-- Hoogle documentation, generated by Haddock
-- See Hoogle, http://www.haskell.org/hoogle/
-- | Linear Algebra
--
-- Types and combinators for linear algebra on free vector spaces
@package linear
@version 0.7
-- | Orphans
module Linear.Instances
instance Traversable1 Complex
instance Foldable1 Complex
instance Traversable Complex
instance Foldable Complex
instance Monad Complex
instance Bind Complex
instance Applicative Complex
instance Apply Complex
instance Functor Complex
instance (Hashable k, Eq k) => Bind (HashMap k)
instance (Hashable k, Eq k) => Apply (HashMap k)
-- | Operations on free vector spaces.
module Linear.Vector
-- | A vector is an additive group with additional structure.
class Bind f => Additive f where zero = pure 0 ^+^ = liftA2 (+) ^-^ = liftA2 (-) lerp alpha u v = alpha *^ u ^+^ (1 - alpha) *^ v
zero :: (Additive f, Num a) => f a
(^+^) :: (Additive f, Num a) => f a -> f a -> f a
(^-^) :: (Additive f, Num a) => f a -> f a -> f a
lerp :: (Additive f, Num a) => a -> f a -> f a -> f a
-- | Compute the negation of a vector
--
--
-- >>> negated (V2 2 4)
-- V2 (-2) (-4)
--
negated :: (Functor f, Num a) => f a -> f a
-- | Compute the right scalar product
--
--
-- >>> V2 3 4 ^* 2
-- V2 6 8
--
(^*) :: (Functor f, Num a) => f a -> a -> f a
-- | Compute the left scalar product
--
--
-- >>> 2 *^ V2 3 4
-- V2 6 8
--
(*^) :: (Functor f, Num a) => a -> f a -> f a
-- | Compute division by a scalar on the right.
(^/) :: (Functor f, Fractional a) => f a -> a -> f a
-- | Produce a default basis for a vector space. If the dimensionality of
-- the vector space is not statically known, see basisFor.
basis :: (Applicative t, Traversable t, Num a) => [t a]
-- | Produce a default basis for a vector space from which the argument is
-- drawn.
basisFor :: (Traversable t, Enum a, Num a) => t a -> [t a]
instance Additive Complex
instance Additive ((->) b)
instance (Eq k, Hashable k) => Additive (HashMap k)
instance Ord k => Additive (Map k)
instance Additive IntMap
-- | Testing for values near zero
module Linear.Epsilon
-- | Provides a fairly subjective test to see if a quantity is near zero.
--
--
-- >>> nearZero (1e-11 :: Double)
-- False
--
--
--
-- >>> nearZero (1e-17 :: Double)
-- True
--
--
--
-- >>> nearZero (1e-5 :: Float)
-- False
--
--
--
-- >>> nearZero (1e-7 :: Float)
-- True
--
class Num a => Epsilon a
nearZero :: Epsilon a => a -> Bool
instance Epsilon Double
instance Epsilon Float
-- | Free metric spaces
module Linear.Metric
-- | A free inner product/metric space
class Applicative f => Metric f where quadrance v = dot v v qd f g = quadrance (liftA2 (-) f g) distance f g = norm (liftA2 (-) f g) norm v = sqrt (dot v v) signorm v = fmap (/ m) v where m = norm v
dot :: (Metric f, Num a) => f a -> f a -> a
quadrance :: (Metric f, Num a) => f a -> a
qd :: (Metric f, Num a) => f a -> f a -> a
distance :: (Metric f, Floating a) => f a -> f a -> a
norm :: (Metric f, Floating a) => f a -> a
signorm :: (Metric f, Floating a) => f a -> f a
-- | Normalize a Metric functor to have unit norm. This
-- function does not change the functor if its norm is 0 or 1.
normalize :: (Floating a, Metric f, Epsilon a) => f a -> f a
-- | Corepresentable functors as vector spaces
module Linear.Core
-- | A Functor f is corepresentable if it is isomorphic to
-- (x -> a) for some x. Nearly all such functors can be
-- represented by choosing x to be the set of lenses that are
-- polymorphic in the contents of the Functor, that is to say
-- x = Rep f is a valid choice of x for
-- (nearly) every Representable Functor.
