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 and live within the APS.

• R is the constructor for the Real Scalar Sub-algebra Grade-0
• V3 is the Vector constructor Grade-1
• BV is the Bivector constructor Grade-2
• 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
• H is the Quaternion constructor it is the Even Sub-algebra with Grade-0 and Grade-2 elements
• C is the Complex constructor it is the Scalar Sub-algebra with Grade-0 and Grade-3 elements
• BPV is the Biparavector constructor with Grade-1 and Grade-2 elements
• ODD is the Odd constructor with Grade-1 and Grade-3 elements
• TPV is the Triparavector constructor with Grade-2 and Grade-3 elements
• 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 some are zero

tol currently 128*eps

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

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 to be calculated and it'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. We have safty problem with unreduced values, so it calls reduce first, as a view pattern.