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

Bivector Transform
    ¦(a)ºa¿b₂ maps one 1-vector a "into" 2-vector b₂ and rotates by ½p "within" b₂. For pure 2-blade b₂=e₁₂ we have ¦(å_i aⁱe_i)=-a²e₁+a¹e₂. It provides 1-vector transformations
    ¦^k(a) º ¦^k-1(a)¿b₂ =2^-kå_i=0^k kC_i(-1)ⁱb₂ⁱab₂^k-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 1-representors for points and structures in Â^N. 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.

    The involution ^# ensures that a_k(x) = x for any x with x¿a_k=0 (ie. "perpendicular" to a_k with xa_k=xÙa_k=(-1)^ka_kx) but means that A(1) = A(A^#)^-1 so scalars are preserved for even A, negated for odd A, and multiplied by potentially nonscalar A(A^#-1) for general 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...Ùa_k) = ± A(a₁)ÙA(a₂)Ù...A(a_k) with the negative case arising when both A and k are odd.
[ Proof :  Follows from aÙb_k = ab_k + b_k^#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^§) . 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 Â^N always has a unit k-rotor representation with k£N.

Higher Dimensional Embeddings
    Suppose we have extended Â_N into Â_N+p,q,r with an associated point embedding ¦ : Â^N ® Â^N+p,q,r. Any multivector A in Â_N+p,q,r represents a transformation G_A,¦ : Â^N ® Â^N defined by G_A,¦(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)¦(G_A,¦(x)) where s(x) is a scalar.
    The involution ^(#) is the "interior" or "unextended" Â_N involution operating within Â_N+p,q,r rather than the proper involution ^# of Â_N+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 Â_N
    = ¦(A(x))     if A is odd or even .

Translations
    Translations in Â^N can only be represented in Â_N 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 Â_N hyperplanes through 0 and then ½d will suffice.

Affine Model

    The representor of x - d is 1 + e₀(x - d) = T_d(1+e₀x)T_d where T_d=(1 - ½ e₀d)=(-½e₀d)^↑ has T_d² = T_2d and T_d^(#)T_d = T_d^§T_d = T_d^§#T_d = 1.
[ Proof : (1-½ e₀d)(1+e₀x)(1-½ e₀d) = (1-½ e₀d)(1+e₀x-½ e₀d) = (1+e₀x-½ e₀d - ½ e₀d) = 1 + e₀(x - d)  .]
    We have T_d^-1 = (1 + ½ e₀d) =  T_-d and T_d^(#) = T_-d = T_d^§ so we have T_d^{# -1} = T_d for all d in Â_N.

    If B is any Â_N multivector representing an "origin centered" transformation of Â^N we can represent the associated operation centered at d with T_-dBT_d . If B is even we have
T_-dBT_d = (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 T_dxT_d^§ where even T_d º (1+½ e_¥d) = (½e_¥d)^↑ = (-½de_¥)^↑ has T_d² = T_2d and T_d^(#)T_d = T_d^§T_d = T_d^§#T_d = 1.
[ Proof : (1-½e_¥d)(x+e₀+½e_¥x²)(1+½e_¥d)   =   (1+½de_¥)(x+e₀+½e_¥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 (T_d^(#))^-1 = T_-d^-1 = T_d (since ^(#) preserves e_¥) for any d Î Â_N. If B is any Â_N multivector representing an "origin centered" transformation of Â^N we can represent the associated operation centered at d with T_-dBT_d. If B is even we have T_-dBT_d = 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+e₀+½d²e_¥.

Reflections
    Consider the transformation Reflect_bl(x) = ¯_bl(x) - ^_bl(x) = 2¯_bl(x) - x where b_l is a nondegenerate m-blade.
   Reflect_bl(x)   = (x¿b_l)b_l^-1 - (xÙb_l)b_l^-1 = ((½(xb_l-b_l^#x) - (½(xb_l+b_l^#x))b_l^-1 = -b_l^#xb_l^-1 = -b_lxb_l^#-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 Reflect_b(x) = -b(x) = -bxb^-1 = -bxb^§ ;
Reflection in the hyperplane through 0 perpendicular to unit vector n is given by Reflect_n^*(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 b_kÎ(e_-u)^* since then sb_k=(-1)^kb_ks. For other b_k, ^(b_k,(e_-u)^*) is a blade of grade <k so we can inductively assume the result for it. Whence sb_ks = s(^(b_k,(e_-u)^*)+¯(b_k,(e_-u)^*)s = s^(b_k,(e_-u)^*)s = 2(s¿^(b_k,(e_-u)^*))s = 2(s¿b_k)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_-mnnT_mn = (1+½ e₀mn)n(1-½ e₀mn) = (1+½ e₀mn)(n+½ e₀m) = n + me₀.