class Functor f => Core f
core :: Core f => ((forall g x. Functor g => (x -> g x) -> f x -> g (f x)) -> a) -> f a
-- | 2-D Vectors
module Linear.V2
-- | A 2-dimensional vector
--
--
-- >>> pure 1 :: V2 Int
-- V2 1 1
--
--
--
-- >>> V2 1 2 + V2 3 4
-- V2 4 6
--
--
--
-- >>> V2 1 2 * V2 3 4
-- V2 3 8
--
--
--
-- >>> sum (V2 1 2)
-- 3
--
data V2 a
V2 :: a -> a -> V2 a
-- | A space that distinguishes 2 orthogonal basis vectors _x and
-- _y, but may have more.
class R2 t where _x = _xy . _x _y = _xy . _y
_x :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_y :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xy :: (R2 t, Functor f) => (V2 a -> f (V2 a)) -> t a -> f (t a)
-- | the counter-clockwise perpendicular vector
--
--
-- >>> perp $ V2 10 20
-- V2 (-20) 10
--
perp :: Num a => V2 a -> V2 a
instance Typeable1 V2
instance Eq a => Eq (V2 a)
instance Ord a => Ord (V2 a)
instance Show a => Show (V2 a)
instance Read a => Read (V2 a)
instance Data a => Data (V2 a)
instance Ix a => Ix (V2 a)
instance Storable a => Storable (V2 a)
instance Epsilon a => Epsilon (V2 a)
instance Distributive V2
instance Core V2
instance R2 V2
instance Metric V2
instance Fractional a => Fractional (V2 a)
instance Num a => Num (V2 a)
instance Monad V2
instance Bind V2
instance Additive V2
instance Applicative V2
instance Apply V2
instance Traversable1 V2
instance Foldable1 V2
instance Traversable V2
instance Foldable V2
instance Functor V2
-- | 3-D Vectors
module Linear.V3
-- | A 3-dimensional vector
data V3 a
V3 :: a -> a -> a -> V3 a
-- | cross product
cross :: Num a => V3 a -> V3 a -> V3 a
-- | scalar triple product
triple :: Num a => V3 a -> V3 a -> V3 a -> a
-- | A space that distinguishes 2 orthogonal basis vectors _x and
-- _y, but may have more.
class R2 t where _x = _xy . _x _y = _xy . _y
_x :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_y :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xy :: (R2 t, Functor f) => (V2 a -> f (V2 a)) -> t a -> f (t a)
-- | A space that distinguishes 3 orthogonal basis vectors: _x,
-- _y, and _z. (It may have more)
class R2 t => R3 t
_z :: (R3 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xyz :: (R3 t, Functor f) => (V3 a -> f (V3 a)) -> t a -> f (t a)
instance Typeable1 V3
instance Eq a => Eq (V3 a)
instance Ord a => Ord (V3 a)
instance Show a => Show (V3 a)
instance Read a => Read (V3 a)
instance Data a => Data (V3 a)
instance Ix a => Ix (V3 a)
instance Epsilon a => Epsilon (V3 a)
instance Storable a => Storable (V3 a)
instance Core V3
instance R3 V3
instance R2 V3
instance Distributive V3
instance Metric V3
instance Fractional a => Fractional (V3 a)
instance Num a => Num (V3 a)
instance Monad V3
instance Bind V3
instance Additive V3
instance Applicative V3
instance Apply V3
instance Traversable1 V3
instance Foldable1 V3
instance Traversable V3
instance Foldable V3
instance Functor V3
-- | 4-D Vectors
module Linear.V4
-- | A 4-dimensional vector.
data V4 a
V4 :: a -> a -> a -> a -> V4 a
-- | Convert a 3-dimensional affine vector into a 4-dimensional homogeneous
-- vector.
vector :: Num a => V3 a -> V4 a
-- | Convert a 3-dimensional affine point into a 4-dimensional homogeneous
-- vector.
point :: Num a => V3 a -> V4 a
-- | A space that distinguishes 2 orthogonal basis vectors _x and
-- _y, but may have more.
class R2 t where _x = _xy . _x _y = _xy . _y
_x :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_y :: (R2 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xy :: (R2 t, Functor f) => (V2 a -> f (V2 a)) -> t a -> f (t a)
-- | A space that distinguishes 3 orthogonal basis vectors: _x,
-- _y, and _z. (It may have more)
class R2 t => R3 t
_z :: (R3 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xyz :: (R3 t, Functor f) => (V3 a -> f (V3 a)) -> t a -> f (t a)
-- | A space that distinguishes orthogonal basis vectors _x,
-- _y, _z, _w. (It may have more.)
