Algebra.Geometric.Cl3

Library implementing standard functions for the Algebra of Physical Space Cl(3,0)

The type for the Algebra of Physical Space

data Cl3 where

Cl3 provides specialized constructors for sub-algebras and other geometric objects contained in the algebra. Cl(3,0), abbreviated to Cl3, is a Geometric Algebra of 3 dimensional space known as the Algebra of Physical Space (APS). Geometric Algebras are Real Clifford Algebras, double precision floats are used to approximate real numbers in this library. Single and Double grade combinations are specialized using algebraic datatypes and live within the APS.

• R is the constructor for the Real Scalar Sub-algebra Grade-0
• V3 is the Three Dimensional Real Vector constructor Grade-1
• BV is the Bivector constructor Grade-2 an Imaginary Three Dimensional Vector
• I is the Imaginary constructor Grade-3 and is the Pseudo-Scalar for APS
• PV is the Paravector constructor with Grade-0 and Grade-1 elements, a Real Scalar plus Vector, (R + V3)
• H is the Quaternion constructor it is the Even Sub-algebra with Grade-0 and Grade-2 elements, a Real Scalar plus Bivector, (R + BV)
• C is the Complex constructor it is the Scalar Sub-algebra with Grade-0 and Grade-3 elements, a Real Scalar plus Imaginar Scalar, (R + I)
• BPV is the Biparavector constructor with Grade-1 and Grade-2 elements, a Real Vector plus Bivector, (V3 + BV)
• ODD is the Odd constructor with Grade-1 and Grade-3 elements, a Vector plus Imaginary Scalar, (V3 + I)
• TPV is the Triparavector constructor with Grade-2 and Grade-3 elements, a Bivector plus Imaginary, (BV + I)
• APS is the constructor for an element in the Algebra of Physical Space with Grade-0 through Grade-3 elements

Constructors

R :: !Double -> Cl3

V3 :: !Double -> !Double -> !Double -> Cl3

BV :: !Double -> !Double -> !Double -> Cl3

I :: !Double -> Cl3

PV :: !Double -> !Double -> !Double -> !Double -> Cl3

H :: !Double -> !Double -> !Double -> !Double -> Cl3

C :: !Double -> !Double -> Cl3

BPV :: !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> Cl3

ODD :: !Double -> !Double -> !Double -> !Double -> Cl3

TPV :: !Double -> !Double -> !Double -> !Double -> Cl3

APS :: !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> !Double -> Cl3

Instances

Eq Cl3

Cl(3,0) has the property of equivalence. Eq is True when all of the grade elements are equivalent.

Defined in Algebra.Geometric.Cl3

Methods

(==) :: Cl3 -> Cl3 -> Bool

(/=) :: Cl3 -> Cl3 -> Bool

Floating Cl3

Cl(3,0) has a Floating instance.

Defined in Algebra.Geometric.Cl3

Methods

pi :: Cl3

exp :: Cl3 -> Cl3

log :: Cl3 -> Cl3

sqrt :: Cl3 -> Cl3

(**) :: Cl3 -> Cl3 -> Cl3

logBase :: Cl3 -> Cl3 -> Cl3

sin :: Cl3 -> Cl3

cos :: Cl3 -> Cl3

tan :: Cl3 -> Cl3

asin :: Cl3 -> Cl3

acos :: Cl3 -> Cl3

atan :: Cl3 -> Cl3

sinh :: Cl3 -> Cl3

cosh :: Cl3 -> Cl3

tanh :: Cl3 -> Cl3

asinh :: Cl3 -> Cl3

acosh :: Cl3 -> Cl3

atanh :: Cl3 -> Cl3

log1p :: Cl3 -> Cl3

expm1 :: Cl3 -> Cl3

log1pexp :: Cl3 -> Cl3

log1mexp :: Cl3 -> Cl3

Fractional Cl3

Cl(3,0) has a Fractional instance

Defined in Algebra.Geometric.Cl3

Methods

(/) :: Cl3 -> Cl3 -> Cl3

recip :: Cl3 -> Cl3

fromRational :: Rational -> Cl3

Data Cl3

Defined in Algebra.Geometric.Cl3

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Cl3 -> c Cl3

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c Cl3

toConstr :: Cl3 -> Constr

dataTypeOf :: Cl3 -> DataType

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c Cl3)

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c Cl3)

gmapT :: (forall b. Data b => b -> b) -> Cl3 -> Cl3

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Cl3 -> r

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Cl3 -> r

gmapQ :: (forall d. Data d => d -> u) -> Cl3 -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Cl3 -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Cl3 -> m Cl3

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Cl3 -> m Cl3

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Cl3 -> m Cl3

Num Cl3

Cl3 has a Num instance. Num is addition, geometric product, negation, abs the largest singular value, and signum.

