Sierpinski (<1K) Multivectors as Transformations
    "It is a profoundly erroneous truism, repeated by all copy books and by eminent people when they are making speeches, that we should cultivate the habit of thinking of what we are doing. The precise opposite is the case. Civilization advances by extending the number of important operations which we can perform without thinking about them." --- Alfred North Whitehead.

    Multivectors constitute an attractive alternative to the conventional matrix representations of transformations in ÂN. We will use uppercase mulitivector labels when we wish to emphasise a multivector's use in this role.

Bivector Transform
    ¦(a)ºa¿b maps one 1-vector a "into" 2-vector b and rotates by ½p "within" b. For pure 2-blade b=e12 we have ¦(åi aiei)=-a2e1+a1e2. It provides 1-vector transformations
    ¦k(a) º ¦k-1(a)¿b =2-kåi=0k kCi(-1)ibiabk-i .

Lorentz Transforms

    We are particularly interested in grade-preserving transformations of multivectors of the form
    A(b) º A#-1(x) º Ax(A#)-1     ;     defined to be Ax(A#) when A# is noninvertible;
and their application to the representions for pointsets in ÂN. For invertible k-versor A# = (-1)k A, such transformations satisfy A(x)¿A(c) = x¿c and are known as Lorentz transforms, or as isometries of inner product ¿. They preserve both magnitudes and subtended angles.    Because they preserve magntude, null points are mapped to to other null points and so Lorentz transforms on UN% induce transfroms on UN.

    The involution # ensures that ak(x) = x for any x with x¿ak=0 (ie. "perpendicular" to ak with xak=xÙak=(-1)kakx) but means that A(1) = A(A#)-1 so scalars are negated for odd A, and multiplied by potentially nonscalar A(A#-1) for mixed A.
    If A is even then A(x) = AxA-1 = A-1(x) in Bellian notation and we have A(bc) = A(b) A(c) . If A is odd so then A = -A-1. If follows that for odd or even A we have A(a1Ùa2...Ùak) = ± A(a1)ÙA(a2)Ù...A(ak) with the negative case arising when both A and k are odd.
[ Proof :  Follows from aÙbk = abk + bk#a  .] Thus A is an outtermorphism, apart from  a possible sign change incurred by the #, and this means that we can apply A for odd or even A to the pureblade GHC representations of geometric objects like k-planes and k-spheres to obtain representors for the Lorentz transformed objects.
    Clearly (-A)#-1 = A#-1 and for even A we can often resolve this ambiguity of sign by specifying A<0>>0.

     Also of interest is A§(x) º Ax(A§) with adjoint A§D(x) = A§-1(x) º (A§)xA if AA§=1. a*AbxA§   =   A§aA*bx .
     For a unit 2k-versor R , since R# = R and R-1 = R§ we have R = R#-1 = R§ .
    We will show below that b§(a) = bab is a pure 1-vector for any 1-vectors a,b. Since this is also so for scalar b we have bab§ = (bab§)<1> for any versor b.  
    Lorentz transforms preserve orthogonality and since it can be shown [  Cartan-Dieudonne Theorem ] that any such transform can be expressed as up to N reflections in hyperplanes, any orthogonality preserving transform can be expressed as an e¥-preserving rotation R§() where R is a nonnull unit k-versor. In particular, the orientation of a continuously rotated rigid body in ÂN always has a unit k-rotor representation with k£N.

    A standard result of complex matrix theory is that any unitary (U^T=U-1) complex matrix can be expressed expressed as U=(iH) where H is Hermitian (H^T=H).
    Within Â4,1, Hermitian conjugation is = §#§ = §[e4] so even R with RR§[e4]=1 has a logarithm ih where h=h.
    Any Hermitian h comprises blades of grades <0;1;4;5;8;...> that exclude e4 and grades <2;3;6;7;..> that include e4. Taking i=e12345 we see that ih comrpises blaaes of grades <1;4;5> that include e4 and grades <2;3> that exclude e4. If we further insist ih be even, we have 4-blades with an e4 factor and 2-blades without.
    Restricting attention to e4* we see that any even unit rotor R in Â4 can be expressed as an exponentiated bivector.the e