class R3 t => R4 t
_w :: (R4 t, Functor f) => (a -> f a) -> t a -> f (t a)
_xyzw :: (R4 t, Functor f) => (V4 a -> f (V4 a)) -> t a -> f (t a)
instance Typeable1 V4
instance Eq a => Eq (V4 a)
instance Ord a => Ord (V4 a)
instance Show a => Show (V4 a)
instance Read a => Read (V4 a)
instance Data a => Data (V4 a)
instance Ix a => Ix (V4 a)
instance Epsilon a => Epsilon (V4 a)
instance Storable a => Storable (V4 a)
instance Core V4
instance R4 V4
instance R3 V4
instance R2 V4
instance Distributive V4
instance Metric V4
instance Fractional a => Fractional (V4 a)
instance Num a => Num (V4 a)
instance Monad V4
instance Bind V4
instance Additive V4
instance Apply V4
instance Applicative V4
instance Traversable1 V4
instance Foldable1 V4
instance Traversable V4
instance Foldable V4
instance Functor V4
-- | Plücker coordinates for lines in 3d homogeneous space.
module Linear.Plucker
-- | Plücker coordinates for lines in a 3-dimensional space.
data Plucker a
Plucker :: a -> a -> a -> a -> a -> a -> Plucker a
-- | Valid Plücker coordinates p will have squaredError
-- p == 0
--
-- That said, floating point makes a mockery of this claim, so you may
-- want to use nearZero.
squaredError :: (Eq a, Num a) => Plucker a -> a
-- | Checks if the line is near-isotropic (isotropic vectors in this
-- quadratic space represent lines in real 3d space)
isotropic :: Epsilon a => Plucker a -> Bool
-- | This isn't th actual metric because this bilinear form gives rise to
-- an isotropic quadratic space
(><) :: Num a => Plucker a -> Plucker a -> a
-- | Given a pair of points represented by homogeneous coordinates generate
-- Plücker coordinates for the line through them.
plucker :: Num a => V4 a -> V4 a -> Plucker a
-- | Checks if the two vectors intersect (or nearly intersect)
intersects :: Epsilon a => Plucker a -> Plucker a -> Bool
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p01 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p02 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p03 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p10 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p12 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p13 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p20 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p21 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p23 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p30 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form a basis for the Plücker space, or the Grassmanian
-- manifold Gr(2,V4).
--
--
-- p01 :: Lens' (Plucker a) a
-- p02 :: Lens' (Plucker a) a
-- p03 :: Lens' (Plucker a) a
-- p23 :: Lens' (Plucker a) a
-- p31 :: Lens' (Plucker a) a
-- p12 :: Lens' (Plucker a) a
--
p31 :: Functor f => (a -> f a) -> Plucker a -> f (Plucker a)
-- | These elements form an alternate basis for the Plücker space, or the
-- Grassmanian manifold Gr(2,V4).
--
--
-- p10 :: Num a => Lens' (Plucker a) a
-- p20 :: Num a => Lens' (Plucker a) a
-- p30 :: Num a => Lens' (Plucker a) a
-- p32 :: Num a => Lens' (Plucker a) a
-- p13 :: Num a => Lens' (Plucker a) a
-- p21 :: Num a => Lens' (Plucker a) a
--
p32 :: (Functor f, Num a) => (a -> f a) -> Plucker a -> f (Plucker a)
instance Eq a => Eq (Plucker a)
instance Ord a => Ord (Plucker a)
instance Show a => Show (Plucker a)
instance Read a => Read (Plucker a)
instance Epsilon a => Epsilon (Plucker a)
instance Metric Plucker
instance Storable a => Storable (Plucker a)
instance Fractional a => Fractional (Plucker a)
instance Num a => Num (Plucker a)
instance Ix a => Ix (Plucker a)
instance Traversable1 Plucker
instance Foldable1 Plucker
instance Traversable Plucker
instance Foldable Plucker
instance Core Plucker
instance Distributive Plucker
instance Monad Plucker
instance Bind Plucker
instance Additive Plucker
instance Applicative Plucker
instance Apply Plucker
instance Functor Plucker
-- | Involutive rings
module Linear.Conjugate
-- | An involutive ring
class Num a => Conjugate a where conjugate = id
conjugate :: Conjugate a => a -> a
instance (Conjugate a, RealFloat a) => Conjugate (Complex a)
instance Conjugate Float
instance Conjugate Double
instance Conjugate Word8
instance Conjugate Word16
instance Conjugate Word32
instance Conjugate Word64
instance Conjugate Word
instance Conjugate Int8
instance Conjugate Int16
instance Conjugate Int32
instance Conjugate Int64
instance Conjugate Int
instance Conjugate Integer
-- | Quaternions
module Linear.Quaternion
-- | Quaternions
data Quaternion a
Quaternion :: a -> {-# UNPACK #-} !(V3 a) -> Quaternion a
-- | A vector space that includes the basis elements _e and
-- _i
class Complicated t
_e :: (Complicated t, Functor f) => (a -> f a) -> t a -> f (t a)
_i :: (Complicated t, Functor f) => (a -> f a) -> t a -> f (t a)
-- | A vector space that includes the basis elements _e, _i,
-- _j and _k
class Complicated t => Hamiltonian t
_j :: (Hamiltonian t, Functor f) => (a -> f a) -> t a -> f (t a)
_k :: (Hamiltonian t, Functor f) => (a -> f a) -> t a -> f (t a)
_ijk :: (Hamiltonian t, Functor f) => (V3 a -> f (V3 a)) -> t a -> f (t a)
-- | Spherical linear interpolation between two quaternions.