Generalised Homogeneous Coordinates

    Reflection in a hyperplane having normal n passing through mn has versor representative
T_-mnnT_mn = (1+½ e_¥mn)n(1-½ e_¥mn) = (1+½ e_¥mn)(n+½ e_¥m) = n + me_¥.

Shears and Strains
S_a,b(x) º x + (x¿a)b with S_a,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 S_a,b(x) º x + bÙ(a¿x) we have
    S_a,b(x_k)ÙS_a,b(c) = S_a,b(x_kÙc)
[ Proof :   x_kÙc + bÙ(a¿x_k)Ùc  + x_kÙbÙ(a¿c)   =   x_kÙc + bÙ((a¿x_k)Ùc  + (-1)^kx_k(a¿c))   =   x_kÙc + bÙ(a¿(x_kÙc)) by the Expanded contraction rule a¿(b_kÙc) = (a¿b_k)Ùc + (-1)^kb_kÙ(a¿c).  .] and S_a,b^-1(y) = S_{-(1+a¿b)^-1a,b}(y) for a¿b¹-1.

    S_c,d(S_a,b(x))   =   x + bÙ(a¿x) + dÙ(c¿(x + bÙ(a¿x))   =   x + bÙ(a¿x) + dÙ(c¿x) + dÙ(c¿(bÙ(a¿x)))

    S_c,d(S_a,b(x))   =   x + b(a¿x) + d(c¿x + (c¿b)(a¿x))   =   x + (b+(c¿b)d)(a¿x) + d(c¿x)

Rotations

    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
    Rotate_n,a(x) = a^-1((n+a)~)^-1x((n+a)~)a = a((n+a)~)x((n+a)~)a = R_n®axR_n®a^§ ,
where R_n®a = a(n+a)~ = (q(aÙ(n+a)^~))^↑ is a unit 1-rotor.
    Since R_n®aR_n®a^§ = 1 we have Rotate_n,a(x)Rotate_n,a(y) = Rotate_n,a(xy) so we can use Rotate_n,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|e^bf
    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.
    Consider

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, R_b,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 e^qb 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.
    R_{a, 0} = e^-½ia = cos(½|a|) - ia/|a| sin(½|a|).
    All 3D rotations 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:

• Combining two rotatations requires the geometric product of two rotors (16Mult) which compares favourably with a matrix product (27Mult).
• The axis and scale of a rotation can be trivially computed as iR; the "decoding" of a rotation matrix is far harder.
• -R_a = -e^-ia/2 = - cos(|a|/2) + ia/|a| sin(|a|/2) = e^ipa/|a|e^-ia/2 = e^{i(2p/|a| - 1)a/2} which is a rotation about a by 2p-|a|.
  But (-R_a)b(-R_a)^§ = R_abR_a^§ so -R_a encodes a rotation with the the same "final" effect as R_a but having opposite sense, (ie."turning the other way"). Rotation matrices cannot do this.
• Easier extension to N > 3 dimensions.

    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 C_{p,q-1 +} as a + bb = ½(a+b) + ½(a-b)b for plussquare unit 2-blade b , with rapidly computable logarithm ln()(a+b)½(1+b) + ln()(a-b))½(1-b)   where 1±b commute with eachother [  The scalar part being determined by required unit magntude from bivector and pseudoscalar ]   
    R_{b,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 have this form. In particular,
    x ® e^qbx = R_b,q,0(x) corresponds to rotation of 2-planes b and b^* by q,
    x ® xe^qb = R_b,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 scalar,bivector combination ae₄ = a¿e₄ + aÙe₄    = e^aa º (aa)^↑ where a=(aÙe₄)~ is a unit bivector 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 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
	=	Rb,q,-f(a)e4

    Let e^bb=be₄ and e^gc=ce₄ . We can express e^bb as (e^dd)_§(c) where e^dd=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 R_{a, c} = R_{a, 0} + ½ e₀(c × R_{a, 0})

Generalised Homogeneous Coordinates

    Rotation by angle |a| about axis c+la is given by R_{a, c} = R_{a, 0} + ½ e_¥(c × R_{a, 0}) = (-½ia)^↑ + ½e_¥(c × (-½ia)^↑) = (-½ia)^↑ + ½e_¥(c ×  sinh(-½ia)) .

Inversion
    Inversion of Â^N is only representable as a Lorentz form in Â_N+1,1.

