Numeric.Coalgebra.Geometric

Geometric coalgebra primitives

newtype BasisCoblade m

Constructors

BasisCoblade

Fields

• runBasisCoblade :: Word64

Instances

RightModule Natural (BasisCoblade m)

Methods

(*.) :: BasisCoblade m -> Natural -> BasisCoblade m

LeftModule Natural (BasisCoblade m)

Methods

(.*) :: Natural -> BasisCoblade m -> BasisCoblade m

Eigenmetric r m => Coalgebra r (BasisCoblade m)

Methods

comult :: (BasisCoblade m -> r) -> BasisCoblade m -> BasisCoblade m -> r

Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m)

Methods

counit :: (BasisCoblade m -> r) -> r

Bounded (BasisCoblade m)

Methods

minBound :: BasisCoblade m

maxBound :: BasisCoblade m

Enum (BasisCoblade m)

Methods

succ :: BasisCoblade m -> BasisCoblade m

pred :: BasisCoblade m -> BasisCoblade m

toEnum :: Int -> BasisCoblade m

fromEnum :: BasisCoblade m -> Int

enumFrom :: BasisCoblade m -> [BasisCoblade m]

enumFromThen :: BasisCoblade m -> BasisCoblade m -> [BasisCoblade m]

enumFromTo :: BasisCoblade m -> BasisCoblade m -> [BasisCoblade m]

enumFromThenTo :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m -> [BasisCoblade m]

Eq (BasisCoblade m)

Methods

(==) :: BasisCoblade m -> BasisCoblade m -> Bool

(/=) :: BasisCoblade m -> BasisCoblade m -> Bool

Integral (BasisCoblade m)

Methods

quot :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

rem :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

div :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

mod :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

quotRem :: BasisCoblade m -> BasisCoblade m -> (BasisCoblade m, BasisCoblade m)

divMod :: BasisCoblade m -> BasisCoblade m -> (BasisCoblade m, BasisCoblade m)

toInteger :: BasisCoblade m -> Integer

Num (BasisCoblade m)

Methods

(+) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

(-) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

(*) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

negate :: BasisCoblade m -> BasisCoblade m

abs :: BasisCoblade m -> BasisCoblade m

signum :: BasisCoblade m -> BasisCoblade m

fromInteger :: Integer -> BasisCoblade m

Ord (BasisCoblade m)

Methods

compare :: BasisCoblade m -> BasisCoblade m -> Ordering

(<) :: BasisCoblade m -> BasisCoblade m -> Bool

(<=) :: BasisCoblade m -> BasisCoblade m -> Bool

(>) :: BasisCoblade m -> BasisCoblade m -> Bool

(>=) :: BasisCoblade m -> BasisCoblade m -> Bool

max :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

min :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

Read (BasisCoblade m)

Methods

readsPrec :: Int -> ReadS (BasisCoblade m)

readList :: ReadS [BasisCoblade m]

readPrec :: ReadPrec (BasisCoblade m)

readListPrec :: ReadPrec [BasisCoblade m]

Real (BasisCoblade m)

Methods

toRational :: BasisCoblade m -> Rational

Show (BasisCoblade m)

Methods

showsPrec :: Int -> BasisCoblade m -> ShowS

show :: BasisCoblade m -> String

showList :: [BasisCoblade m] -> ShowS

Ix (BasisCoblade m)

Methods

range :: (BasisCoblade m, BasisCoblade m) -> [BasisCoblade m]

index :: (BasisCoblade m, BasisCoblade m) -> BasisCoblade m -> Int

unsafeIndex :: (BasisCoblade m, BasisCoblade m) -> BasisCoblade m -> Int

inRange :: (BasisCoblade m, BasisCoblade m) -> BasisCoblade m -> Bool

rangeSize :: (BasisCoblade m, BasisCoblade m) -> Int

unsafeRangeSize :: (BasisCoblade m, BasisCoblade m) -> Int

Bits (BasisCoblade m)

Methods

(.&.) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

(.|.) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

xor :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

complement :: BasisCoblade m -> BasisCoblade m

shift :: BasisCoblade m -> Int -> BasisCoblade m

rotate :: BasisCoblade m -> Int -> BasisCoblade m

zeroBits :: BasisCoblade m

bit :: Int -> BasisCoblade m

setBit :: BasisCoblade m -> Int -> BasisCoblade m

clearBit :: BasisCoblade m -> Int -> BasisCoblade m

complementBit :: BasisCoblade m -> Int -> BasisCoblade m

testBit :: BasisCoblade m -> Int -> Bool

bitSizeMaybe :: BasisCoblade m -> Maybe Int

bitSize :: BasisCoblade m -> Int

isSigned :: BasisCoblade m -> Bool

shiftL :: BasisCoblade m -> Int -> BasisCoblade m

unsafeShiftL :: BasisCoblade m -> Int -> BasisCoblade m

shiftR :: BasisCoblade m -> Int -> BasisCoblade m

unsafeShiftR :: BasisCoblade m -> Int -> BasisCoblade m

rotateL :: BasisCoblade m -> Int -> BasisCoblade m

rotateR :: BasisCoblade m -> Int -> BasisCoblade m

popCount :: BasisCoblade m -> Int

Abelian (BasisCoblade m)

