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

bar is a Clifford Conjugate, the vector grades are negated

dag is the Complex Conjugate, the imaginary grades are negated

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

toR takes any Cliffor and returns the R portion

toV3 takes any Cliffor and returns the V3 portion

toBV takes any Cliffor and returns the BV portion

toI takes any Cliffor and returns the I portion

toPV takes any Cliffor and returns the PV poriton

toH takes any Cliffor and returns the H portion

toC takes any Cliffor and returns the C portion

toBPV takes any Cliffor and returns the BPV portion

toODD takes any Cliffor and returns the ODD portion

toTPV takes any Cliffor and returns the TPV portion

toAPS takes any Cliffor and returns the APS portion

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)

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

tol currently 128*eps

Cl3_R a compact storable data type for R.

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

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

Cl3_V3 a compact storable data type for V3.

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

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

Cl3_BV a compact storable data type for BV.

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

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

Cl3_I a compact storable data type for I.

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

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

Cl3_PV a compact storable data type for PV.

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

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

Cl3_H a compact storable data type for H.

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

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

Cl3_C a compact storable data type for C.

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

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

Cl3_BPV a compact storable data type for BPV.

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

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

Cl3_ODD a compact storable data type for ODD.

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

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

Cl3_TPV a compact storable data type for TPV.

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

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

Cl3_APS a compact storable data type for APS.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

randC random combination of Grade 0 and Grade 3

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

randBPV random combination of Grade 1 and Grade 2

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

randODD random combination of Grade 1 and Grade 3

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

randTPV random combination of Grade 2 and Grade 3

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

randAPS random combination of all 4 grades

rangeAPS random combination of all 4 grades within a range

randUnitV3 a unit vector with a random direction

randProjector a projector with a random direction

randNilpotent a nilpotent element with a random orientation

randUnitary a unitary element with a random orientation

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

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

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

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