Generalised Homogeneous Coordinates
    We have seen that the Â_N+1,1 1-vector s=c-½ r²e_¥ represents a sphere or radius r centre c in Â^N. What happens if we reflect in such a 1-vector? It can be shown that -sxs =   -i²s^*xs^* = ((x-c)/r)²(g(x)+e₀+½ g(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 + (r/l)²n so mapping the exterior of the sphere into its interior, and vice versa.
    Thus if a_k+2 represents a k-sphere then -a_k+2xa_k+2^#-1 prserves xÎa_k+2 and acts as reflection (inversion) in the k-sphere over k-plane xÎe_¥Ùa_k+2 . It also embodies a reflection in the k-plane e_¥Ùa_k+2 .
    This also holds if a_k+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 a_k+2 and B_l+1 will intersect and it is natural to think of a_k+2 Ç B_l+1 as the "axis" of rotation (a_k+2B_l+1)_#-1 . If there is no intersection then the trsnform may includes a translation component, eg. when successively reflecting in two parallel hyperplanes.  

Transversion
    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 ® (e₀-½e_¥)(1+½e_¥d)(e₀-½e_¥) x (e₀-½e_¥)(1-½e_¥d)(e₀-½e_¥)   =   (1+½e₀d) x (1-½e₀d) so transversion by d has Lorentz spinor form (1+½e₀d)_§=(½e₀d)^↑_§.

Dilation and Involution
    "Rescaling" of Â^N is only representable as a Lorentz form in Â_N+1,1.

Generalised Homogeneous Coordinates
    Consider (fe_¥0)^↑ º e^e_¥0f º  cosh(f)+ sinh(f)e_¥0.
    (fe_¥0)^↑_§(e₀) º e^e_¥0fe₀e^-e_¥0f = ( coshf+ sinhf)e₀( coshf- sinhfe_¥0) = ( coshf+ sinhf)( coshf+ sinhf)e₀ = e^2fe₀. Similarly e^e_¥0fe_¥e^-e_¥0f = e^-2fe_¥.
    Thus e^e_¥0fxe^-e_¥0f = e^e_¥0f(x+e₀+½ x²e_¥)e^-e_¥0f   =   x + e^2fe₀ + ½x²e^-2fe_¥   =   e^2f( e^-2fx + e₀ + ½(e^-2fx)²e_¥)   =   e^2f¦(e^-2fx).
    Hence the multivector D_l = (-½ e_¥0 ln(l))^↑ = ½l^-½((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^-½((1-l) + e_¥0(1+l)) .

Summary of GHC Transformations

Transform  [ A^(x º AxA^ ]	Effect	s(x)
(½e¥d)↑§	Translation by dÎi	1
(½e0d)↑§	Transversion by dÎi	1
(-½fe¥0)↑§	Dilation (scaling) by f↑	(-f)↑
(e¥0(-½fe¥0)↑))§	Dilation (scaling) by -f↑	-(-f)↑
((-½iqa)↑ + ½e¥(c × (-½iqa)↑)§	Rotatation of q about axis c+la (assuming a2=1)	1
R§	Rotatation by even rotar R with RR§=1	1
-(ak+2)#-1	Reflection 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 A^↑x(-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" of the discrete rotation of x at q=0 to bxb^{^} at q=½p.
    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) .

    (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

    With AB = e_¥Ù(e₀+b)Ùe₃    ;     ABi = e_¥Ùb' + e₁₂ where b' = f(e₁ + ge₂)^~ = ¯_w(b)w .

    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 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 e₁₂+e_¥(de₄+fe₁) + ge_¥0 and i = e_¥0e₁₂₃₄₅ we have 3-blade A = e₃₄₅ + dg^-1e_¥e₃₅ (so Ai= e_¥0e₁₂) and 2-blade B = e_¥Ùb where null b is the embedding of b = gf(1+g²)^-1e₁ - f(1+g²)^-1e₂ + dg^-1e₄ . For large g, b » e_¥Ù(e₀ + g^-1(fe₁+de₄) while for small g we have B » e_¥Ù(e₀ + fe₂ + dg^-1e₄) .

    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 a_ij 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.

Multivector Shapes


k-tubes
    Our formal discussion of paths, surfaces, and other M-curves must await consideration of   multivector calculus since we require notions of smoothness and continuity, but we can nontheless describe and manipulate a wide variety of closed "blobby shapes" of interest to programmers as k-blade trajectories through U_N^%.  A toroidal surface may be constructed by sweeping a circle around a circle , for example, or a cylinder by translating a cirle along a line. More generally, we can define a k-tube as a union È_lÎ[a,b] a_k+1(l) of multiple (k-1)-spheres , with an implied intersection with the U^N% horosophere. Note that this is not the cummulative join (which is likely to be i) but a set of hyperphere duals (restricted to a point set). If we intersect È_lÎ[a,b] a_k+1(l) with the U_N^% horosphere and disembedd we recover the set of U^N points which lie in at least one a_k+1(l).    

