r1 &- s1) (r2 &- s2) (r3 &- s3) (r4 &- s4)
neg  (Mat4 r1 r2 r3 r4)                    = Mat4 (neg r1) (neg r2) (neg r3) (neg r4)
zero = Mat4 zero zero zero zero

instance Vector Mat4 where
scalarMul s (Mat4 r1 r2 r3 r4) = Mat4 (g r1) (g r2) (g r3) (g r4) where g = scalarMul s
mapVec    f (Mat4 r1 r2 r3 r4) = Mat4 (g r1) (g r2) (g r3) (g r4) where g = mapVec f

instance MultSemiGroup Mat4 where
(.*.) (Mat4 r1 r2 r3 r4) n =
let (Mat4 c1 c2 c3 c4) = transpose n
in Mat4 (Vec4 (r1 &. c1) (r1 &. c2) (r1 &. c3) (r1 &. c4))
(Vec4 (r2 &. c1) (r2 &. c2) (r2 &. c3) (r2 &. c4))
(Vec4 (r3 &. c1) (r3 &. c2) (r3 &. c3) (r3 &. c4))
(Vec4 (r4 &. c1) (r4 &. c2) (r4 &. c3) (r4 &. c4))
one = idmtx

instance Ring Mat4

instance LeftModule Mat4 Vec4 where
lmul (Mat4 row1 row2 row3 row4) v = Vec4 (row1 &. v) (row2 &. v) (row3 &. v) (row4 &. v)

instance RightModule Vec4 Mat4 where
rmul v mt = lmul (transpose mt) v

instance Diagonal Vec4 Mat4 where
diag (Vec4 x y z w) = Mat4 (Vec4 x 0 0 0) (Vec4 0 y 0 0) (Vec4 0 0 z 0) (Vec4 0 0 0 w)

instance Tensor Mat4 Vec4 where
outer (Vec4 a b c d) (Vec4 x y z w) = Mat4
(Vec4 (a*x) (a*y) (a*z) (a*w))
(Vec4 (b*x) (b*y) (b*z) (b*w))
(Vec4 (c*x) (c*y) (c*z) (c*w))
(Vec4 (d*x) (d*y) (d*z) (d*w))

instance Determinant Mat4 where
det = error "det/Mat4: not implemented yet"
-- det (Mat4 r1 r2 r3 r4) =

{-
instance Show Mat4 where
show (Mat4 r1 r2 r3 r4) = show r1 ++ "\n" ++ show r2 ++ "\n" ++ show r3 ++ "\n" ++ show r4
-}

instance Storable Mat4 where
sizeOf    _ = 4 * sizeOf (undefined::Vec4)
alignment _ = alignment  (undefined::Vec4)

peek q = do
let p = castPtr q :: Ptr Vec4
k = sizeOf (undefined::Vec4)
r1 <- peek        p
r2 <- peekByteOff p (k  )
r3 <- peekByteOff p (k+k)
r4 <- peekByteOff p (3*k)
return (Mat4 r1 r2 r3 r4)

poke q (Mat4 r1 r2 r3 r4) = do
let p = castPtr q :: Ptr Vec4
k = sizeOf (undefined::Vec4)
poke        p       r1
pokeByteOff p (k  ) r2
pokeByteOff p (k+k) r3
pokeByteOff p (3*k) r4

instance Random Mat4 where
random = randomR (Mat4 v1 v1 v1 v1, Mat4 v2 v2 v2 v2) where
v1 = Vec4 (-1) (-1) (-1) (-1)
v2 = Vec4   1    1    1    1
randomR (Mat4 a b c d, Mat4 e f g h) gen =
let (x,gen1) = randomR (a,e) gen
(y,gen2) = randomR (b,f) gen1
(z,gen3) = randomR (c,g) gen2
(w,gen4) = randomR (d,h) gen3
in (Mat4 x y z w, gen4)

instance Dimension Mat4 where dim _ = 4

instance MatrixNorms Mat4 where
frobeniusNorm (Mat4 r1 r2 r3 r4) =
sqrt \$
normsqr r1 +
normsqr r2 +
normsqr r3 +
normsqr r4

instance Pointwise Mat4 where
pointwise (Mat4 x1 y1 z1 w1) (Mat4 x2 y2 z2 w2) = Mat4 (x1 &! x2) (y1 &! y2) (z1 &! z2) (w1 &! w2)

--------------------------------------------------------------------------------
-- Extend instances

instance Extend Vec2 Vec3 where
extendZero   (Vec2 x y) = Vec3 x y 0
extendWith t (Vec2 x y) = Vec3 x y t
trim (Vec3 x y _)       = Vec2 x y

instance Extend Vec2 Vec4 where
extendZero   (Vec2 x y) = Vec4 x y 0 0
extendWith t (Vec2 x y) = Vec4 x y t t
trim (Vec4 x y _ _)     = Vec2 x y

instance Extend Vec3 Vec4 where
extendZero   (Vec3 x y z) = Vec4 x y z 0
extendWith t (Vec3 x y z) = Vec4 x y z t
trim (Vec4 x y z _)       = Vec3 x y z

instance Extend Mat2 Mat3 where
extendZero   (Mat2 p q) = Mat3 (extendZero p) (extendZero q) zero
extendWith w (Mat2 p q) = Mat3 (extendZero p) (extendZero q) (Vec3 0 0 w)
trim (Mat3 p q _) = Mat2 (trim p) (trim q)

instance Extend Mat2 Mat4 where
extendZero   (Mat2 p q) = Mat4 (extendZero p) (extendZero q) zero zero
extendWith w (Mat2 p q) = Mat4 (extendZero p) (extendZero q) (Vec4 0 0 w 0) (Vec4 0 0 0 w)
trim (Mat4 p q _ _) = Mat2 (trim p) (trim q)

instance Extend Mat3 Mat4 where
extendZero   (Mat3 p q r) = Mat4 (extendZero p) (extendZero q) (extendZero r) zero
extendWith w (Mat3 p q r) = Mat4 (extendZero p) (extendZero q) (extendZero r) (Vec4 0 0 0 w)
trim (Mat4 p q r _) = Mat3 (trim p) (trim q) (trim r)

--------------------------------------------------------------------------------

```