Data.Algebra.Quaternion

See the spatial-math package for usage.

Documentation

type QuatF = Quaternion Float

type QuatD = Quaternion Double

type QuatR = Quaternion Rational

type QuatM = Quaternion Micro

type QuatN = Quaternion Nano

type QuatP = Quaternion Pico

data Quaternion a

Constructors

Quaternion !a !(V3 a)

Instances

Functor Quaternion

Defined in Data.Algebra.Quaternion

Methods

fmap :: (a -> b) -> Quaternion a -> Quaternion b

(<$) :: a -> Quaternion b -> Quaternion a

Foldable Quaternion

Defined in Data.Algebra.Quaternion

Methods

fold :: Monoid m => Quaternion m -> m

foldMap :: Monoid m => (a -> m) -> Quaternion a -> m

foldr :: (a -> b -> b) -> b -> Quaternion a -> b

foldr' :: (a -> b -> b) -> b -> Quaternion a -> b

foldl :: (b -> a -> b) -> b -> Quaternion a -> b

foldl' :: (b -> a -> b) -> b -> Quaternion a -> b

foldr1 :: (a -> a -> a) -> Quaternion a -> a

foldl1 :: (a -> a -> a) -> Quaternion a -> a

toList :: Quaternion a -> [a]

null :: Quaternion a -> Bool

length :: Quaternion a -> Int

elem :: Eq a => a -> Quaternion a -> Bool

maximum :: Ord a => Quaternion a -> a

minimum :: Ord a => Quaternion a -> a

sum :: Num a => Quaternion a -> a

product :: Num a => Quaternion a -> a

Distributive Quaternion

Defined in Data.Algebra.Quaternion

Methods

distribute :: Functor f => f (Quaternion a) -> Quaternion (f a)

collect :: Functor f => (a -> Quaternion b) -> f a -> Quaternion (f b)

distributeM :: Monad m => m (Quaternion a) -> Quaternion (m a)

collectM :: Monad m => (a -> Quaternion b) -> m a -> Quaternion (m b)

Representable Quaternion

Defined in Data.Algebra.Quaternion

Associated Types

type Rep Quaternion :: Type

Methods

tabulate :: (Rep Quaternion -> a) -> Quaternion a

index :: Quaternion a -> Rep Quaternion -> a

Foldable1 Quaternion

Defined in Data.Algebra.Quaternion

Methods

fold1 :: Semigroup m => Quaternion m -> m

foldMap1 :: Semigroup m => (a -> m) -> Quaternion a -> m

toNonEmpty :: Quaternion a -> NonEmpty a

Semiring a => Semimodule a (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

(*.) :: a -> Quaternion a -> Quaternion a

(.*) :: Quaternion a -> a -> Quaternion a

Eq a => Eq (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

(==) :: Quaternion a -> Quaternion a -> Bool

(/=) :: Quaternion a -> Quaternion a -> Bool

Ord a => Ord (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

compare :: Quaternion a -> Quaternion a -> Ordering

(<) :: Quaternion a -> Quaternion a -> Bool

(<=) :: Quaternion a -> Quaternion a -> Bool

(>) :: Quaternion a -> Quaternion a -> Bool

(>=) :: Quaternion a -> Quaternion a -> Bool

max :: Quaternion a -> Quaternion a -> Quaternion a

min :: Quaternion a -> Quaternion a -> Quaternion a

Show a => Show (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

showsPrec :: Int -> Quaternion a -> ShowS

show :: Quaternion a -> String

showList :: [Quaternion a] -> ShowS

Generic (Quaternion a)

Defined in Data.Algebra.Quaternion

Associated Types

type Rep (Quaternion a) :: Type -> Type

Methods

from :: Quaternion a -> Rep (Quaternion a) x

to :: Rep (Quaternion a) x -> Quaternion a

Ring a => Semigroup (Multiplicative (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

(<>) :: Multiplicative (Quaternion a) -> Multiplicative (Quaternion a) -> Multiplicative (Quaternion a)

sconcat :: NonEmpty (Multiplicative (Quaternion a)) -> Multiplicative (Quaternion a)

stimes :: Integral b => b -> Multiplicative (Quaternion a) -> Multiplicative (Quaternion a)

(Additive - Semigroup) a => Semigroup (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

(<>) :: Additive (Quaternion a) -> Additive (Quaternion a) -> Additive (Quaternion a)

sconcat :: NonEmpty (Additive (Quaternion a)) -> Additive (Quaternion a)

stimes :: Integral b => b -> Additive (Quaternion a) -> Additive (Quaternion a)

(Additive - Semigroup) a => Semigroup (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

(<>) :: Quaternion a -> Quaternion a -> Quaternion a

sconcat :: NonEmpty (Quaternion a) -> Quaternion a

stimes :: Integral b => b -> Quaternion a -> Quaternion a

Ring a => Monoid (Multiplicative (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

mempty :: Multiplicative (Quaternion a)

mappend :: Multiplicative (Quaternion a) -> Multiplicative (Quaternion a) -> Multiplicative (Quaternion a)

mconcat :: [Multiplicative (Quaternion a)] -> Multiplicative (Quaternion a)

(Additive - Monoid) a => Monoid (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

mempty :: Additive (Quaternion a)

mappend :: Additive (Quaternion a) -> Additive (Quaternion a) -> Additive (Quaternion a)

mconcat :: [Additive (Quaternion a)] -> Additive (Quaternion a)