    A k-tube is thus the k-curve formed by sweeping a (k-1)-sphere through U^N , varying its radius, centre, and orientation. Setting a_k+1(a)=a_k+1(b) or r(a)=r(b)=0 ensures a closed k-curve.         Thus:
    A 0-tube is a 1-curve, ie. a path through U^N pointspace.
    A 1-tube is the twin trajectories formed by sweeping a bipoint through U^N , varying its radius, centre, and orientation.
    A 2-tube is the tubelike surface formed by sweeping a circle through U^N , varying its radius, centre, and orientation according to taste. If a_k+1(b)=a_k+1(a) we have a closed surface such as a torus, otherwise we have "open ends" such as for an uncapped cyclinder. 2-tubes are useful as "enclosing surfaces" for more complicated objects.
    A 3-tube is the filled tubelike volume formed by sweeping a 2-sphere through U^N , varying its radius, centre, and orientation. 3-tubes are closed irresepective of whetere the 2-sphere tarverses a closed path and like 2-tubes that have been "filled in" and the ends.

    If we vary l discretely rather than smoothly from a to b the k-tube becomes a union of (k-1)-spheres, with gradient discontinuites in the enclosing (k-1)-curve which may fragment into seperate sections. We might have K hyperspheres defining a "collision envelope" for a more complicated object, for example.

    b Ç È_lÎ[a,b] a_k+1(l)   =   È_lÎ[a,b]  (bÇa_k+1(l)) .
    For the sake of definiteless, let us take k=1 and suppose  {a_k+1(l)} representes a deformed k-curve while b is a 3-blade representing a line or a circle. Each a_k+1(l) is an undeformed 1-sphere that can intersect in the horosphere with b at most twice. From a programming standpoint, calculating the intersection of b with the 2-tube ammounts to seeking all the l solving (bÇa_k+1(l))²=0 and even this is insufficient to pick up the exceptional double intersections when b lies along a diameter of a_k+1(l) . Even though the meet bÇa_k+1(l) for a nonintersective l provides useful measures and directions for seeking a solution, the problem of construting such tubular meets  is potentially computationally intensive. The discreteness of the geometric interpretation (the ray either intersects a circle or it doesn;t) means that we have to find the exact solving l, and (depending on how wobbly the k-tube is) there might be a large integer number of solution l within [a,b]. If {a_k+1(l)} is closed there will be an even number of intersection points which we can think of as arising in "entry and exit" bipoints, but we might only be actually interested in one (the closest to a ray source, for example).
    If a_k+1(l) = ¦_l(_Akp10) for some invertible outtermorphic tranform ¦_l   we have

    bÇ È_lÎ[a,b] ¦_l(_Akp10)   =   È_lÎ[a,b] (¦_l( ¦_l^-1(b)_intrsc_Akp10) ) which replaces "tracing" a straight 1-curve past a k-tube by tracing a "wobbly" 1-curve past a k-sphere.

    More generally we might intersect l-sphere b_l+1  with k-tube {a_k+1(l)} to obtain a collection of (k-(N-l))-curvea which we can reformulate as intersecting an l-tube with a single k-sphere. More generally still, we can intersect a k-tube with an l-tube
    È_lÎ[a,b] ¦_l(_Akp10) Ç È_mÎ[c,d] g_m(_Blp10)   =   È_l,m ¦_l(_Akp10) Ç g_m(_Blp10)     taking the union accross all possible combinations of l and m   . This is an intensive computation but we can generalise still further, constructing M-curves by sweeping (M-1)-tubes or more complicated (M-1)-curves through U_N^%.

    The problem is of course that while the k-tube   È_lÎ[a,b] a_k+1(l) is well defined symbolically and may be algebraically manipulated as above,   the representation and manipulation of k-tubes requires linked lists and parameterisations of multivectors rather than simple multivectors. We might for example store K+1 "spotvalue" (k+1)-blades for l = jK^-1 .

    Clearly k-tubes can intersect themselves. The l parameter is not interpreted as a "time", all the a_k+1(l) are considered to "exist together" and a given U^N point can lie within more than one of them. A d-extruded k-tube is a k-tube whose centre c(l) satisfuies d^-1¿c'(l) > 0 and b(l) Î d^* . The k-sphere is thus travelling "along" d and aligned perpendicular to it so that all points in a particular a_k+1(l) have shared d coordinate unique to that l. Typically we might have d=e₄ and regard a_k+1(l) as representing the position and orientation of a spacial k-sphere at time t=l.

Transforming k-blades
    Suppose a₁ and a₂ are distinct k-blades in U_N. We seek to construct a "rotation" ¦_l(a₁) taking a₁ at l=0 to a₂ at l=1 that takes (½d)-blade (a₂Da₁)Ça₁ into (½d)-blade (a₂Da₁)Ça₂ but does not effect directions in (a₁Da₂)^* and during which ¦_l(a₁) remains within the join a₁Èa₂.
    We thus seek a transformation of the form ¦_l(x) = f^lxf^-l for some even multivector f within a₁Da₂ . Insisiting f be even ensures that it commutes with everything in (a₁Èa₂)^* and in a₁Ça₂ so that fxf^-1 = x for x Î (a₁Da₂)^* . Thus our "rotation" will preserve the meet a₁Ça₂ as well as everything entirely outside the join.