Additive (BasisCoblade m)

Methods

(+) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

sinnum1p :: Natural -> BasisCoblade m -> BasisCoblade m

sumWith1 :: Foldable1 f => (a -> BasisCoblade m) -> f a -> BasisCoblade m

Monoidal (BasisCoblade m)

Methods

zero :: BasisCoblade m

sinnum :: Natural -> BasisCoblade m -> BasisCoblade m

sumWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m

Semiring (BasisCoblade m)

Multiplicative (BasisCoblade m)

Methods

(*) :: BasisCoblade m -> BasisCoblade m -> BasisCoblade m

pow1p :: BasisCoblade m -> Natural -> BasisCoblade m

productWith1 :: Foldable1 f => (a -> BasisCoblade m) -> f a -> BasisCoblade m

Unital (BasisCoblade m)

Methods

one :: BasisCoblade m

pow :: BasisCoblade m -> Natural -> BasisCoblade m

productWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m

DecidableAssociates (BasisCoblade m)

Methods

isAssociate :: BasisCoblade m -> BasisCoblade m -> Bool

DecidableUnits (BasisCoblade m)

Methods

recipUnit :: BasisCoblade m -> Maybe (BasisCoblade m)

isUnit :: BasisCoblade m -> Bool

(^?) :: Integral n => BasisCoblade m -> n -> Maybe (BasisCoblade m)

DecidableZero (BasisCoblade m)

Methods

isZero :: BasisCoblade m -> Bool

Rig (BasisCoblade m)

Methods

fromNatural :: Natural -> BasisCoblade m

Commutative (BasisCoblade m)

type Comultivector r m = Covector r (BasisCoblade m)

Operations over an eigenbasis

class Eigenbasis m where

Minimal complete definition

euclidean, antiEuclidean, v, e

Methods

euclidean :: proxy m -> Bool

antiEuclidean :: proxy m -> Bool

v :: m -> BasisCoblade m

e :: Int -> m

Instances

Eigenbasis Euclidean

Methods

euclidean :: proxy Euclidean -> Bool

antiEuclidean :: proxy Euclidean -> Bool

v :: Euclidean -> BasisCoblade Euclidean

e :: Int -> Euclidean

class (Ring r, Eigenbasis m) => Eigenmetric r m where

Minimal complete definition

metric

Methods

metric :: m -> r

Instances

Ring r => Eigenmetric r Euclidean

Methods

metric :: Euclidean -> r

newtype Euclidean

Constructors

Euclidean Int

Instances

Enum Euclidean

Methods

succ :: Euclidean -> Euclidean

pred :: Euclidean -> Euclidean

toEnum :: Int -> Euclidean

fromEnum :: Euclidean -> Int

enumFrom :: Euclidean -> [Euclidean]

enumFromThen :: Euclidean -> Euclidean -> [Euclidean]

enumFromTo :: Euclidean -> Euclidean -> [Euclidean]

enumFromThenTo :: Euclidean -> Euclidean -> Euclidean -> [Euclidean]

Eq Euclidean

Methods

(==) :: Euclidean -> Euclidean -> Bool

(/=) :: Euclidean -> Euclidean -> Bool

Integral Euclidean

Methods

quot :: Euclidean -> Euclidean -> Euclidean

rem :: Euclidean -> Euclidean -> Euclidean

div :: Euclidean -> Euclidean -> Euclidean

mod :: Euclidean -> Euclidean -> Euclidean

quotRem :: Euclidean -> Euclidean -> (Euclidean, Euclidean)

divMod :: Euclidean -> Euclidean -> (Euclidean, Euclidean)

toInteger :: Euclidean -> Integer

Data Euclidean

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Euclidean -> c Euclidean

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Euclidean

toConstr :: Euclidean -> Constr

dataTypeOf :: Euclidean -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c Euclidean)

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Euclidean)

gmapT :: (forall b. Data b => b -> b) -> Euclidean -> Euclidean

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Euclidean -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Euclidean -> r

gmapQ :: (forall d. Data d => d -> u) -> Euclidean -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Euclidean -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Euclidean -> m Euclidean