Defined in Algebra.Geometric.Cl3

Methods

(+) :: Cl3 -> Cl3 -> Cl3

(-) :: Cl3 -> Cl3 -> Cl3

(*) :: Cl3 -> Cl3 -> Cl3

negate :: Cl3 -> Cl3

abs :: Cl3 -> Cl3

signum :: Cl3 -> Cl3

fromInteger :: Integer -> Cl3

Ord Cl3

Cl3 has a total preorder ordering in which all pairs are comparable by two real valued functions. Comparison of two reals is just the typical real compare function. Comparison of to imaginary numbers is just the typical comparison function. When reals are compared to anything else it will compare the absolute value of the reals to the magnitude of the other cliffor. Compare of two complex values compares the polar magnitude of the complex numbers. Compare of two vectors compares the vector magnitudes. The Ord instance for the general case is based on the singular values of each cliffor and this Ordering compares the largest singular value abs and then the littlest singular value lsv. Some arbitrary cliffors may return EQ for Ord but not be exactly == equivalent, but they are related by a right and left multiplication of two unitary elements. For instance for the Cliffors A and B, A == B could be False, but compare A B is EQ, because A * V = U * B, where V and U are unitary.

Defined in Algebra.Geometric.Cl3

Methods

compare :: Cl3 -> Cl3 -> Ordering

(<) :: Cl3 -> Cl3 -> Bool

(<=) :: Cl3 -> Cl3 -> Bool

(>) :: Cl3 -> Cl3 -> Bool

(>=) :: Cl3 -> Cl3 -> Bool

max :: Cl3 -> Cl3 -> Cl3

min :: Cl3 -> Cl3 -> Cl3

Read Cl3

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3

readList :: ReadS [Cl3]

readPrec :: ReadPrec Cl3

readListPrec :: ReadPrec [Cl3]

Show Cl3

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3 -> ShowS

show :: Cl3 -> String

showList :: [Cl3] -> ShowS

Generic Cl3

Defined in Algebra.Geometric.Cl3

Associated Types

type Rep Cl3 :: Type -> Type

Methods

from :: Cl3 -> Rep Cl3 x

to :: Rep Cl3 x -> Cl3

Storable Cl3

Cl3 instance of Storable uses the APS constructor as its standard interface. "peek" returns a cliffor constructed with APS. "poke" converts a cliffor to APS. For a more compact storing of constructors other than APS use the storable subtypes Cl3_R, Cl3_V3, Cl3_BV, Cl3_I, Cl3_PV, Cl3_H, Cl3_C, Cl3_BPV, Cl3_ODD, Cl3_TPV.

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3 -> Int

alignment :: Cl3 -> Int

peekElemOff :: Ptr Cl3 -> Int -> IO Cl3

pokeElemOff :: Ptr Cl3 -> Int -> Cl3 -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3

pokeByteOff :: Ptr b -> Int -> Cl3 -> IO ()

peek :: Ptr Cl3 -> IO Cl3

poke :: Ptr Cl3 -> Cl3 -> IO ()

NFData Cl3

Cl3 can be reduced to a normal form.