    As pure blades, a₁ and a₂ have scalar square and so (since a₁² commutes with a₂ and a₂² commutes with a₁) both a₁ and a₂ anticommute with a₁×a₂. If we can find a particular logarithm (a₂a₁)^↓ that anticommutes with a₁ and a₂ (which is by no means certain) then we can take f^l = (l(a₁a₂)^↓)^↑ .
    The delta product (a₁Da₂) provides an even d-blade of signature (a₁Da₂)² that in many cases of interest provides a decentralisation blade allowing analytic computation of a particular (a₁a₂)^↓ and checking this log for anticommutation with b provides a comparatively rapid check of whether such a rotor form can be constructed.   
    We have a₂a₁=a_2*a₁ + b + a₂Da₁ where b is is <2;4;..;d-2>-vector whose square has grade <0;4;8;..;d> with the d-grade component a multiple of a₂Da₁.  

      d=0 arises when a₁=aa₂ . No rotation is necessary.

      d=2 arises when k=1, and when m=k-2 . We have only grades <0;2> in the geometric product corresponding to a₂a₁ = a_2*a₁ + a₂×a₁. Since a₂×a₁ anticommutes with a₁ and a₂ it provides a rotor blade if it has negative square.

    When d=4 the delta product commutes with a₁ and a₂ and may provide a suitable decentralisation blade. Now a₂×a₁ has grade <2;6;10;...;d-2> while (a₂×a₁)² = a₂a₁a₂a₁ + a₁a₂a₁a₂ - 2a₁²a₂² has (a₂×a₁)^2§ = (a₂×a₁)² and so must have grade <0;4;8;...d> and the d-blade component must be a multiple of a₂Da₁. For N£7 we thus have (a₂×a₁)² = a + ba₂Da₁ when d=4, enabling us to decentralise on a₂Da₁ and form a logarithm though if (a₂Da₁)² ³ 0 we may require an i.

    d=6 requires N³6, k³3 so we already have a constructive approach for rotating one k-blade into another for N£5.   For d=6 we have
    a₂a₁ = a₀ + a₂A₂ + a₄A₄ + a₆A₆ with a₂×a₁ = a₂A₂ + a₆A₆ and a₂~a₁ = a₀ + a₄A₄ and a₆A₆=a₂Da₁ a pure 6-blade. Note that A₄² and A₂² are <0;4>-vectors rather han scalars.
    When k=3 so that A₆=(a₂Ùa₁)^~ spans A₂ and A₄ and so commutes with them, then (since a₂×a₁ commutes with a₂~a₂) A₂ commutes with A₄.
    Though the 4-grade componet of A₂² is a multiple of A₄, the 4-grade component of A₄² is not. We have only (A₄²-A_4*A₄)² = b_4,0+b_4,1A₄ .

Transforming k-spheres
      Since (aÙb)² = ¼(a-b)⁴ 2-blade aÙb always has postive square so we cannot naturally "rotate" point a to point b without either imposing an arbitary centre codistant from a and b or "translating" from a to b which can be regarded as rotation about e_¥. This "problem" persists for nonintersecting hyperspheres s₁^* and s₂^*.  

    Suppose we have two distinct k-spheres represented by two unit (k+2)-blades a₁ =  Q_b1,s1 and a₂ =  Q_b2,s2 in (N+2)-dimensional space U^N% for 1 £ k<N . In a typical application we might have a₁ = p₁Ùp₁Ù...Ùp_k+2 ; a₂ = q₁Ùq₁Ù...Ùq_k+2 for particular 2(k+2) points p₁,..,p_k+2,q₁,..,q_k+2 specifying the "portions" of the k-spheres we are most interested in. We might represent a₁ and a₂ in such "factored form" rather than as an expanded k-vector with regard to a particular extended basis; we might have have s₁,b₁ and s₂,b₂ explicitly; or we might have a₁ and a₂ in extended basis coordinate form.

    For k=1, we might regard the 1-sphere as passing through the points in the particular order p₁,p₂,p₃ and regard p₁ and p₃ as "endpoints" of a "region" of interest. More generally we regard the (k+2) control points as specifying a "spotsampling" of a particular "region" of the (k-1)-sphere in which we are particularly interested.
    In an intrinsically distinct class of problems, we merely seek to "rotate" or "transform" unit k-blade a₁ to unit k-blade a₂ in as "natural" a manner as possible, independant of any choice of control points and in such cases the use or imposition of particular control points may be regarded as undesirable. It is this class of problem that we address first.