Higher Dimensional Embeddings
    Suppose we have extended ÂN into ÂN+p,q,r with an associated point embedding ¦ : ÂN ® ÂN+p,q,r. Any multivector A in ÂN+p,q,r represents a transformation GA,¦ : ÂN ® ÂN defined by GA,¦(x) = ¦-1(A¦(x)(A#)-1).
    In the three embeddings we consider here, ¦-1(sx) = ¦-1(x) for any nonzero scalar s so we have A¦(x)(A(#))-1 = s(x)¦(GA,¦(x)) where s(x) is a scalar.
    The involution (#) is the "interior" or "unextended" ÂN involution operating within ÂN+p,q,r rather than the proper involution # of ÂN+p,q,r ; so that, for example, in the affine model we take e0(#) = e0 rather than e0# = -e0.

 Homogeneous Coordinates
    A(¦(x)) = A(x+e0) = A(x) + Ae0(A(#))-1 = A(x) + e0A#(A(#))-1 = A(x) + e0 = ¦(A(x))

Affine Model
    A(x) = A(1 + e0x) = A(A(#))-1 + Ae0x(A#)-1 = A(A(#))-1 + e0A#x(A(#))-1
    If A is even/odd we have A(¦(x)) = ± ¦(A(x)).

Generalised Homogeneous Coordinates

    A(¦(x)) = A(x + e0 + ½ x2e¥) = A(x) + A(e0) + ½ x2A(e¥)
    = A(x) + e0 + ½ x2e¥     if A is within ÂN
    = ¦(A(x))     if A is odd or even .

    Translations in ÂN can only be represented in ÂN by addition of a 1-vector in the conventional manner. In some higher dimensional emebeddings we can exploit the fact that a translation by displacement vector d is equivalent to reflection in any two hyperplanes normal to d spaced ½ |d| apart. In particular, the ÂN hyperplanes through 0 and then ½d will suffice.

Affine Model

    The representor of x - d is 1 + e0(x - d) = Td(1+e0x)Td where Td=(1 - ½ e0d)=(-½e0d) has Td2 = T2d and Td(#)Td = Td§Td = Td§#Td = 1.
[ Proof : (1-½ e0d)(1+e0x)(1-½ e0d) = (1-½ e0d)(1+e0xe0d) = (1+e0xe0d - ½ e0d) = 1 + e0(x - d)  .]
    We have Td-1 = (1 + ½ e0d) =  T-d and Td(#) = T-d = Td§ so we have Td# -1 = Td for all d in ÂN.

    If B is any ÂN multivector representing an "origin centered" transformation of ÂN we can represent the associated operation centered at d with T-dBTd . If B is even we have
T-dBTd = (1+½ e0d)B(1-½ e0d) = B - ½(Be0d-e0dB) = B + ½ e0 (dB - Bd) = B + ½ e0(d × B) .

Generalised Homogeneous Coordinates

    The representor of x + d   is TdxTd§ = TdxTd(#) where even Td º (1+½ e¥d) = (½e¥d) = (-½de¥) has Td2 = T2d and Td(#)Td = Td§Td = Td§#Td = 1.
[ Proof : (1-½e¥d)(x+e0e¥x2)(1+½e¥d)   =   (1+½de¥)(x+e0e¥x2)(1-½ de¥)   =   (x+e0+ ½e¥x2 + ½ de¥x + ½ d(-1+e¥0)) (1-½de¥)
    = x+e0+ ½e¥x2 + ½ de¥x - ½d + ½de¥0 -   ½(xde¥ + e0de¥ - ½d2e¥ - ½d2e¥) = x+e0+ ½e¥x2 + ½ de¥x - d -   ½xde¥
    = x-d+e0+ ½e¥x2 - ½(dx+xd)e¥   =   x-d + e0 +½(x-d)2e¥  .]
    We can thus represent translations with spinor transforms that preserve the e0 coordinate (s(x)=1). As with the affine model, we have (Td(#))-1 = T-d-1 = Td (since (#) preserves e¥) for any d Î ÂN. If B is any ÂN multivector representing an "origin centered" transformation of ÂN we can represent the associated operation centered at d with T-dBTd. If B is even we have T-dBTd = B + ½ e¥(d × B).