Defined in Algebra.Geometric.Cl3

Methods

rnf :: Cl3 -> ()

Random Cl3

Random instance for the Random library

Defined in Algebra.Geometric.Cl3

Methods

randomR :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

random :: RandomGen g => g -> (Cl3, g)

randomRs :: RandomGen g => (Cl3, Cl3) -> g -> [Cl3]

randoms :: RandomGen g => g -> [Cl3]

type Rep Cl3

Defined in Algebra.Geometric.Cl3

type Rep Cl3 = D1 ('MetaData "Cl3" "Algebra.Geometric.Cl3" "cl3-3.0.0.0-2fedqkWsj2XGSvAWIGAfaG" 'False) (((C1 ('MetaCons "R" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "V3" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)))) :+: (C1 ('MetaCons "BV" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: (C1 ('MetaCons "I" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "PV" 'PrefixI 'False) ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)))))) :+: ((C1 ('MetaCons "H" 'PrefixI 'False) ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: (C1 ('MetaCons "C" 'PrefixI 'False) (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :+: C1 ('MetaCons "BPV" 'PrefixI 'False) ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)))))) :+: (C1 ('MetaCons "ODD" 'PrefixI 'False) ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: (C1 ('MetaCons "TPV" 'PrefixI 'False) ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :+: C1 ('MetaCons "APS" 'PrefixI 'False) (((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))) :*: ((S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double)) :*: (S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double) :*: S1 ('MetaSel ('Nothing :: Maybe Symbol) 'NoSourceUnpackedness 'SourceStrict 'DecidedStrict) (Rec0 Double))))))))

Clifford Conjugate and Complex Conjugate

bar :: Cl3 -> Cl3

bar is a Clifford Conjugate, the vector grades are negated

dag :: Cl3 -> Cl3

dag is the Complex Conjugate, the imaginary grades are negated

The littlest singular value

lsv :: Cl3 -> Cl3

lsv the littlest singular value. Useful for testing for invertability.

Constructor Selectors - For optimizing and simplifying calculations

toR :: Cl3 -> Cl3

toR takes any Cliffor and returns the R portion

toV3 :: Cl3 -> Cl3

toV3 takes any Cliffor and returns the V3 portion

toBV :: Cl3 -> Cl3

toBV takes any Cliffor and returns the BV portion

toI :: Cl3 -> Cl3

toI takes any Cliffor and returns the I portion

toPV :: Cl3 -> Cl3

toPV takes any Cliffor and returns the PV poriton

toH :: Cl3 -> Cl3

toH takes any Cliffor and returns the H portion

toC :: Cl3 -> Cl3

toC takes any Cliffor and returns the C portion

toBPV :: Cl3 -> Cl3

toBPV takes any Cliffor and returns the BPV portion

toODD :: Cl3 -> Cl3

toODD takes any Cliffor and returns the ODD portion

toTPV :: Cl3 -> Cl3

toTPV takes any Cliffor and returns the TPV portion

toAPS :: Cl3 -> Cl3

toAPS takes any Cliffor and returns the APS portion

Pretty Printing for use with Octave

showOctave :: Cl3 -> String

showOctave for useful for debug purposes. The additional octave definition is needed:

e0 = [1,0;0,1]; e1=[0,1;1,0]; e2=[0,-i;i,0]; e3=[1,0;0,-1];

This allows one to take advantage of the isomorphism between Cl3 and M(2,C)

Eliminate grades that are less than tol to use a simpler Constructor

reduce :: Cl3 -> Cl3

reduce function reduces the number of grades in a specialized Cliffor if they are zero-ish

tol :: Cl3

tol currently 128*eps

Compact Storable types for the Cl3 Constructors with smart constructors

data Cl3_R

Cl3_R a compact storable data type for R.

Instances

Read Cl3_R

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_R

readList :: ReadS [Cl3_R]

readPrec :: ReadPrec Cl3_R

readListPrec :: ReadPrec [Cl3_R]

Show Cl3_R

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_R -> ShowS

show :: Cl3_R -> String

showList :: [Cl3_R] -> ShowS

Storable Cl3_R

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_R -> Int

alignment :: Cl3_R -> Int

peekElemOff :: Ptr Cl3_R -> Int -> IO Cl3_R

pokeElemOff :: Ptr Cl3_R -> Int -> Cl3_R -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_R

pokeByteOff :: Ptr b -> Int -> Cl3_R -> IO ()

peek :: Ptr Cl3_R -> IO Cl3_R

poke :: Ptr Cl3_R -> Cl3_R -> IO ()

toCl3_R :: Cl3 -> Cl3_R

toCl3_R converts a Cl3 value constructed with R to its compact form.

fromCl3_R :: Cl3_R -> Cl3

fromCl3_R converts the compact Cl3_R type back to a Cl3 type.

data Cl3_V3

Cl3_V3 a compact storable data type for V3.

