Data.Semifield.Quaternion

Documentation

data Quaternion a

Constructors

Quaternion !a !(V3 a)

Instances

Functor Quaternion

Defined in Data.Semifield.Quaternion

Methods

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

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

Foldable Quaternion

Defined in Data.Semifield.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.Semifield.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.Semifield.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.Semifield.Quaternion

Methods

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

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

toNonEmpty :: Quaternion a -> NonEmpty a

Eq a => Eq (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

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

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

Ord a => Ord (Quaternion a)

Defined in Data.Semifield.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.Semifield.Quaternion

Methods

showsPrec :: Int -> Quaternion a -> ShowS

show :: Quaternion a -> String

showList :: [Quaternion a] -> ShowS

Generic (Quaternion a)

Defined in Data.Semifield.Quaternion

Associated Types

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

Methods

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

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

Semigroup a => Semigroup (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

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

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

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

Monoid a => Monoid (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

mempty :: Quaternion a

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

mconcat :: [Quaternion a] -> Quaternion a

Ring a => Semiring (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

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

fromBoolean :: Bool -> Quaternion a

Group a => Group (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

negate :: Quaternion a -> Quaternion a

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

Ring a => Ring (Quaternion a)

Defined in Data.Semifield.Quaternion

Methods

fromInteger :: Integer -> Quaternion a

abs :: Quaternion a -> Quaternion a

signum :: Quaternion a -> Quaternion a

Field a => Semifield (Quaternion a)

(Right) division ring over a.

Note that, due to non-commutativity, it is possible to divide in two different ways.

See https://en.wikipedia.org/wiki/Quaternion#Unit_quaternion.

>>> q = Quaternion 1 (V3 1 2 3) :: QuatR
>>> q // q
Quaternion (15 % 1) (V3 (0 % 1) (0 % 1) (0 % 1))

Defined in Data.Semifield.Quaternion

Methods

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

recip :: Quaternion a -> Quaternion a

fromRational :: Rational -> Quaternion a

fromPositive :: Positive -> Quaternion a

Generic1 Quaternion

Defined in Data.Semifield.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.Semifield.Quaternion

type Rep Quaternion = Maybe I3

type Rep (Quaternion a)

Defined in Data.Semifield.Quaternion

type Rep (Quaternion a) = D1 (MetaData "Quaternion" "Data.Semifield.Quaternion" "rings-0.0.2.4-9dBg3spVdomqRKHFxJllu" 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.Semifield.Quaternion

type Rep1 Quaternion = D1 (MetaData "Quaternion" "Data.Semifield.Quaternion" "rings-0.0.2.4-9dBg3spVdomqRKHFxJllu" 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)))

type QuatF = Quaternion Float

type QuatD = Quaternion Double

type QuatR = Quaternion Rational

type QuatM = Quaternion Micro

type QuatN = Quaternion Nano

type QuatP = Quaternion Pico

refpos :: Field a => V3 a

iter4 :: Ring a => Quaternion a -> V3 a -> V3 a

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

Obtain a Quaternion from 4 base field elements.

real :: Quaternion a -> a

Real or scalar part of a quaternion.

imag :: Quaternion a -> V3 a

Imaginary or vector part of a quaternion.

norm :: Semiring a => Quaternion a -> a

Squared norm of a quaternion.

 norm x == real $ x >< conj x

normalize :: QuatD -> QuatD

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 1 (-1) 0

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

conj :: Group a => Quaternion a -> Quaternion a

qi :: Unital 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 :: Unital 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 :: Unital 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)

cosAngle :: QuatP -> V3 Pico -> Pico

q00 :: Field a => Quaternion a

Identity (empty) rotation.

All rotations written according to the right-hand rule.

q01 :: Field a => Quaternion a

A \( \pi/2 \) radian rotation about the y axis.

>>> m33 0 0 1 0 1 0 (-1) 0 0 #> V3 0 1 0 :: V3 Micro
V3 0.000000 1.000000 0.000000

q02 :: Field a => Quaternion a

A \( \pi \) radian rotation about the y axis.

q03 :: Field a => Quaternion a

A \( 3 \pi/2 \) radian rotation about the y axis.

q04 :: Field a => Quaternion a

A \( \pi/2 \) radian rotation about the z axis.