    Note that (1+½ e¥d)e¥(1-½ e¥d)=e¥ so e¥ is preserved by translations while (1+½ e¥d)e0(1-½ e¥d) = d+e0d2e¥.

    Note that since e¥0 and e¥d anticommute we have (½e¥d) = (½e¥d-½+½e¥0) = (-½)e¥de¥0) = (-½)e¥de¥0) = (-½)cosh(½)+2(½e¥de¥0sinh(½) = ½(1+(-1)) + (½e¥de¥0)(1-(-1)) .

    Consider the transformation Reflectbl(x) = ¯bl(x) - ^bl(x) = 2¯bl(x) - x where bl is a nondegenerate m-blade.
   Reflectbl(x)   = (x¿bl)bl-1 - (xÙbl)bl-1 = ((½(xbl-bl#x) - (½(xbl+bl#x))bl-1 = -bl#xbl-1 = -blxbl#-1 .
    [ b(x) = ^(x,b) - ¯(x,b) corresponds to a rotation of subspace b by p ],
    In particular:
Reflection in the 2-plane (bivector) b is given by Reflectb(x) = -b(x) = -bxb-1 = -bxb§ ;
Reflection in the hyperplane through 0 perpendicular to unit vector n is given by Reflectn*(x) = -nxn = n(x).
[  Verification : -nxn = -(nx)n = -(nx)n-1 = -(n¿x + nÙx)n-1 = -(n¿x)n-1 + (xÙn)n-1 = -¯n(x) + ^n(x) ]
    When n2=-1 we have -nxn = ¯n(x) - ^n(x) .

    To reflect in a displaced subspace we apply a translation bringing the subspace to the origin, reflect, and then reverse the translation.

Null Reflection Rule

    For any nullvector s ,     s§(b) = s=(b) º sbs = 2(s¿b)s        We will call this the null reflection rule.
[ Proof :  Trivially true for scalar b. True for 1-vectors since s(bÙs) = s¿(bÙs) = (s¿b)s - (s¿s)b = s(s¿b)   Þ sbs = s(b¿s+bÙs)=2s(b¿s). True in particular (zerovalued) for b2 = e-u which anticommutes with s. Thus the result holds " b Î e-u . Also true for any bkÎ(e-u)* since then sbk=(-1)kbks. For other bk, ^(bk,(e-u)*) is a blade of grade <k so we can inductively assume the result for it. Whence sbks = s(^(bk,(e-u)*)+¯(bk,(e-u)*)s = s^(bk,(e-u)*)s = 2(s¿^(bk,(e-u)*))s = 2(s¿bk)s  .]
    In particular, s§(b2) = 2(s¿b2)Ùs .

    We also have the null reflected 2-blade rule       sabs = s(aÙb)s = sbsa - sasb .
[ Proof : sabs = sbsa + sabs - (sbs)a = sbsa + sabs - 2(s¿b)sa = sbsa + sa(bs-2(s¿b)) = sbsa + sa(-sb)  .]

Affine Model

    -n(1+e0x)n = -1 + e0nxn = -1 - e0x'
    Reflection in a hyperplane having normal n passing through mn has versor representative
T-mnnTmn = (1+½ e0mn)n(1-½ e0mn) = (1+½ e0mn)(ne0m) = n + me0.

Generalised Homogeneous Coordinates

    Reflection in a hyperplane having normal n passing through mn has versor representative
T-mnnTmn = (1+½ e¥mn)n(1-½ e¥mn) = (1+½ e¥mn)(ne¥m) = n + me¥.

Shears and Strains
Sa,b(x) º x + (x¿a)b with Sa,b-1(y) = y -   (1+a¿b)-1(y¿a)b = S-(1+a¿b)-1a,b(y) constitutes a shear in plane aÙb when a¿b=0 sending aa+bb+gc to aa + (b+aa2)b + gc where c Î (aÙb)* ; and a strain along a when b=la , sending aa+gc to a(1+la2)a  + c where c Î a* .
    Any linear transformation can be expressed as a succession of reflections and shears.