Instances

Read Cl3_V3

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_V3

readList :: ReadS [Cl3_V3]

readPrec :: ReadPrec Cl3_V3

readListPrec :: ReadPrec [Cl3_V3]

Show Cl3_V3

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_V3 -> ShowS

show :: Cl3_V3 -> String

showList :: [Cl3_V3] -> ShowS

Storable Cl3_V3

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_V3 -> Int

alignment :: Cl3_V3 -> Int

peekElemOff :: Ptr Cl3_V3 -> Int -> IO Cl3_V3

pokeElemOff :: Ptr Cl3_V3 -> Int -> Cl3_V3 -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_V3

pokeByteOff :: Ptr b -> Int -> Cl3_V3 -> IO ()

peek :: Ptr Cl3_V3 -> IO Cl3_V3

poke :: Ptr Cl3_V3 -> Cl3_V3 -> IO ()

toCl3_V3 :: Cl3 -> Cl3_V3

toCl3_V3 converts a Cl3 value constructed with V3 to its compact form.

fromCl3_V3 :: Cl3_V3 -> Cl3

fromCl3_V3 converts the compact Cl3_V3 type back to a Cl3 type.

data Cl3_BV

Cl3_BV a compact storable data type for BV.

Instances

Read Cl3_BV

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_BV

readList :: ReadS [Cl3_BV]

readPrec :: ReadPrec Cl3_BV

readListPrec :: ReadPrec [Cl3_BV]

Show Cl3_BV

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_BV -> ShowS

show :: Cl3_BV -> String

showList :: [Cl3_BV] -> ShowS

Storable Cl3_BV

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_BV -> Int

alignment :: Cl3_BV -> Int

peekElemOff :: Ptr Cl3_BV -> Int -> IO Cl3_BV

pokeElemOff :: Ptr Cl3_BV -> Int -> Cl3_BV -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_BV

pokeByteOff :: Ptr b -> Int -> Cl3_BV -> IO ()

peek :: Ptr Cl3_BV -> IO Cl3_BV

poke :: Ptr Cl3_BV -> Cl3_BV -> IO ()

toCl3_BV :: Cl3 -> Cl3_BV

toCl3_BV converts a Cl3 value constructed with BV to its compact form.

fromCl3_BV :: Cl3_BV -> Cl3

fromCl3_BV converts the compact Cl3_BV type back to a Cl3 type.

data Cl3_I

Cl3_I a compact storable data type for I.

Instances

Read Cl3_I

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_I

readList :: ReadS [Cl3_I]

readPrec :: ReadPrec Cl3_I

readListPrec :: ReadPrec [Cl3_I]

Show Cl3_I

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_I -> ShowS

show :: Cl3_I -> String

showList :: [Cl3_I] -> ShowS

Storable Cl3_I

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_I -> Int

alignment :: Cl3_I -> Int

peekElemOff :: Ptr Cl3_I -> Int -> IO Cl3_I

pokeElemOff :: Ptr Cl3_I -> Int -> Cl3_I -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_I

pokeByteOff :: Ptr b -> Int -> Cl3_I -> IO ()

peek :: Ptr Cl3_I -> IO Cl3_I

poke :: Ptr Cl3_I -> Cl3_I -> IO ()

toCl3_I :: Cl3 -> Cl3_I

toCl3_I converts a Cl3 value constructed with I to its compact form.

fromCl3_I :: Cl3_I -> Cl3

fromCl3_I converts the compact Cl3_I type back to a Cl3 type.

data Cl3_PV

Cl3_PV a compact storable data type for PV.

Instances

Read Cl3_PV

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_PV

readList :: ReadS [Cl3_PV]

readPrec :: ReadPrec Cl3_PV

readListPrec :: ReadPrec [Cl3_PV]

Show Cl3_PV

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_PV -> ShowS

show :: Cl3_PV -> String

showList :: [Cl3_PV] -> ShowS

Storable Cl3_PV

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_PV -> Int

alignment :: Cl3_PV -> Int

peekElemOff :: Ptr Cl3_PV -> Int -> IO Cl3_PV

pokeElemOff :: Ptr Cl3_PV -> Int -> Cl3_PV -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_PV

pokeByteOff :: Ptr b -> Int -> Cl3_PV -> IO ()

peek :: Ptr Cl3_PV -> IO Cl3_PV

poke :: Ptr Cl3_PV -> Cl3_PV -> IO ()

toCl3_PV :: Cl3 -> Cl3_PV

toCl3_PV converts a Cl3 value constructed with PV to its compact form.

fromCl3_PV :: Cl3_PV -> Cl3

fromCl3_PV converts the compact Cl3_PV type back to a Cl3 type.

data Cl3_H

Cl3_H a compact storable data type for H.