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Euclidean -> m Euclidean

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Euclidean -> m Euclidean

Num Euclidean

Methods

(+) :: Euclidean -> Euclidean -> Euclidean

(-) :: Euclidean -> Euclidean -> Euclidean

(*) :: Euclidean -> Euclidean -> Euclidean

negate :: Euclidean -> Euclidean

abs :: Euclidean -> Euclidean

signum :: Euclidean -> Euclidean

fromInteger :: Integer -> Euclidean

Ord Euclidean

Methods

compare :: Euclidean -> Euclidean -> Ordering

(<) :: Euclidean -> Euclidean -> Bool

(<=) :: Euclidean -> Euclidean -> Bool

(>) :: Euclidean -> Euclidean -> Bool

(>=) :: Euclidean -> Euclidean -> Bool

max :: Euclidean -> Euclidean -> Euclidean

min :: Euclidean -> Euclidean -> Euclidean

Read Euclidean

Methods

readsPrec :: Int -> ReadS Euclidean

readList :: ReadS [Euclidean]

readPrec :: ReadPrec Euclidean

readListPrec :: ReadPrec [Euclidean]

Real Euclidean

Methods

toRational :: Euclidean -> Rational

Show Euclidean

Methods

showsPrec :: Int -> Euclidean -> ShowS

show :: Euclidean -> String

showList :: [Euclidean] -> ShowS

Ix Euclidean

Methods

range :: (Euclidean, Euclidean) -> [Euclidean]

index :: (Euclidean, Euclidean) -> Euclidean -> Int

unsafeIndex :: (Euclidean, Euclidean) -> Euclidean -> Int

inRange :: (Euclidean, Euclidean) -> Euclidean -> Bool

rangeSize :: (Euclidean, Euclidean) -> Int

unsafeRangeSize :: (Euclidean, Euclidean) -> Int

Abelian Euclidean

Additive Euclidean

Methods

(+) :: Euclidean -> Euclidean -> Euclidean

sinnum1p :: Natural -> Euclidean -> Euclidean

sumWith1 :: Foldable1 f => (a -> Euclidean) -> f a -> Euclidean

Monoidal Euclidean

Methods

zero :: Euclidean

sinnum :: Natural -> Euclidean -> Euclidean

sumWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean

Semiring Euclidean

Multiplicative Euclidean

Methods

(*) :: Euclidean -> Euclidean -> Euclidean

pow1p :: Euclidean -> Natural -> Euclidean

productWith1 :: Foldable1 f => (a -> Euclidean) -> f a -> Euclidean

Group Euclidean

Methods

(-) :: Euclidean -> Euclidean -> Euclidean

negate :: Euclidean -> Euclidean

subtract :: Euclidean -> Euclidean -> Euclidean

times :: Integral n => n -> Euclidean -> Euclidean

Unital Euclidean

Methods

one :: Euclidean

pow :: Euclidean -> Natural -> Euclidean

productWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean

Rig Euclidean

Methods

fromNatural :: Natural -> Euclidean

Ring Euclidean

Methods

fromInteger :: Integer -> Euclidean

Commutative Euclidean

TriviallyInvolutive Euclidean

InvolutiveSemiring Euclidean

InvolutiveMultiplication Euclidean

Methods

adjoint :: Euclidean -> Euclidean

Eigenbasis Euclidean

Methods

euclidean :: proxy Euclidean -> Bool

antiEuclidean :: proxy Euclidean -> Bool

v :: Euclidean -> BasisCoblade Euclidean

e :: Int -> Euclidean

RightModule Integer Euclidean

Methods

(*.) :: Euclidean -> Integer -> Euclidean

RightModule Natural Euclidean

Methods

(*.) :: Euclidean -> Natural -> Euclidean

LeftModule Integer Euclidean

Methods

(.*) :: Integer -> Euclidean -> Euclidean

LeftModule Natural Euclidean

Methods

(.*) :: Natural -> Euclidean -> Euclidean

Ring r => Eigenmetric r Euclidean

Methods

metric :: Euclidean -> r

Grade

grade :: BasisCoblade m -> Int

filterGrade :: Monoidal r => BasisCoblade m -> Int -> Comultivector r m

Inversions

reverse :: Group r => BasisCoblade m -> Comultivector r m

gradeInversion :: Group r => BasisCoblade m -> Comultivector r m

cliffordConjugate :: Group r => BasisCoblade m -> Comultivector r m

Products

geometric :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

outer :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

Inner products

contractL :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

contractR :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

hestenes :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

dot :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m

liftProduct :: (BasisCoblade m -> BasisCoblade m -> Comultivector r m) -> Comultivector r m -> Comultivector r m -> Comultivector r m