    More generally, defining Sa,b(x) º x + bÙ(a¿x) we have
    Sa,b(xk)ÙSa,b(c) = Sa,b(xkÙc)
[ Proof :   xkÙc + bÙ(a¿xk)Ùc  + xkÙbÙ(a¿c)   =   xkÙc + bÙ((a¿xk)Ùc  + (-1)kxk(a¿c))   =   xkÙc + bÙ(a¿(xkÙc)) by the Expanded contraction rule a¿(bkÙc) = (a¿bk)Ùc + (-1)kbkÙ(a¿c).  .] and Sa,b-1(y) = S-(1+a¿b)-1a,b(y) for a¿b¹-1.

    Sc,d(Sa,b(x))   =   x + bÙ(a¿x) + dÙ(c¿(x + bÙ(a¿x))   =   x + bÙ(a¿x) + dÙ(c¿x) + dÙ(c¿(bÙ(a¿x)))

    Sc,d(Sa,b(x))   =   x + b(a¿x) + d(c¿x + (c¿b)(a¿x))   =   x + (b+(c¿b)d)(a¿x) + d(c¿x)


    Let a and n be N-dimensional vectors with a2 = n2 = ±1. If a¹-n, the rotation with axis through 0 taking n to a can be considered as a reflection in the hyperplane r.(a+n) = 0 followed by reflection in the hyperplane r.a = 0. We can thus express the rotation geometrically as
    Rotaten,a(x) = a-1((n+a)~)-1x((n+a)~)a = a((n+a)~)x((n+a)~)a = Rn®axRn®a§ ,
    Now Rn®a = a(n+a)~ = |n+a|-1(1 + cosq + sinq(aÙn)~) = (½q(aÙn)~) is a unit 1-rotor, with q the scalar angle subtended between n and a.
    Since Rn®aRn®a§ = 1 we have Rotaten,a(x)Rotaten,a(y) = Rotaten,a(xy) so we can use Rotaten,a to rotate multivectors.
    If a2 = n2 = 1 we have a(n+a)~ = (1+an) / Ö(2+2a.n) . If a2 = n2 = -1 we have a(n+a)~ = (-1+an) / Ö(2-2a.n) .
    Any sequence of k such rotations can be represented by a k-rotor given by the geometric product of the contributory 1-rotors, this product being itself an even unit versor. Thus a rotor can also specify an N-D orientation. The operation a-1(b) = aba-1 is called rotation by rotor a. Geometrically, a rotation corresponds to a sequence of 1-rotor rotations and so can also represent the orientation of a rotated rigid N-D body or reference frame.
    For N=3, any bivector is a 2-blade so every even multivector has form a + aÙb = a((a-a¿b)a/a2 + b) , hence every even 3D multivector is a 2-versor and a 1-rotor.

    For general N we have    na   =   (a.n)  + nÙa.   =   (a.n)  + |nÙa|b where b =  (nÙa)~ = (n×a)~ an even unit bivector with b2=-1.
    Writing f = cos-1(n*a) for the angle subtended by unitvector n and a we have na = |a|( cosf + b sinf) = |a|ebf
    We can thus express any vector a in the form (nn)a = n(na) = |a|ne(nÙa)~f where n is a given ("base direction") unit vector.

    Let b be a unit bivector (b2 = -1).  Since b=-b#-1 we have -byb = b(y) =  ^(y,b)-¯(y,b) and b¯(y,b) = b(y¿b) = -y¿b.
Rb,q, f(y)º eqby efb§ = ( cosq+ sinqb)y( cosf- sinfb) = ... [algebra ommitted] ...
= cos(q+f)¯(y,b) - sin(q+f)y¿b + cos(q-f)^(y,b) - sin(q-f)^(y,b)b-1 .

    In particular, Rb,q,q(y) = cos(q+f)¯(y,b) - sin(q+f)y¿b     which is a rotatation of 2-plane b by 2q (preserving b*). Thus the spinor eqb rotates 2-plane b by angle 2q when used as a rotor.

3D Rotations

    In N=3, all rotors are unit magnitude scalar-bivector pairs. There is an equivalence between orientations and scaled-directions. Any orthornmal coordinate frame can be characterised by three Eulerian angles or by a unit Â3 rotor. Because there are as many bivectors as 1-vectors, orientations and scaled-directions require the same number of variables to   specify (eg. three Eulerian angles). When multiplying rotors, the result as an even multivector is constrained to a 3D space.