Instances

Read Cl3_H

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_H

readList :: ReadS [Cl3_H]

readPrec :: ReadPrec Cl3_H

readListPrec :: ReadPrec [Cl3_H]

Show Cl3_H

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_H -> ShowS

show :: Cl3_H -> String

showList :: [Cl3_H] -> ShowS

Storable Cl3_H

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_H -> Int

alignment :: Cl3_H -> Int

peekElemOff :: Ptr Cl3_H -> Int -> IO Cl3_H

pokeElemOff :: Ptr Cl3_H -> Int -> Cl3_H -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_H

pokeByteOff :: Ptr b -> Int -> Cl3_H -> IO ()

peek :: Ptr Cl3_H -> IO Cl3_H

poke :: Ptr Cl3_H -> Cl3_H -> IO ()

toCl3_H :: Cl3 -> Cl3_H

toCl3_H converts a Cl3 value constructed with H to its compact form.

fromCl3_H :: Cl3_H -> Cl3

fromCl3_H converts the compact Cl3_H type back to a Cl3 type.

data Cl3_C

Cl3_C a compact storable data type for C.

Instances

Read Cl3_C

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_C

readList :: ReadS [Cl3_C]

readPrec :: ReadPrec Cl3_C

readListPrec :: ReadPrec [Cl3_C]

Show Cl3_C

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_C -> ShowS

show :: Cl3_C -> String

showList :: [Cl3_C] -> ShowS

Storable Cl3_C

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_C -> Int

alignment :: Cl3_C -> Int

peekElemOff :: Ptr Cl3_C -> Int -> IO Cl3_C

pokeElemOff :: Ptr Cl3_C -> Int -> Cl3_C -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_C

pokeByteOff :: Ptr b -> Int -> Cl3_C -> IO ()

peek :: Ptr Cl3_C -> IO Cl3_C

poke :: Ptr Cl3_C -> Cl3_C -> IO ()

toCl3_C :: Cl3 -> Cl3_C

toCl3_C converts a Cl3 value constructed with C to its compact form.

fromCl3_C :: Cl3_C -> Cl3

fromCl3_C converts the compact Cl3_C type back to a Cl3 type.

data Cl3_BPV

Cl3_BPV a compact storable data type for BPV.

Instances

Read Cl3_BPV

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_BPV

readList :: ReadS [Cl3_BPV]

readPrec :: ReadPrec Cl3_BPV

readListPrec :: ReadPrec [Cl3_BPV]

Show Cl3_BPV

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_BPV -> ShowS

show :: Cl3_BPV -> String

showList :: [Cl3_BPV] -> ShowS

Storable Cl3_BPV

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_BPV -> Int

alignment :: Cl3_BPV -> Int

peekElemOff :: Ptr Cl3_BPV -> Int -> IO Cl3_BPV

pokeElemOff :: Ptr Cl3_BPV -> Int -> Cl3_BPV -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_BPV

pokeByteOff :: Ptr b -> Int -> Cl3_BPV -> IO ()

peek :: Ptr Cl3_BPV -> IO Cl3_BPV

poke :: Ptr Cl3_BPV -> Cl3_BPV -> IO ()

toCl3_BPV :: Cl3 -> Cl3_BPV

toCl3_BPV converts a Cl3 value constructed with BPV to its compact form.

fromCl3_BPV :: Cl3_BPV -> Cl3

fromCl3_BPV converts the compact Cl3_BPV type back to a Cl3 type.

data Cl3_ODD

Cl3_ODD a compact storable data type for ODD.