>>> rotate (q04 :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 0.999996
>>> rotate (q04 :: QuatM) $ V3 1 1 0
V3 -0.999998 0.999997 0.000000

q05 :: Field a => Quaternion a

A \( 2 \pi/3 \) radian rotation about the x-y-z axis.

 q05 ≡ q01 >< q04

>>> rotate (q05 :: QuatM) $ V3 1 1 1
V3 1.000000 1.000000 1.000000
>>> cosAngle q05 (fmap (irt2><) $ V3 1 (-1) 0)
-0.499999999991
>>> cosAngle q05 (fmap (irt2><) $ V3 0 1 (-1))
-0.499999999991
>>> cosAngle q05 (fmap (irt2><irt3><) $ V3 1 (-2) 1)
-0.499999999993

q06 :: Field a => Quaternion a

A \( \pi \) radian rotation about the x-y axis.

 q06 ≡ q02 >< q04

>>> rotate (q06 :: QuatM) $ V3 1 1 0
V3 0.999997 0.999997 0.000000
>>> rotate (q06 :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 -0.999997

q07 :: Field a => Quaternion a

A \( 2 \pi/3 \) radian rotation about the (-x)-(-y)-z axis.

Equivalent to a \( 3 \pi/2 \) radian rotation about y axis, followed by \( \pi/2 \) radian rotation about z axis:

 q07 ≡ q03 >< q04

>>> rotate (q07 :: QuatM) $ V3 1 1 (-1)
V3 1.000000 1.000000 -1.000000
>>> cosAngle q07 ((irt2><irt3><) <$> V3 (-1) 2 1)
-0.499999999993

q08 :: Field a => Quaternion a

A \( 3 \pi/2 \) radian rotation about the z axis.

>>> rotate (q08 :: QuatM) $ V3 0 0 1
V3 0.000000 0.000000 0.999996
>>> rotate (q08 :: QuatM) $ V3 1 1 0
V3 0.999997 -0.999997 0.000000

q09 :: Field a => Quaternion a

A \( 2 \pi/3 \) radian rotation about the (-x)-y-(-z) axis.

 q09 ≡ q01 >< q08

>>> rotate (q09 :: QuatM) $ V3 1 (-1) 1
V3 1.000000 -1.000000 1.000000
>>> cosAngle q09 ((irt2><irt3><) <$> V3 1 2 1)
-0.499999999993

q10 :: Field a => Quaternion a

 q10 ≡ q02 >< q08

q11 :: Field a => Quaternion a

A \( 2 \pi/3 \) radian rotation about the x-(-y)-(-z) axis.

 q11 ≡ q03 >< q08

>>> rotate (q11 :: QuatM) $ V3 (-1) 1 1
V3 1.000000 1.000000 -1.000000
>>> cosAngle q11 ((irt2><irt3><) <$> V3 1 2 (-1))
-0.499999999993

q12 :: Field a => Quaternion a

A \( \pi/2 \) radian rotation about the x axis.

>>> rotate (q12 :: QuatM) $ V3 1 0 0
V3 0.999996 0.000000 0.000000
>>> rotate (q12 :: QuatM) $ V3 0 1 1
V3 0.000000 -0.999998 0.999997

q13 :: Field a => Quaternion a

 q13 ≡ q01 >< q12

q14 :: Field a => Quaternion a

 q14 ≡ q02 >< q12

q15 :: Field a => Quaternion a

 q15 ≡ q03 >< q12

q16 :: Field a => Quaternion a

A \( \pi \) radian rotation about the x axis.

λ> rotate (q16 :: QuatM) $ V3 1 0 0 V3 1.000000 0.000000 0.000000 λ> rotate (q16 :: QuatM) $ V3 0 1 1 V3 0.000000 -1.000000 -1.000000

q17 :: Field a => Quaternion a

 q17 ≡ q01 >< q16

q18 :: Field a => Quaternion a

 q18 ≡ q02 >< q16

q19 :: Field a => Quaternion a

 q19 ≡ q03 >< q16

q20 :: Field a => Quaternion a

A \( 3 \pi/2 \) radian rotation about the x axis.

>>> rotate (q20 :: QuatM) $ V3 1 0 0
V3 0.999996 0.000000 0.000000
>>> rotate (q20 :: QuatM) $ V3 0 1 1
V3 0.000000 0.999997 -0.999997

q21 :: Field a => Quaternion a

 q21 ≡ q01 >< q20

q22 :: Field a => Quaternion a

 q22 ≡ q02 >< q20

q23 :: Field a => Quaternion a

 q23 ≡ q03 >< q20

irt2 :: Field a => a

irt3 :: Field a => a

divq :: Semiring a => (a -> a -> a) -> (a -> a) -> Quaternion a -> Quaternion a -> Quaternion a

(/) :: QuatD -> QuatD -> QuatD