    Transformations mapping a k-sphere to another k-sphere can embody aribitary rotations or "twist" within the k-sphere or indeed any "stretching" or "remapping" of the target k-sphere to itself. Even if we insist on retaining a uniform "spread" of points, we still have an arbitary twists (a single angle in the case of k=1 or a rotation in SO_k more generally).

The Slide
    The simplest way to "morph" a₁ to a₂ is to simulataneously and independantly "translate" c₁ to c₂, "ramp" r₁ to r₂, and "rotate" b₁ to b₂. but this may be unsatisfactory if a₁ and a₂ lie in a common (j-2)-sphere a₁Èa₂ because the intermediary k-spheres will not in general remain within the join. We will refer to this "directmost" morphing of a₁ to a₂ as the slide from a₁ to a₂. We interpolate r,c, and b seperately on l and recombine them to form ¦_l(a₁).
    The splitting of a₁ into c, r and b is determined by e₀ and e_¥. Regarding a₁ as a blade in Â_p+1,q+1 we can pick any e₊ and e_- to conduct a "general homegenised disembedding" into Â_p,_q. In fact, we can pick any null e₀ and e_¥ with e₀¿e_¥=-1 The slide is always available, but risks traversing potentially forbidden subspaces.

    Rather than ramp r₁ to r₂ with r(l)=r₁+l(r₂-r₁) we can ramp on r² with r(l) = (r₁²+l(r₂²-r₁²))^½ which has r'(l)=½(r₂²-r₁²)r(l)^-1 rather than r'(l)=r₂-r₁.

    We also have s(l)=s₁+l(s₂-s₁) with squared radius r(l)² = l(l-1)(c₁+c₂)² + lr₁²+(1-l)r₂².

The Inside Slide
    If a₁ and a₂ intersect it will be in a (m-2)-sphere for some m£k+2 equal to the grade of the meet a₁Ça₂. The delta product a₁Da₂ has even grade d = 2(k+2-m) and the join is either a (2k+4-m)-blade rpresenting an encompassing (2k+2-m)-sphere having a particular centre c , or the (N+2)-blade pseudoscalar e_¥0i when m £ 2k+2-N .    [  Previous webposted drafts of this document falsely asserted join e_-^* ]

    For N£3 the extended space has dimension N+2£5 so we can use a₂Da₁ as a decentralisation blade as described above.

      d=2 arises when k=1, and when m=k (eg. if p_j = q_j for all but one j) corresponding to k-spheres a₁ and a₂ having k common points and intersecting in a (k-2)-sphere. Then we have the particularly simple case d= 2, the geometric product a₂a₁ has grades <0;2> only, corresponding to a₂a₁ = a_2*a₁ + a₂×a₁ with a₂×a₁ anticommuting with a₁ and a₂ providing a rotor blade.

    If d=4 then the delta product commutes with a₁ and a₂ and may provide a suitable decentralisation blade. We can have two nonintersecting (m=0) bipoints (k=0) residing in a common 2-sphere (j=4) when N³3. For N ³ 4 we can have two 1-spheres (k=1) meeting in a point (m=1)  and residing in a common 3-sphere (j=5); or two 2-spheres (k=2) meeting in a bipoint (m=2) and residing in a common 4-sphere (j=6) if N³ 5. For N³7 we can have two 2-spheres (k=2) meeting in a 1-sphere (m=3) and residing in a common 3-sphere.

    Any tranformation ¦ taking a k-sphere a₁ to a₂ also defines a transformation F taking (N-1)-sphere s₁^* to s₂^* where s₁ = a₁¿(e_¥Ùa₁) and s₂=a₂¿(e_¥Ùa₂). We can see F as an "enclosing transformation" that is "refined"to ¦ by being forced to map a particular k-plane slice of s₁^* to a particular k-plane slice of s₂^*.
    We can expect particular behaviour of ¦ (such as radii r passing through ¥) to also arise in F. If we can keep the radii encountered in moving from s₁^* to s₂^* , the radii of ¦ will also be so limited.

    Suppose for now that (s₁Ùs₂)² < 0 so that enclosing hyperspheres s₁^* and s₂^* intersect in a (k-1)-sphere m =  Q_b,s of centre c, radius r £ Min(r₁,r₂), tangency (k-1)-blade b.
    This means that s₁^*·Ès₂^*·  has one, rather than two seperate, "interiors"   and we are interested in transformations of a₁ to a₂ that remain either always inside, always outside, or always within   s₁^*·Ès₂^*· .

