-- Hoogle documentation, generated by Haddock -- See Hoogle, http://www.haskell.org/hoogle/ -- | Clifford Algebra of three dimensional space. -- -- Haskell Library implementing standard functions for the Algebra of -- Physical Space Cl(3,0) @package cl3 @version 2.0.0.0 -- | Library implementing standard functions for the Algebra of Physical -- Space Cl(3,0) module Algebra.Geometric.Cl3 -- | 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. -- -- data Cl3 [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 -- | bar is a Clifford Conjugate, the vector grades are negated bar :: Cl3 -> Cl3 -- | dag is the Complex Conjugate, the imaginary grades are negated dag :: Cl3 -> Cl3 -- | lsv the littlest singular value. Useful for testing for -- invertability. lsv :: Cl3 -> Cl3 -- | toR takes any Cliffor and returns the R portion toR :: Cl3 -> Cl3 -- | toV3 takes any Cliffor and returns the V3 portion toV3 :: Cl3 -> Cl3 -- | toBV takes any Cliffor and returns the BV portion toBV :: Cl3 -> Cl3 -- | toI takes any Cliffor and returns the I portion toI :: Cl3 -> Cl3 -- | toPV takes any Cliffor and returns the PV poriton toPV :: Cl3 -> Cl3 -- | toH takes any Cliffor and returns the H portion toH :: Cl3 -> Cl3 -- | toC takes any Cliffor and returns the C portion toC :: Cl3 -> Cl3 -- | toBPV takes any Cliffor and returns the BPV portion toBPV :: Cl3 -> Cl3 -- | toODD takes any Cliffor and returns the ODD portion toODD :: Cl3 -> Cl3 -- | toTPV takes any Cliffor and returns the TPV portion toTPV :: Cl3 -> Cl3 -- | toAPS takes any Cliffor and returns the APS portion toAPS :: Cl3 -> Cl3 -- | 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) showOctave :: Cl3 -> String -- | reduce function reduces the number of grades in a specialized -- Cliffor if they are zero-ish reduce :: Cl3 -> Cl3 -- | tol currently 128*eps tol :: Cl3 -- | randR random Real Scalar (Grade 0) with random magnitude and -- random sign randR :: RandomGen g => g -> (Cl3, g) -- | rangeR random Real Scalar (Grade 0) with random magnitude -- within a range and a random sign rangeR :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randV3 random Vector (Grade 1) with random magnitude and random -- direction the direction is using spherical coordinates randV3 :: RandomGen g => g -> (Cl3, g) -- | rangeV3 random Vector (Grade 1) with random magnitude within a -- range and a random direction rangeV3 :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randBV random Bivector (Grade 2) with random magnitude and -- random direction the direction is using spherical coordinates randBV :: RandomGen g => g -> (Cl3, g) -- | rangeBV random Bivector (Grade 2) with random magnitude in a -- range and a random direction rangeBV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randI random Imaginary Scalar (Grade 3) with random magnitude -- and random sign randI :: RandomGen g => g -> (Cl3, g) -- | rangeI random Imaginary Scalar (Grade 3) with random magnitude -- within a range and random sign rangeI :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randPV random Paravector made from random Grade 0 and Grade 1 -- elements randPV :: RandomGen g => g -> (Cl3, g) -- | rangePV random Paravector made from random Grade 0 and Grade 1 -- elements within a range rangePV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randH random Quaternion made from random Grade 0 and Grade 2 -- elements randH :: RandomGen g => g -> (Cl3, g) -- | rangeH random Quaternion made from random Grade 0 and Grade 2 -- elements within a range rangeH :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randC random combination of Grade 0 and Grade 3 randC :: RandomGen g => g -> (Cl3, g) -- | rangeC random combination of Grade 0 and Grade 3 within a range rangeC :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randBPV random combination of Grade 1 and Grade 2 randBPV :: RandomGen g => g -> (Cl3, g) -- | rangeBPV random combination of Grade 1 and Grade 2 within a -- range rangeBPV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randODD random combination of Grade 1 and Grade 3 randODD :: RandomGen g => g -> (Cl3, g) -- | rangeODD random combination of Grade 1 and Grade 3 within a -- range rangeODD :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randTPV random combination of Grade 2 and Grade 3 randTPV :: RandomGen g => g -> (Cl3, g) -- | rangeTPV random combination of Grade 2 and Grade 3 within a -- range rangeTPV :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randAPS random combination of all 4 grades randAPS :: RandomGen g => g -> (Cl3, g) -- | rangeAPS random combination of all 4 grades within a range rangeAPS :: RandomGen g => (Cl3, Cl3) -> g -> (Cl3, g) -- | randUnitV3 a unit vector with a random direction randUnitV3 :: RandomGen g => g -> (Cl3, g) -- | randProjector a projector with a random direction randProjector :: RandomGen g => g -> (Cl3, g) -- | randNilpotent a nilpotent element with a random orientation randNilpotent :: RandomGen g => g -> (Cl3, g) -- | randUnitary a unitary element with a random orientation randUnitary :: RandomGen g => g -> (Cl3, g) -- | eigvals calculates the eignenvalues of the cliffor. This is -- useful for determining if a cliffor is the pole of a function. eigvals :: Cl3 -> (Cl3, Cl3) -- | 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. hasNilpotent :: Cl3 -> Bool -- | 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
