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Language	Haskell98

Numeric.Coalgebra.Geometric

Contents

Geometric coalgebra primitives
Operations over an eigenbasis
Grade
Inversions
Products
Inner products

Synopsis

newtype BasisCoblade m = BasisCoblade {
- runBasisCoblade :: Word64
}
type Comultivector r m = Covector r (BasisCoblade m)
class Eigenbasis m where
class (Ring r, Eigenbasis m) => Eigenmetric r m where
newtype Euclidean = Euclidean Int
grade :: BasisCoblade m -> Int
filterGrade :: Monoidal r => BasisCoblade m -> Int -> Comultivector r m
reverse :: Group r => BasisCoblade m -> Comultivector r m
gradeInversion :: Group r => BasisCoblade m -> Comultivector r m
cliffordConjugate :: Group r => BasisCoblade m -> Comultivector r m
geometric :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
outer :: Eigenmetric r m => BasisCoblade m -> BasisCoblade m -> Comultivector r m
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

Geometric coalgebra primitives

newtype BasisCoblade m #

Constructors

BasisCoblade
Fields runBasisCoblade :: Word64

Instances

RightModule Natural (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (*.) :: BasisCoblade m -> Natural -> BasisCoblade m #
LeftModule Natural (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (.*) :: Natural -> BasisCoblade m -> BasisCoblade m #
Eigenmetric r m => Coalgebra r (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods comult :: (BasisCoblade m -> r) -> BasisCoblade m -> BasisCoblade m -> r #
Eigenmetric r m => CounitalCoalgebra r (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods counit :: (BasisCoblade m -> r) -> r #
Bounded (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods minBound :: BasisCoblade m # maxBound :: BasisCoblade m #
Enum (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (==) :: BasisCoblade m -> BasisCoblade m -> Bool # (/=) :: BasisCoblade m -> BasisCoblade m -> Bool #
Integral (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods readsPrec :: Int -> ReadS (BasisCoblade m) # readList :: ReadS [BasisCoblade m] # readPrec :: ReadPrec (BasisCoblade m) # readListPrec :: ReadPrec [BasisCoblade m] #
Real (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods toRational :: BasisCoblade m -> Rational #
Show (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods showsPrec :: Int -> BasisCoblade m -> ShowS # show :: BasisCoblade m -> String # showList :: [BasisCoblade m] -> ShowS #
Ix (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric
Additive (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods zero :: BasisCoblade m # sinnum :: Natural -> BasisCoblade m -> BasisCoblade m # sumWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m #
Semiring (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric
Multiplicative (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric 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) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods one :: BasisCoblade m # pow :: BasisCoblade m -> Natural -> BasisCoblade m # productWith :: Foldable f => (a -> BasisCoblade m) -> f a -> BasisCoblade m #
Commutative (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric
DecidableAssociates (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods isAssociate :: BasisCoblade m -> BasisCoblade m -> Bool #
DecidableUnits (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods recipUnit :: BasisCoblade m -> Maybe (BasisCoblade m) # isUnit :: BasisCoblade m -> Bool # (^?) :: Integral n => BasisCoblade m -> n -> Maybe (BasisCoblade m) #
DecidableZero (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods isZero :: BasisCoblade m -> Bool #
Rig (BasisCoblade m) #
Instance details Defined in Numeric.Coalgebra.Geometric Methods fromNatural :: Natural -> 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 #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric Methods metric :: Euclidean -> r #

newtype Euclidean #

Constructors

Euclidean Int

Instances

Enum Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (==) :: Euclidean -> Euclidean -> Bool # (/=) :: Euclidean -> Euclidean -> Bool #
Integral Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (+) :: Euclidean -> Euclidean -> Euclidean # (-) :: Euclidean -> Euclidean -> Euclidean # (*) :: Euclidean -> Euclidean -> Euclidean # negate :: Euclidean -> Euclidean # abs :: Euclidean -> Euclidean # signum :: Euclidean -> Euclidean # fromInteger :: Integer -> Euclidean #
Ord Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric Methods readsPrec :: Int -> ReadS Euclidean # readList :: ReadS [Euclidean] # readPrec :: ReadPrec Euclidean # readListPrec :: ReadPrec [Euclidean] #
Real Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods toRational :: Euclidean -> Rational #
Show Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods showsPrec :: Int -> Euclidean -> ShowS # show :: Euclidean -> String # showList :: [Euclidean] -> ShowS #
Ix Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #
Instance details Defined in Numeric.Coalgebra.Geometric
Additive Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (+) :: Euclidean -> Euclidean -> Euclidean # sinnum1p :: Natural -> Euclidean -> Euclidean # sumWith1 :: Foldable1 f => (a -> Euclidean) -> f a -> Euclidean #
Monoidal Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods zero :: Euclidean # sinnum :: Natural -> Euclidean -> Euclidean # sumWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean #
Semiring Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric
Multiplicative Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (*) :: Euclidean -> Euclidean -> Euclidean # pow1p :: Euclidean -> Natural -> Euclidean # productWith1 :: Foldable1 f => (a -> Euclidean) -> f a -> Euclidean #
Group Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (-) :: Euclidean -> Euclidean -> Euclidean # negate :: Euclidean -> Euclidean # subtract :: Euclidean -> Euclidean -> Euclidean # times :: Integral n => n -> Euclidean -> Euclidean #
Unital Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods one :: Euclidean # pow :: Euclidean -> Natural -> Euclidean # productWith :: Foldable f => (a -> Euclidean) -> f a -> Euclidean #
Commutative Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric
TriviallyInvolutive Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric
InvolutiveSemiring Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric
InvolutiveMultiplication Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods adjoint :: Euclidean -> Euclidean #
Rig Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods fromNatural :: Natural -> Euclidean #
Ring Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods fromInteger :: Integer -> Euclidean #
Eigenbasis Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods euclidean :: proxy Euclidean -> Bool # antiEuclidean :: proxy Euclidean -> Bool # v :: Euclidean -> BasisCoblade Euclidean # e :: Int -> Euclidean #
RightModule Integer Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (*.) :: Euclidean -> Integer -> Euclidean #
RightModule Natural Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (*.) :: Euclidean -> Natural -> Euclidean #
LeftModule Integer Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (.*) :: Integer -> Euclidean -> Euclidean #
LeftModule Natural Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric Methods (.*) :: Natural -> Euclidean -> Euclidean #
Ring r => Eigenmetric r Euclidean #
Instance details Defined in Numeric.Coalgebra.Geometric 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 #