    The rotor for a rotation by angle |a| about axis 0 + la is thus the q=½ |a| spinor for the 2-plane a* = -ia.
    Ra, 0 = eia = cos(½|a|) - ia/|a| sin(½|a|).
    All 3D rotations preserving 0 have this form.
    Applying such a rotor (sparsity 4) twice to a vector (sparsity 5) by twice calling a general geometic product routine incurrs 28 nonzero multiplications. Computationally, this operation is precisely equivalent to a rotation by quaternian so a specific routine requires 19Mult. Both of these compare poorly with the 9Mult for application of a rotation matrix.
    However, 3D rotors provide the same advantages over rotation matrices as quaternions do, namely:

4D Rotations

    For N=p+q=4 we have 4 1-vectors but 6 bivectors . As <0;2;4>-vectors, even versor 4D rotors have a pseudoscalar component, even multivectors have eight degrees of freedom, seven if normalised. If i2=-1 (requires q>0) we can express them as a <0;2>-vectors in Cp,q-1 + as a + bb = ½(a+b) + ½(a-b)b for plussquare unit 2-blade b , with rapidly computable logarithm (a+b)½(1+b) + (a-b)½(1-b)   where the idempotents ½(1±b) commute with eachother [  The scalar part being determined by required unit magntude from bivector and pseudoscalar ]   
    Rb,q, f(y) as defined above   corresponds to a simultaneous rotation of 2-plane b by q+f and the orthogonal 2-plane b* by q-f. All 4D rotations preserving 0 have this form. In particular,
    x ® eqbx = Rb,q,0(x) corresponds to rotation of 2-planes b and b* by q,
    x ® xeqb = Rb,0,q(x) corresponds to rotation of 2-planes b by q and b* by -q,
    We can represent any unitvector a Î Â4 by the <0;2>-vector 2-versor ae4 = a¿e4 + aÙe4    = eaa º (aa) where a=(aÙe4)~ is a unit 2-blade and a =   cos-1(a¿e4). Note that e4 anti-commutes with a. We also have a = (a½a)e4(a§½a)
    This represents each point of the unit 3-sphere (ie. the unit sphere in 4D) by a zero-centred rotation that takes the "preferred" or "polar" direction e4 to a.
    Once again we encounter quaternions, this time as rotational representors for 4D points.
      Letting a=(aÙe4)~ and a= cos-1(a¿e4) so that ae4 = (aa) we have
Rb,b, q,f(eaa) =( cosq+ sinqb)ae4( cosf- sinfb) = (( cosq+ sinqb)a( cosf+ sinfb))e4

    Let ebb=be4 and egc=ce4 . We can express ebb as (edd)§(c) where edd=de4 . We then have
Rb,c,b,g(y)º ebdyegc = edd egc edd§ yegc = eddRc,c,g,g (edd§y) = eddRc,g,-g (edd§y)
corresponding to a rotation of 2-planes d and d* by -d, followed by a rotation of 2-plane c* by 2g, followed by rotations of d by d and d* by -d.

Affine Model

    Rotation by angle |a| about axis c+la is given by Ra, c = Ra, 0 + ½ e0(c × Ra, 0)

Generalised Homogeneous Coordinates

    Rotation by angle |a| about axis c+la is given by Ra, c = Ra, 0 + ½ e¥(c × Ra, 0) = (-½ia) + ½e¥(c × (-½ia)) = (-½ia) + ½e¥(c ×  sinh(-½ia)) .

    Inversion of ÂN is only representable as a Lorentz form in ÂN+1,1.