Instances

Read Cl3_ODD

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_ODD

readList :: ReadS [Cl3_ODD]

readPrec :: ReadPrec Cl3_ODD

readListPrec :: ReadPrec [Cl3_ODD]

Show Cl3_ODD

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_ODD -> ShowS

show :: Cl3_ODD -> String

showList :: [Cl3_ODD] -> ShowS

Storable Cl3_ODD

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_ODD -> Int

alignment :: Cl3_ODD -> Int

peekElemOff :: Ptr Cl3_ODD -> Int -> IO Cl3_ODD

pokeElemOff :: Ptr Cl3_ODD -> Int -> Cl3_ODD -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_ODD

pokeByteOff :: Ptr b -> Int -> Cl3_ODD -> IO ()

peek :: Ptr Cl3_ODD -> IO Cl3_ODD

poke :: Ptr Cl3_ODD -> Cl3_ODD -> IO ()

toCl3_ODD :: Cl3 -> Cl3_ODD

toCl3_ODD converts a Cl3 value constructed with ODD to its compact form.

fromCl3_ODD :: Cl3_ODD -> Cl3

fromCl3_ODD converts the compact Cl3_ODD type back to a Cl3 type.

data Cl3_TPV

Cl3_TPV a compact storable data type for TPV.

Instances

Read Cl3_TPV

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_TPV

readList :: ReadS [Cl3_TPV]

readPrec :: ReadPrec Cl3_TPV

readListPrec :: ReadPrec [Cl3_TPV]

Show Cl3_TPV

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_TPV -> ShowS

show :: Cl3_TPV -> String

showList :: [Cl3_TPV] -> ShowS

Storable Cl3_TPV

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_TPV -> Int

alignment :: Cl3_TPV -> Int

peekElemOff :: Ptr Cl3_TPV -> Int -> IO Cl3_TPV

pokeElemOff :: Ptr Cl3_TPV -> Int -> Cl3_TPV -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_TPV

pokeByteOff :: Ptr b -> Int -> Cl3_TPV -> IO ()

peek :: Ptr Cl3_TPV -> IO Cl3_TPV

poke :: Ptr Cl3_TPV -> Cl3_TPV -> IO ()

toCl3_TPV :: Cl3 -> Cl3_TPV

toCl3_TPV converts a Cl3 value constructed with TPV to its compact form.

fromCl3_TPV :: Cl3_TPV -> Cl3

fromCl3_TPV converts the compact Cl3_TPV type back to a Cl3 type.

data Cl3_APS

Cl3_APS a compact storable data type for APS.

Instances

Read Cl3_APS

Defined in Algebra.Geometric.Cl3

Methods

readsPrec :: Int -> ReadS Cl3_APS

readList :: ReadS [Cl3_APS]

readPrec :: ReadPrec Cl3_APS

readListPrec :: ReadPrec [Cl3_APS]

Show Cl3_APS

Defined in Algebra.Geometric.Cl3

Methods

showsPrec :: Int -> Cl3_APS -> ShowS

show :: Cl3_APS -> String

showList :: [Cl3_APS] -> ShowS

Storable Cl3_APS

Defined in Algebra.Geometric.Cl3

Methods

sizeOf :: Cl3_APS -> Int

alignment :: Cl3_APS -> Int

peekElemOff :: Ptr Cl3_APS -> Int -> IO Cl3_APS

pokeElemOff :: Ptr Cl3_APS -> Int -> Cl3_APS -> IO ()

peekByteOff :: Ptr b -> Int -> IO Cl3_APS

pokeByteOff :: Ptr b -> Int -> Cl3_APS -> IO ()

peek :: Ptr Cl3_APS -> IO Cl3_APS

poke :: Ptr Cl3_APS -> Cl3_APS -> IO ()

toCl3_APS :: Cl3 -> Cl3_APS

toCl3_APS converts a Cl3 value constructed with APS to its compact form.

fromCl3_APS :: Cl3_APS -> Cl3

fromCl3_APS converts the compact Cl3_APS type back to a Cl3 type.