    The natural grade-preserving transformation taking s₁ to s₂ is F_l(x) = (l½p(s₁Ùs₂)^~)^↑_§(x) º (l½p(s₁Ùs₂)^~)^↑ x((l½p(s₁Ùs₂)^~)^↑)_§ . As we take l from 0 to 4, c(l) goes from c₁ at l=0 through c₂ at l=1 back to c₁ (aquiring an irrelevant sign flip of s) at l=2, then back to c₂ at l=3, returning to c₁ again at l=4.
    In one of the intervals [0,1] and [2,3] (equivalent to taking l from 0 to -1) , the sphere s(l)^* reamins insidie s₁^*· È s₂^*· and s(l) coincides with the meet encloser s, for some l. We can think of s(l) as "sweeping the interior" of s₁^*· È s₂^*· . For the other interval, r(l) increases from r₁ to ¥ (s(l) instantaneusly   the hyperplane through c of tangent b) and then drops back r₂. We think of s(l)^* as "sweeping the exterior". As with many rotations, we have a choice of a "short way" and a "long way" round, and it is the half of the lÎ[-1,1]  transformational "circuit" that attains minimal r(l) rather than the one taking r through ¥ that principly interests us, since it provides a smooth bounded and continuous distortion, and this is determined by whether r(l)² = s(l)² is falling or rising with l at l=0.
    We will henceforth assume that the "inside" (r minmising) of the two (±lw(s₁Ùs₂)^~)^↑_§ rotors (where w = sin^-1(r₁^-1r₂^-1 |(s₁Ùs₂)²|^½) is chosen and denote it by R^l_s1®s2 .

    We can meaningfully write
    s₁^*· È s₂^*·   =   È_lÎ[0,1] R^l_s1^*®s2^* (s₁^*) because the s(l)^* hyperspheres sweep through the interiors of both hyperspheres.
    Also s₁^*·   =   È_lÎ[0,1] R^l_c1^*®s1^* (c₁^*) .

    Given an inside transform F_l taking s₁^* to s₂^* with For r¹0 we have a_k+2 = s= a_k+2¿(e_¥Ùa_k+2) and b_k+1 = e_¥0¿(sa_k+2) so an obvious s₁®a₂ transformation is provided by
    ¦_l(a₁) =  Q _{R^lb1®b2 , R^ls1^*®s2^*} in which we transfrom s₁ to s₂ and b₁ to b₂ independantly.
    Though b₁ and b₂ are in U_N rather than _Udn^% there may still remain a lot of freedom in choosing R^l_b1®b2 . The intermediate r(l) are independant of this choice, however, dependant purely on F_l.
    If a₁ and a₂ lie in a common (j+2)-sphere, this transformation is not in general confined to that join. Rather, the centre travels directly from c₁ to c₂; r adjusts to keep m within s(l); while the tangency efficiently realigns to the target orientation. There is no attempt to remain within a higher sphere so we will refer to such a transformation as an inside slide.   The k-sphere centre does not take a direct route, but rather a curve passing near the meet centre c.

The Inside Swing
    A natural restriction to impose on ¦ is to insist that k-sphere ¦_l(a₁) either contains or lies within the meet (N-2)-sphere m=s₁^*Çs₂^* for some intermediary l Î [0,1]. If k=m=N-2 (such as when transforming circles in 3D) then we can split ¦ into two transformations a₁®m®a₂   . We are still sliding the centre directly, but now the orientation is aligned to "orthogonal to travel" on passing through the meet. We can regard this as an inside slide but making a "token effort" to keep the orientaion orthogonal to the direction of travel of the k-sphere centre. We will call this transform (for k=N-2) the inside swing .

    If a₁ and a₂ are 2-blades representing bipoints then in general they will have scalar meet and 4-blade 2-sphere join j with a₁a₂ decomposing as a_2*a₁ + a₂×a₁ + a₂Ùa₁ into the sum of a scalar , a 2-vector, and a minussquare 4-blade.
    The 4-blade commutes with a₁ and a₂ and because (a₂×a₁)² is scalar provides a decentralisation blade and so a logarithm (a₁a₂)^↓. 5-blade c_jÙj where c_j is the embedded centre of the join 2-sphere providing a suitable i.

    If a₁ and a₂ are 3-blades representing 1-spheres , then for N=3 we again have a₁a₂ decomposing <0;2;4> as a_2*a₁ + a₂×a₁ + a₂Da₁ but this time both a₂Da₁ and a₂×a₁ are minussquare blades. If a₁ and a₂  lie in a common 2-sphere we are dual to the bipoints case, and can form a roatation taking a₁ to a₂ within a₁Èa₂ but more generally the join is 5-blade e_¥0e₁₂₃ . So a₁ and a₂ must meet in at least a 1-blade m which for non-intersecting a₁ and a₂ will represent an "imaginary" (m-2)-antisphere with negative r² but a particular centre c.

    Thus for N=3 we can construct rotors to transform k-spheres .

    Moving to N=4 means that unless two 1-spheres intersect or lie in a common 2-sphere, we lose the scalar squaredness of a₁a₂ - a₁da₂ - a₁Da₂ .