Generalised Homogeneous Coordinates
    We have seen that the ÂN+1,1 1-vector s=cr2e¥ represents a sphere or radius r centre c in ÂN. What happens if we reflect in such a 1-vector? It can be shown that -sxs =   -i2s*xs* = ((x-c)/r)2(g(x)+e0g(x)2e0) where g(x)=(r2/(x-c)2)(x-c)+c is known as inversion in the sphere centre c radius r. Inversion takes c+ln to c + l-1r2n so mapping the exterior of the sphere into its interior, and vice versa.
    Thus if ak+2 represents a k-sphere then -ak+2xak+2#-1 prserves xÎak+2 and acts as reflection (inversion) in the k-sphere over k-plane xÎe¥Ùak+2 . It also embodies a reflection in the k-plane e¥Ùak+2 .
    This also holds if ak+2 representsd a k-plane and we can formulate the notion of a generalised rotation consisting of an even number of successive reflections in k-planes and k-spheres. In general, blades ak+2 and Bl+1 will intersect and it is natural to think of ak+2 Ç Bl+1 as the "axis" of rotation (ak+2Bl+1)#-1 . If there is no intersection then the trsnform may includes a translation component, eg. when successively reflecting in two parallel hyperplanes.  

    A transversion by displacement d is an inversion in the unit hypersphere e+*, followed by translation by d, followed by another inversion in e+*. In GHC we can form this as Lorentz form x ® (e0e¥)(1+½e¥d)(e0e¥) x (e0e¥)(1-½e¥d)(e0e¥)   =   (1+½e0d) x (1-½e0d) so transversion by d has Lorentz spinor form (1+½e0d)§=(½e0d)§.

Dilation and Involution
    "Rescaling" of ÂN is only representable as a Lorentz form in ÂN+1,1.

Generalised Homogeneous Coordinates
    Consider (fe¥0) º ee¥0f º  cosh(f)+ sinh(f)e¥0.
    (fe¥0)§(e0) º ee¥0fe0e-e¥0f = ( coshfsinhf)e0coshfsinhfe¥0) = ( coshfsinhf)( coshfsinhf)e0 = e2fe0. Similarly ee¥0fe¥e-e¥0f = e-2fe¥.
    Thus ee¥0fxe-e¥0f = ee¥0f(x+e0x2e¥)e-e¥0f   =   x + e2fe0 + ½x2e-2fe¥   =   e2f( e-2fx + e0 + ½(e-2fx)2e¥)   =   e2f¦(e-2fx).
    Hence the multivector Dl = (-½ e¥0 ln(l)) = ½l((1+l) + e¥0(1-l)) corresponds to the dilation g(x) = lx for l > 0.
    Since e¥0¦(x)e¥0 = x - e0 - ½ x2e¥ = -¦(-x) we can dilate by -l < 0 with D-l = e¥0(-½ e¥0 ln(l))   =   ½l((1-l) + e¥0(1+l)) .

Perspective Projection
    Perspective projection into the z=h plane p ® h(e3¿p)-1p can be expressed in GHC somewhat inefficiently as p ® e¥¿( (e¥Ùe0Ùp)Ç(e¥Ù(e0+he3)Ù(e3*))) =   -e0¿((e¥0p)Ç(e¥(e0+he3)(e3*))) but noting that perspective projection into z=h corresponds to dilation by h(e3¿p) then assuming e3¿p>0 perspective projection is provided by the rotor application (-½e¥0((e0¿p)-1(e3¿p)-1h))§ provided h>0 -=(-½e¥0(h - (e0¿p) - (e3¿p))§ provided h>0 or (e¥0(-½e¥0((e0¿p)-1(e3¿p)-1h)))§ when h<0. Note that unlike all the perviosuly discussed transformation rotors, this one depends on the p to which it is to be applied.

Summary of GHC Transformations
Transform  [ A^(x) º AxA^ ] Effects(x)
e¥d)§Translation by dÎi1
e0d)§Transversion by dÎi1
(-½fe¥0)§Dilation (scaling) by f(-f)
(e¥0(-½fe¥0)))§Dilation (scaling) by -f-(-f)
((½q(aÙn)~))§Minimal rotation taking n to a1
((-½iqa) + ½e¥(c × (-½iqa))§ Rotatation of q about axis c+la (assuming a2=1)1
R§Rotatation by even rotor R with RR§=11
-(ak+2)#-1Reflection in k-sphere or k-plane ak+2-1

Spinor Transforms
    We see above that transforms of perhaps greatest interest are spinor transforms of the form A-↑(x) º (A)-1(x) º Ax(-A) where A is usually even and often satisfies (-A) = (A)^ for some reversing conjugation ^ so that A-↑ = (A)^ .