Random Instances

randR :: RandomGen g => g -> (Cl3, g)

randR random Real Scalar (Grade 0) with random magnitude and random sign

rangeR :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeR random Real Scalar (Grade 0) with random magnitude within a range and a random sign

randV3 :: RandomGen g => g -> (Cl3, g)

randV3 random Vector (Grade 1) with random magnitude and random direction the direction is using spherical coordinates

rangeV3 :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeV3 random Vector (Grade 1) with random magnitude within a range and a random direction

randBV :: RandomGen g => g -> (Cl3, g)

randBV random Bivector (Grade 2) with random magnitude and random direction the direction is using spherical coordinates

rangeBV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeBV random Bivector (Grade 2) with random magnitude in a range and a random direction

randI :: RandomGen g => g -> (Cl3, g)

randI random Imaginary Scalar (Grade 3) with random magnitude and random sign

rangeI :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeI random Imaginary Scalar (Grade 3) with random magnitude within a range and random sign

randPV :: RandomGen g => g -> (Cl3, g)

randPV random Paravector made from random Grade 0 and Grade 1 elements

rangePV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangePV random Paravector made from random Grade 0 and Grade 1 elements within a range

randH :: RandomGen g => g -> (Cl3, g)

randH random Quaternion made from random Grade 0 and Grade 2 elements

rangeH :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeH random Quaternion made from random Grade 0 and Grade 2 elements within a range

randC :: RandomGen g => g -> (Cl3, g)

randC random combination of Grade 0 and Grade 3

rangeC :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeC random combination of Grade 0 and Grade 3 within a range

randBPV :: RandomGen g => g -> (Cl3, g)

randBPV random combination of Grade 1 and Grade 2

rangeBPV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeBPV random combination of Grade 1 and Grade 2 within a range

randODD :: RandomGen g => g -> (Cl3, g)

randODD random combination of Grade 1 and Grade 3

rangeODD :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeODD random combination of Grade 1 and Grade 3 within a range

randTPV :: RandomGen g => g -> (Cl3, g)

randTPV random combination of Grade 2 and Grade 3

rangeTPV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeTPV random combination of Grade 2 and Grade 3 within a range

randAPS :: RandomGen g => g -> (Cl3, g)

randAPS random combination of all 4 grades

rangeAPS :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g)

rangeAPS random combination of all 4 grades within a range

randUnitV3 :: RandomGen g => g -> (Cl3, g)

randUnitV3 a unit vector with a random direction

randProjector :: RandomGen g => g -> (Cl3, g)

randProjector a projector with a random direction

randNilpotent :: RandomGen g => g -> (Cl3, g)

randNilpotent a nilpotent element with a random orientation

randUnitary :: RandomGen g => g -> (Cl3, g)

randUnitary a unitary element with a random orientation

Helpful Functions

eigvals :: Cl3 -> (Cl3, Cl3)

eigvals calculates the eignenvalues of the cliffor. This is useful for determining if a cliffor is the pole of a function.

hasNilpotent :: Cl3 -> Bool

hasNilpotent takes a Cliffor and determines if the vector part and the bivector part are orthoganl and equal in magnitude, i.e. that it is simular to a nilpotent BPV.

spectraldcmp :: (Cl3 -> Cl3) -> (Cl3 -> Cl3) -> Cl3 -> Cl3

spectraldcmp the spectral decomposition of a function to calculate analytic functions of cliffors in Cl(3,0). This function requires the desired function's R, I, and C instances to be calculated and the function's derivative. If multiple functions are being composed, its best to pass the composition of the funcitons to this function and the derivative to this function. Any function with a Taylor Series approximation should be able to be used. A real, imaginary, and complex version of the function to be decomposed must be provided and spectraldcmp will handle the case for an arbitrary Cliffor.

It may be possible to add, in the future, a RULES pragma like:

"spectral decomposition function composition"
forall f f' g g' cliff.
spectraldcmp f f' (spectraldcmp g g' cliff) = spectraldcmp (f.g) (f'.g') cliff

project :: Cl3 -> Cl3

project makes a projector based off of the vector content of the Cliffor.

mIx :: Cl3 -> Cl3

mIx a more effecient 'x -> I (-1) * x' typically useful for converting a Bivector to a Vector in the same direction. Related to Hodge Dual and/or Inverse Hodge Star.

timesI :: Cl3 -> Cl3

timesI is a more effecient 'x -> I 1 * x'

abssignum :: Cl3 -> (Cl3, Cl3)

abssignum is a more effecient 'cl3 -> (abs cl3, signum cl3)' So abs is always R and signum is the same type of constructor as the input signum is the element divided by its largest singular value abs