--
spectraldcmp :: (Cl3 -> Cl3) -> (Cl3 -> Cl3) -> Cl3 -> Cl3 -- | project makes a projector based off of the vector content of -- the Cliffor. project :: 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. mIx :: Cl3 -> Cl3 -- | timesI is a more effecient 'x -> I 1 * x' timesI :: Cl3 -> Cl3 instance GHC.Enum.Enum Algebra.Geometric.Cl3.ConCl3 instance GHC.Enum.Bounded Algebra.Geometric.Cl3.ConCl3 instance GHC.Generics.Generic Algebra.Geometric.Cl3.Cl3 instance Data.Data.Data Algebra.Geometric.Cl3.Cl3 instance GHC.Read.Read Algebra.Geometric.Cl3.Cl3 instance GHC.Show.Show Algebra.Geometric.Cl3.Cl3 instance System.Random.Random Algebra.Geometric.Cl3.Cl3 instance Control.DeepSeq.NFData Algebra.Geometric.Cl3.Cl3 instance GHC.Classes.Eq Algebra.Geometric.Cl3.Cl3 instance GHC.Classes.Ord Algebra.Geometric.Cl3.Cl3 instance GHC.Num.Num Algebra.Geometric.Cl3.Cl3 instance GHC.Real.Fractional Algebra.Geometric.Cl3.Cl3 instance GHC.Float.Floating Algebra.Geometric.Cl3.Cl3 instance Foreign.Storable.Storable Algebra.Geometric.Cl3.Cl3 -- | Library implementing standard functions for the Jones Calculus in the -- Cl3 Library. This implementation of the Jones Calculus is based on the -- convensions of SPIE's Field Guide to Polarization (ϕ = ω t − k z). -- -- -- --

Jones Vectors

-- -- Within the system of the Bloch Sphere, the Jones Vectors in Cl3 are -- calculated by generating the left ideal of the rotation of a unit -- vector to the e3 basis. Standard form for for a versor is 'rot -- = exp $ (-i/2) * theta * u' for angle theta and the -- rotational axis unit vector u. -- -- Bloch Sphere Coordinates: -- -- 
--      e3
--      |
--      |____e2
--     / 
--    /
--   e1
--
module Algebra.Geometric.Cl3.JonesCalculus -- | hpv horizontally polarized Jones vector hpv :: Cl3 -- | vpv vertically polarized Jones vector vpv :: Cl3 -- | dpv diagonally polarized Jones vector dpv :: Cl3 -- | apv anti-diagonally polarized Jones vector apv :: Cl3 -- | rpv right hand circularly polarized Jones vector rpv :: Cl3 -- | lpv left hand circularly polarized Jones vector lpv :: Cl3 -- | jv function that returns Jones vector from input vector unit -- vector currently converts the input to a unit vector jv :: Cl3 -> Cl3 -- | hpm Horizontal Polarizer Jones Matrix hpm :: Cl3 -- | vpm Vertical Polarizer Jones Matrix vpm :: Cl3 -- | dpm Diagonal Polarizer Jones Matrix dpm :: Cl3 -- | apm Anti-diagonal Polarizer Jones Matrix apm :: Cl3 -- | rpm Right Hand Circular Polarizer Jones Matrix rpm :: Cl3 -- | lpm Left Hand Circular Polarizer Jones Matrix lpm :: Cl3 -- | jm funciton that returns a Jones Matrix from an input Bloch -- Vector currently converts the input to a unit vector jm :: Cl3 -> Cl3 -- | hpmRot Jones matrix for a rotated ideal Linear Horizontal -- Polarizer. Input value should be a scalar angle in Radians. hpmRot :: Cl3 -> Cl3 -- | qwp Quarter Wave Plate Jones Matrix qwp :: Cl3 -- | hwp Half Wave Plate Jones Matrix hwp :: Cl3 -- | qwpRot Rotated Quarter Wave Plate Jones Matrix. Input value -- should be a scalar angle in Radians. qwpRot :: Cl3 -> Cl3 -- | hwpRot Rotated Half Wave Plate Jones Matrix. Input value should -- be a scalar angle in Radians. hwpRot :: Cl3 -> Cl3 -- | wp a Wave Plate with phase shift of phi Jones Matrix. Input -- value should be a scalar angle in Radians. wp :: Cl3 -> Cl3 -- | wpRot a Rotated Wave Plate with phase shift of phi and rotation -- theta Jones Matrix. The first input value is phi the phase shift as a -- scalar value in Radians. The second input value is theta the rotation -- a scalar angle in Radians. wpRot :: Cl3 -> Cl3 -> Cl3 -- | refl a Refelection Jones Matrix refl :: Cl3 -- | randJonesVec a Random Jones Vector. randJonesVec :: RandomGen g => g -> (Cl3, g) -- | randOrthogonalJonesVec a Random Orthogonal Complementary pair -- of Jones Vectors. randOrthogonalJonesVec :: RandomGen g => g -> ((Cl3, Cl3), g) -- | factorize is a function that takes an Jones Vector after -- transformation by an optical chain, and returns the amplitude (amp), -- phase (phi), and normalized Jones Vector (vec), by the factorization -- of the input such that: amp * exp (i*phi/2) * vec factorize :: Cl3 -> (Cl3, Cl3, Cl3)