(Additive - Monoid) a => Monoid (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

mempty :: Quaternion a

mappend :: Quaternion a -> Quaternion a -> Quaternion a

mconcat :: [Quaternion a] -> Quaternion a

(Additive - Group) a => Group (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

inv :: Additive (Quaternion a) -> Additive (Quaternion a)

greplicate :: Integer -> Additive (Quaternion a) -> Additive (Quaternion a)

(Additive - Group) a => Group (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

inv :: Quaternion a -> Quaternion a

greplicate :: Integer -> Quaternion a -> Quaternion a

(Additive - Group) a => Loop (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

lempty :: Additive (Quaternion a)

lreplicate :: Natural -> Additive (Quaternion a) -> Additive (Quaternion a)

(Additive - Group) a => Loop (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

lempty :: Quaternion a

lreplicate :: Natural -> Quaternion a -> Quaternion a

(Additive - Group) a => Quasigroup (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

(//) :: Additive (Quaternion a) -> Additive (Quaternion a) -> Additive (Quaternion a)

(\\) :: Additive (Quaternion a) -> Additive (Quaternion a) -> Additive (Quaternion a)

(Additive - Group) a => Quasigroup (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

(//) :: Quaternion a -> Quaternion a -> Quaternion a

(\\) :: Quaternion a -> Quaternion a -> Quaternion a

(Additive - Group) a => Magma (Additive (Quaternion a))

Defined in Data.Algebra.Quaternion

Methods

(<<) :: Additive (Quaternion a) -> Additive (Quaternion a) -> Additive (Quaternion a)

(Additive - Group) a => Magma (Quaternion a)

Defined in Data.Algebra.Quaternion

Methods

(<<) :: Quaternion a -> Quaternion a -> Quaternion a

Ring a => Ring (Quaternion a)

Defined in Data.Algebra.Quaternion

Ring a => Semiring (Quaternion a)

Defined in Data.Algebra.Quaternion

Ring a => Presemiring (Quaternion a)

Defined in Data.Algebra.Quaternion

Generic1 Quaternion

Defined in Data.Algebra.Quaternion

Associated Types

type Rep1 Quaternion :: k -> Type

Methods

from1 :: Quaternion a -> Rep1 Quaternion a

to1 :: Rep1 Quaternion a -> Quaternion a

type Rep Quaternion

Defined in Data.Algebra.Quaternion

type Rep Quaternion = Maybe I3

type Rep (Quaternion a)

Defined in Data.Algebra.Quaternion

type Rep (Quaternion a) = D1 (MetaData "Quaternion" "Data.Algebra.Quaternion" "rings-0.0.3-21Z3wlGnIMkBc1cJ09ZojY" False) (C1 (MetaCons "Quaternion" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) (Rec0 a) :*: S1 (MetaSel (Nothing :: Maybe Symbol) SourceUnpack SourceStrict DecidedStrict) (Rec0 (V3 a))))

type Rep1 Quaternion

Defined in Data.Algebra.Quaternion

type Rep1 Quaternion = D1 (MetaData "Quaternion" "Data.Algebra.Quaternion" "rings-0.0.3-21Z3wlGnIMkBc1cJ09ZojY" False) (C1 (MetaCons "Quaternion" PrefixI False) (S1 (MetaSel (Nothing :: Maybe Symbol) NoSourceUnpackedness SourceStrict DecidedStrict) Par1 :*: S1 (MetaSel (Nothing :: Maybe Symbol) SourceUnpack SourceStrict DecidedStrict) (Rec1 V3)))

quat :: a -> a -> a -> a -> Quaternion a

Obtain a Quaternion from 4 base field elements.

scal :: Quaternion a -> a

Real or scalar part of a quaternion.

vect :: Quaternion a -> V3 a

rotate :: Ring a => Quaternion a -> V3 a -> V3 a

Use a quaternion to rotate a vector.

>>> rotate qk . rotate qj $ V3 1 1 0 :: V3 Int
V3 1 (-1) 0

normalize :: QuatD -> QuatD

Scale a QuatD to unit length.

>>> normalize $ normalize $ quat 2.0 2.0 2.0 2.0
Quaternion 0.5 (V3 0.5 0.5 0.5)

qe :: Semiring a => Quaternion a

The real quaternion.

Represents no rotation.

qe = unit

qi :: Semiring a => Quaternion a

The i quaternion.

Represents a \( \pi \) radian rotation about the x axis.

>>> rotate (qi :: QuatM) $ V3 1 0 0
V3 1.000000 0.000000 0.000000
>>> rotate (qi :: QuatM) $ V3 0 1 0
V3 0.000000 -1.000000 0.000000
>>> rotate (qi :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 -1.000000

>>> qi * qj
Quaternion 0 (V3 0 0 1)

qj :: Semiring a => Quaternion a

The j quaternion.

Represents a \( \pi \) radian rotation about the y axis.

>>> rotate (qj :: QuatM) $ V3 1 0 0
V3 -1.000000 0.000000 0.000000
>>> rotate (qj :: QuatM) $ V3 0 1 0
V3 0.000000 1.000000 0.000000
>>> rotate (qj :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 -1.000000

>>> qj * qk
Quaternion 0 (V3 1 0 0)

qk :: Semiring a => Quaternion a

The k quaternion.

Represents a \( \pi \) radian rotation about the z axis.

>>> rotate (qk :: QuatM) $ V3 1 0 0
V3 -1.000000 0.000000 0.000000
>>> rotate (qk :: QuatM) $ V3 0 1 0
V3 0.000000 -1.000000 0.000000
>>> rotate (qk :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 1.000000

>>> qk * qi
Quaternion 0 (V3 0 1 0)
>>> qi * qj * qk
Quaternion (-1) (V3 0 0 0)