slerp :: RealFloat a => Quaternion a -> Quaternion a -> a -> Quaternion a
-- | asin with a specified branch cut.
asinq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | acos with a specified branch cut.
acosq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | atan with a specified branch cut.
atanq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | asinh with a specified branch cut.
asinhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | acosh with a specified branch cut.
acoshq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | atanh with a specified branch cut.
atanhq :: RealFloat a => Quaternion a -> Quaternion a -> Quaternion a
-- | norm of the imaginary component
absi :: Floating a => Quaternion a -> a
-- | raise a Quaternion to a scalar power
pow :: RealFloat a => Quaternion a -> a -> Quaternion a
-- | Apply a rotation to a vector.
rotate :: (Conjugate a, RealFloat a) => Quaternion a -> V3 a -> V3 a
-- | axisAngle axis theta builds a Quaternion
-- representing a rotation of theta radians about axis.
axisAngle :: (Epsilon a, Floating a) => V3 a -> a -> Quaternion a
instance Typeable1 Quaternion
instance Eq a => Eq (Quaternion a)
instance Ord a => Ord (Quaternion a)
instance Read a => Read (Quaternion a)
instance Show a => Show (Quaternion a)
instance Data a => Data (Quaternion a)
instance (RealFloat a, Epsilon a) => Epsilon (Quaternion a)
instance RealFloat a => Floating (Quaternion a)
instance (Conjugate a, RealFloat a) => Conjugate (Quaternion a)
instance Distributive Quaternion
instance Hamiltonian Quaternion
instance Complicated Quaternion
instance Complicated Complex
instance Metric Quaternion
instance RealFloat a => Fractional (Quaternion a)
instance RealFloat a => Num (Quaternion a)
instance Storable a => Storable (Quaternion a)
instance Traversable Quaternion
instance Foldable Quaternion
instance Core Quaternion
instance Ix a => Ix (Quaternion a)
instance Monad Quaternion
instance Bind Quaternion
instance Additive Quaternion
instance Applicative Quaternion
instance Apply Quaternion
instance Functor Quaternion
-- | Simple matrix operation for low-dimensional primitives.
module Linear.Matrix
-- | Matrix product. This can compute mixed dense-dense, sparse-dense and
-- sparse-sparse matrix products.