    If b2=b^2=-1 then (qb)x(qb^)   =   ( cosq+ sinqb)x( cosq+ sinqb^)   =   ( cosq)2x + ( sinq)2bxb^ + sin(2q)½(bx+xb^) which we can regard as the natural "subdivision" or "interpolation" of the discrete rotation of x at q=0 to bxb^ at qp.
    When ^ is negation we have (qb)x(-qb)   =   ( cosq)2x - ( sinq)2bxb + sin(2q)(b×x)   =   x+ + cos(2q)x- + sin(2q)bx-   =   x+ + (2qb)x- where x+ commutes with b and x- anticommutes with b and x=x++x- .

    If a2=la and b2=mb then c(a)=(ab)a(-ab)   =   (1+m-1((am)-1)b)a (1+m-1((-am)-1)b) for central a has c2= lc independant of m and a.
    When l=m=1 this is
    c(a)= (1+(a-1)b)a(1+((-a)-1)b   =   a+(a-1)ba+((-a)-1)ab + (a-1)((-a)-1)bab

    If ^ is a reversing conjugation that preserves a ,b, and a then c(a)^ = c(-a) so
    ½(c(a)+c(a)^)   =   a+( cosha-1)(ba+ab)+ (a-1)((-a)-1)bab and ½(c(a)-c(a)^)   =    sinh(a)b×a .

    For large scalar a we thus have c(a) decomposing as approximately (a)(½(ba+ab) - bab) preserved by ^ and (a)b×a negated by ^ .

    For a = ai we have c(a)   =   a + ( cosa-1)(ba+ab) + sinai(ba-ab) + 2(1- cosa)bab .

Example (l(b+e¥d+ge¥0))-1

      Recall from our discussion of exponentiation that e12+e¥(de3+fe1) + ge¥0   =   (g+Ai)B     where unit A = e3 + dg-1e¥ (so Ai=e¥0e12+dg-1e¥e123) and B = e¥Ùb = e¥Ù(e0+b) (with b   =   f(1+g2)-1(ge1-e2) + dg-1e3     =   f(ge1-e2)~ + dg-1e3 ) commute ; providing idempotised form (g+i)½(1+A)½(1+B) + (g-i)½(1-A)½(1+B) - (g+i)½(1+A)½(1-B) - (g-i)½(1-A)½(1-B) . This yields

    (l(e12+fe1+de3+ge¥0)) =  cosh(lg) cos(l) +  sinh(lg) sin(l)Ai +  sinh(lg) cos(l)B + ((lg) sin(l) + (-lg) cos(l))ABi
    where AB = e¥Ù(e0+b)Ùe3    ;     ABi = e¥Ùb' + e12 where b' = f(e1 + ge2)~ = ¯e12(b)e12 .
    For e12+e¥(de4+fe1) + ge¥0 and i = e¥0e12345 we have 3-blade A = e345 + dg-1e¥e35 (so Ai= e¥0e12 + dg-1e¥e124) and 2-blade B = e¥Ùb where b=f(ge1 - e2)~ + dg-1e4. so that AB = e¥ÙbÙe345 and ABi remains the same.

    For the more general (lw+e¥d+ge¥0) we have Ai = e¥0w + g-1e¥(wÙd) = (e¥0 + g-1e¥^w(d))w ; B = e¥Ù(e0+b) where b = (1+g2)-1(g+w)¯w(d) + g-1^w(d) ; ABi = (1 + e¥¯w(b))w .

    Thus, if B» = -B then for l >> 1 , (½l(b+e¥d+ge¥0))» will tend to amplify  ^B(x) and reduce ¯B(x) . so driving x towards a multiple of ^B(x)½(1-B) , wheras l << -1 drives towards a multiple of ^Bx½(1+B). Here ¯ and ^ denote decomposition x = ¯q(x) + ^q(x) into parts commuting and anticommuting with q respectively.

    For g=0 we have (e12 + de¥3 + fe¥1)   =   i(1-dA)B   =   (i+de¥)B where i=e¥0e123 ; A = e¥e123 ; and b = e¥03 - fe¥e23 = e¥Ù(e0-fe2)Ùe3 , or i=e¥0e12345 and B = e¥0e345 - fe¥e2345 = b* + (e¥¯b(d))* for minussquare e4 replacing plussquare e3.