    As a degenerate example, let a₂ = -4.25e_¥e₁₂ - 2e₀e₁₂ - 4e₁₂₃ representing the 1-sphere of radius ½, centre 2e₃ parallel to the e₁₂ plane; and a₁ = -4.25e_¥e₁₂ - 2e₀e₁₂ + 4e₁₂₃ representing the similar 1-sphere centred on -4e₃. Their join is the 2-sphere centre 0 of radius 2.06155, and the product a₂a₁ = 33 + 34e_¥e₃  + 16e₀e₃ = 33 + 32.984895h where h=1.3077e_¥e₃+0.48507e₀e₃ has h²=1, yielding (a₁a₂)^↓ = 4.31836e_¥e₃ + 2.03217e₀e₃ . This provides appropriate fractional powers of a₂a₁ for the transformation   x ® (a₂a₁)^½lx(a₂a₁)^-½l that for l=1 maps a₁ to a₂ and for l=½ maps a₁ to the "join equator" 1-sphere having centre 0 and radius 2.01625 parallel to e₁₂.
    Thus is typical of transforms obtainied by logging the product. The k-sphere remains in the join but its radius can expand and contract considerably.

Mutating k-spheres
    So far we have "rotated" k-spheres into k-spheres, the the logaithm based method described generalises immediately to "transmogifying" a (k+2)-sphere a_k into an (l+2)-sphere b_l.  We now explicitly do not want a grade preserving transformation and this coincides with a failure of (b_la_k)^↓ to undergo reversion when commuted across a_k. Our transformation is x ® (l(b_la_k))^↑ x rather than (½l(b_la_k))^↑ x (½l(b_la_k))^§ x . The former preserves everything wholly outside the join a_kÈb_l but the latter does not. However, if we are only interested in morphing a particular (k+2)-sphere a_k into a particluar (l+2)-sphere b_l then we are essntially only intersted in <k;l>-grade multivector (l(b_la_k)^↓)^↑ a_k which we can regard as a "union" or "superposition" of a (k+2)-sphere and a (l+2)-sphere.

Transforming using control points
    Returning to our degenerate example above, we also have the more "direct" translation by d=4e₃ of a₁ to a₂. Letting a₁=p+1Ùp₂Ùp₃ where p+1, p₂ and p₃ are any three distinct points in a₁ we can express the translation as ¦_l(a₁) = ((-½le_¥d)^↑p+1(-½le_¥d)^↑) Ù((-½le_¥d)^↑p₂(-½le_¥d)^↑) Ù((-½le_¥d)^↑p₃(-½le_¥d)^↑) with l=1 giving a₂ .   The intermediary 1-spheres now all have a common radius rather than lieing in a common 2-sphere.

    More generally, if we know that particular control points p_i map to particular target points q_i then we can construct operations that translate all the control points simultaneously in this way, but there is a danger that control trajectories will cross eachother, making the intermediary (k+1)-blade vanish for particular l Î (0,1).

    Each of the d e_i comprising the maximal absolute coordinate d-blade in a particlar extended frame representation of a₁Da₂ can be projected into a₁ and a₂. Chosing the one which minimises ¯_a1(e_i)² + ^_a1(e_i)² ensures that distinct 1-vectors p=¯_a1(e_i) and q=¯_a2(e_i) are both non small. The rotation ((1-g(l))½ cos^-1(q¿p)(qÙp)^~)^↑_§ will then rotate a₁ into a (k-1)-sphere a₁' contaning q instead of p. (g is a dropoff function of choice). We can then rotate a₁' into a₂ by recursion, because the dimension of the meet will have risen by one, and d fallen by two.

    Thus we have the basis for an algorithm that rotates a₁ to a₂ in ½d distinct "stages", one "control point" at a time; implicitly chosing particular frame-dependant but generally appropriate control points.  At the i^th stage, the algorith will typically pick a point p_i within a₁ ⁽ⁱ⁾ that has the "furthest" to rotate to its destination q_i in a₂ and then "bend it out" towards q_i. Once q_i is reached it is "locked" into the (k-1)-sphere and preserved within it by subsequent elemental rotations. The composite "rotation" (½q_½d-1(p_½d-1Ùq_½d-1^~)^↑_§... (½q₀(p₁Ùq₁)^~)^↑_§ rotates a₁ into a₂ through d intermediary (k-1)-spheres a₁ ⁽¹⁾ with a₁ ⁽⁰⁾ = a₁ and a₁ ^(d) = a₂. The control point p_i lies within a₁ ⁽ⁱ⁾ rather than a₁ but we can always "backrotate" it by (½q_i-1(p_i-1Ùq_i-1)^~)^↑_§... (½q₀(p₀Ùq₀)^~)^↑_§... into a control point p' within a₁ that is rotated to q_i by the final composite rotation.