--
--
-- >>> V2 (V3 1 2 3) (V3 4 5 6) !*! V3 (V2 1 2) (V2 3 4) (V2 4 5)
-- V2 (V2 19 25) (V2 43 58)
--
--
--
-- >>> V2 (fromList [(1,2)]) (fromList [(2,3)]) !*! fromList [(1,V3 0 0 1), (2, V3 0 0 5)]
-- V2 (V3 0 0 2) (V3 0 0 15)
--
(!*!) :: (Functor m, Foldable r, Apply r, Distributive n, Num a) => m (r a) -> r (n a) -> m (n a)
-- | Matrix * column vector
--
--
-- >>> V2 (V3 1 2 3) (V3 4 5 6) !* V3 7 8 9
-- V2 50 122
--
(!*) :: (Functor m, Metric r, Num a) => m (r a) -> r a -> m a
-- | Row vector * matrix
--
--
-- >>> V2 1 2 *! V2 (V3 3 4 5) (V3 6 7 8)
-- V3 15 18 21
--
(*!) :: (Metric r, Distributive n, Num a) => r a -> r (n a) -> n a
-- | Matrix-scalar product
--
--
-- >>> V2 (V2 1 2) (V2 3 4) !!* 5
-- V2 (V2 5 10) (V2 15 20)
--
(!!*) :: (Functor m, Functor r, Num a) => m (r a) -> a -> m (r a)
-- | Scalar-matrix product
--
--
-- >>> 5 *!! V2 (V2 1 2) (V2 3 4)
-- V2 (V2 5 10) (V2 15 20)
--
(*!!) :: (Functor m, Functor r, Num a) => a -> m (r a) -> m (r a)
-- | Hermitian conjugate or conjugate transpose
--
--
-- >>> adjoint (V2 (V2 (1 :+ 2) (3 :+ 4)) (V2 (5 :+ 6) (7 :+ 8)))
-- V2 (V2 (1.0 :+ (-2.0)) (5.0 :+ (-6.0))) (V2 (3.0 :+ (-4.0)) (7.0 :+ (-8.0)))
--
adjoint :: (Functor m, Distributive n, Conjugate a) => m (n a) -> n (m a)
-- | A 2x2 matrix with row-major representation
type M22 a = V2 (V2 a)
-- | A 3x3 matrix with row-major representation
type M33 a = V3 (V3 a)
-- | A 4x4 matrix with row-major representation
type M44 a = V4 (V4 a)
-- | A 4x3 matrix with row-major representation
type M43 a = V4 (V3 a)
-- | Convert a 3x3 matrix to a 4x4 matrix extending it with 0's in the new
-- row and column.
m33_to_m44 :: Num a => M33 a -> M44 a
-- | Convert from a 4x3 matrix to a 4x4 matrix, extending it with the [
-- 0 0 0 1 ] column vector
m43_to_m44 :: Num a => M43 a -> M44 a
-- | 2x2 matrix determinant.
--
--
-- >>> det22 (V2 (V2 a b) (V2 c d))
-- a * d - b * c
--
det22 :: Num a => M22 a -> a
-- | 3x3 matrix determinant.
--
--
-- >>> det33 (V3 (V3 a b c) (V3 d e f) (V3 g h i))
-- a * (e * i - f * h) - d * (b * i - c * h) + g * (b * f - c * e)
--
det33 :: Num a => M33 a -> a
-- | 2x2 matrix inverse.
--
--
-- >>> inv22 $ V2 (V2 1 2) (V2 3 4)
-- Just (V2 (V2 (-2.0) 1.0) (V2 1.5 (-0.5)))
--
inv22 :: (Epsilon a, Floating a) => M22 a -> Maybe (M22 a)
-- | 3x3 matrix inverse.
--
--
-- >>> inv33 $ V3 (V3 1 2 4) (V3 4 2 2) (V3 1 1 1)
-- Just (V3 (V3 0.0 0.5 (-1.0)) (V3 (-0.5) (-0.75) 3.5) (V3 0.5 0.25 (-1.5)))
--
inv33 :: (Epsilon a, Floating a) => M33 a -> Maybe (M33 a)
-- | 3x3 identity matrix.
--
--
-- >>> eye3
-- V3 (V3 1 0 0) (V3 0 1 0) (V3 0 0 1)
--
eye3 :: Num a => M33 a
-- | 4x4 identity matrix.
--
--
-- >>> eye4
-- V4 (V4 1 0 0 0) (V4 0 1 0 0) (V4 0 0 1 0) (V4 0 0 0 1)
--
eye4 :: Num a => M44 a
-- | Compute the trace of a matrix
--
--
-- >>> trace (V2 (V2 a b) (V2 c d))
-- a + d
--
trace :: (Monad f, Foldable f, Num a) => f (f a) -> a
-- | Extract the translation vector (first three entries of the last
-- column) from a 3x4 or 4x4 matrix
translation :: (R3 t, R4 v, Functor f, Functor t) => (V3 a -> f (V3 a)) -> t (v a) -> f (t a)
-- | Build a rotation matrix from a unit Quaternion.
fromQuaternion :: Num a => Quaternion a -> M33 a
-- | Build a transformation matrix from a rotation expressed as a
-- Quaternion and a translation vector.
mkTransformation :: Num a => Quaternion a -> V3 a -> M44 a
-- | This module simply re-exports everything from the various modules that
-- make up the linear package.
module Linear