Implementation via idempotentised forms

    Suppose we have a = a00U00 +   a10U10 + a01U01 + a11U11
    = (a00+a10+a01+a11) + (a00-a10+a01-a11)A + (a00+a10-a01-a11)B + (a00-a10-a01+a11)AB
where U00=½(1+A)½(1+B) ; U10=½(1-A)½(1+B) ; U01=½(1+A)½(1-B) ; U11=½(1-A)½(1-B) for commuting unit squared A and B and complex aij for some i commuting with A and B.

    Suppose further that we seek to evaluate a§(x) º axa§ , equivalent to axa§# if a is even.

    In both the above example cases, B negates under §# and § and (anti)commutes with ^(p) ( ¯B(p) ) while A (anti)commutes with ¯A(x) (^A(x)) is preserved by §# and negated by § in the 3-blade case (vice versa in the 1-blade case). 7-blade i is presrved  by §# but negated by §.
    Since for even a we can choose either § or §# it is natural to favour », the reversing conjugation that negates our choice of i, which will tend to preserve A.

    (a00U00 +   a10U10 + a01U01 + a11U11)x (a00U00 +   a10U10 + a01U01 + a11U11)»
    = (a00U00 +   a10U10 + a01U01 + a11U11) (¯B(x) + ^B(x)) (a00»U01 +   a10»U11 + a01»U00 + a11»U10)
    = ( ¯A¯B(x)(a00U00 +   a10U10 + a01U01 + a11U11) + ^A¯B(x)(a00U10 +   a10U00 + a01U11 + a01U01)
        + ¯A^B(x)(a00U01 +   a10U11 + a01U00 + a11U10)     + ^A^B(x)(a00U11 +   a10U01 + a01U10 + a11U00)
        ) ( a01»U00 + a11»U10 + a00»U01 + a10»U11 )
    = ¯A¯B(x) (a00a01»U00   + a10a11»U10 +a01a00»U01 + a11a10»U11) + ^A¯B(x) (a10a01»U00   + a00a11»U10 +a01a00»U01 + a01a10»U11)
    + ¯A^B(x) (|a01|2U00 + |a11|2U10 + |a00|2U01 + |a10|2U11) + ^A^B(x) (a11a01»U00   + a01a11»U10 +a10a00»U01 + a00a10»»U11)
    = ¯A¯B(x)L00 + ^A¯B(x)L10 + ¯A^B(x)L10 + ^A^B(x)L11     for four multivectors L_ij determined by a independant of x.
    = ( a00a01»¯A¯B(x) + a10a01»^A¯B(x) + |a01|2¯A^B(x) + a11a01»^A^B(x) )U00 +  (...)U10 + (...)U01 + (...)U11
    Here ¯A¯B(x)L00  abbreviates (¯A(¯B(x)))L00 .

    For 1-vector A we know ¯A(y) commutes with both A and B so that ^B¯A(y) = 0.

    If we are going to compute axa§ for multiple x we can compute and store the L00, L10, L10 and L11 and use them as right geometric multipliers for the four ¯A¯B(x) , ^A¯B(x), ¯A^B(x), ^A^B(x).
    Because we did the ^A and ¯A second, if we expand the L_ij out in terms of 1,i,A,B,etc. then the products with the A reduce to either ¿ or Ù.

    Alternatively, for k-vector x we can favour four <k;G-k>-vector "idempotent coordinate" left-multipliers for the U00,U10,U01,U11 like a00a01»¯A¯B(x) + a10a01»^A¯B(x) + |a01|2¯A^B(x) + a11a01»^A^B(x) where G is the grade of i; easily additively recombined into <k,G-k>-vector  left multipliers for 1, A, B, and AB.

    We can add and subtract such representations by adding the seperate coordinates, but multiplications are complicated by the potential non (anti)commutivity of the ¯A¯B(x), ¯A^B(x),... with B.

Next : Multivector Arcana

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Copyright (c) Ian C G Bell 1998, 2014
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