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Introduction

    Geometric algebra provides a practical alternative to conventional 3D vector methods which extends far more readily to higher dimensions. It also provides a coordinate independant symbolic geometry (an "algebra of directions") extendable into a geometric calculus of profound relevance to areas as diverse as quantum physics and computer vision.
    The purpose of this work is to provide a concise but comprehensive introduction and broad reference for geometric algebra for those interested in it as a powerful computational and theoretical resource that spans and unifies a diverse range of fields. Fuller more formal mathematical treatments exist elsewhere and this document can serve as a primer for tackling such works. It assumes familiarity with "conventional" 3D constructs such as vectors and matrices and such basic mathematical functions as cos(q) and e^x.  Mathmatical notations used are defined in the glossary.

    Multivectors, the "elements" of Geometric Algbera, are a generalisation of traditional vectors that provide a far richer mathematical structure than vectors alone. Many programmers will have encountered particular multivectors before in the form of complex numbers and quaternians and the closest analogy for the generalisation of vectors to multivectors is perhaps complex numbers as a generalisation of real numbers. Regarding real numbers as a special case of the more general "class" or "field" of complex numbers allows logarithms of negative numbers and additional functions such as "complex conjugation". By generalising vectors to a particular subset of multivectors not only can vectors be multiplied and divided by eachother, but we obtain a multitude of useful conjugations and bilinear products and can usefully define logarithmic and trigonometric functions of them. Since particular N+2 dimensional multivectors    can represent arbitary N-dimesional lines, circles, planes, and spheres we can, for example, speak of taking the log of a sphere. Geometric objects become algebra elements ameniable to both purely symbolic and computational (numerical) manipulation.

    The notion of multiplying two 3D vectors together is familiar to programmers as the so called vector cross product a×b being a vector of direction perpendicular to vectors a and b with magnitude |a||b| sin(q) where q is the angle subtended by a and b. However, this only works in 3D. The fundamental essence of geometric algebra is the geometric vector product ab = a.b + aÙb where . is the traditional scalar-valued vector dot product and "2-blade" aÙb is the wedge or outter product described later. Everything else follows from this.   
    This section describes what multivectors are mathematically and lists the many operations and products that can be usefully applied to them in considerable detail. Of necessity some of this material is mathematically intensive and the reader is is encouraged to "skim" rather than absorbing every product, conjugation, normalisation technique and logarithmic computation strategum on their first pass. The "point" of multivectors is what you can do with them, and this is addressed in the later sections once the basics of the symbolic and computational manipulations are covered.

    When first encountering multivectors they can seem bewildering, with a plethora of products and an overload of operators provided for their manipulation. But familiarity breeds respect. Given that multivectors are the "language" of (dynamic) geometry and by implication of nature itself, a half-dozen new symbols does not seem excessive.

    In the Multivectors Programming chapter we describe how to impliment multivectors of both low and high dimension N in C/C++.
    In the Multivectors as Geometric Objects we will see some of the applications of multivectors in the elegant representation and manipulation of N-dimensional spheres, planes, lines and conics.
    In the Multivectors as Transformations we will see how multivectors can also be used to transform, distort, displace, and morph such geometric constructs.
    In Multivector Arcana we cover some more esoteric mathematical aspects of multivectors of less general interest.
    In later sections we cover Multivector Calculus and the uses of multivectors in physics, in particular Relativity and Quantum Mechanics.
    This treatment favours the contractive "computer scientist's" inner product a¿b over the semisymmetric "physicist's" inner product a.b where possible because this is arguably the more fundamental and has certain functional advantages.

Multivectors
"And therefore in geometry (which is the only science that it hath pleased God hitherto to bestow on mankind), men begin at settling the significations of their words; which settling of significations, they call definitions, and place them in the beginning of their reckoning." ---  Thomas Hobbes, Leviathan

    Although a notable advantage of geometric algebra is coordinate independance, we will initially take an orthonormal coordinate (basis) based approach here since this is likely to be the most practicable in a programming context.

    Geometric algebra is essentially a set of arithmetical techniques for manipulating N-dimensional vectors and to see how to properly multiply (and divide!) vectors we must first generalise the concept of vectors and scalars.
    Given k linearly independant N-dimensional vectors a₁,a₂,..,a_k from a vector space U^N their outer product a_k = a₁Ùa₂...Ùa_k (not to be confused with the 3D vector "cross" product)  is known as a blade of grade (aka. step or degree) k, or a k-blade. [  We are interested initially in U^N=Â^N, the space of real coordinate N-D vectors, but will refer to U^N to emphasise applicability to alternate (eg. nonEuclidean) spaces which are of interest to us, particularly with regard to relativistic physics. By "linearly independant" we here mean that no one of the a₁,a₂,..a_k can be expressed as a real-wighted sum of the others ]
    The fundamental rules for Ù are
    Antisymmetry: aÙb = -bÙa ( and so aÙa=0) for any vectors a and b ;
    Linearity: aÙ(b+c) = aÙb + aÙc ; and
    Associativity: aÙ(bÙc) = (aÙb)Ùc.
     A k-blade a_k can be thought of as representing an orientated and scaled k-dimensional subspace of U^N, one in which all the vectors satisfy aÙa_k = 0. We say that a₁,a₂,..,a_k are a k-frame for this subspace.
    A linear "weighted additive" combination of k-blades is known as a k-vector. An example 4D 3-vector is ½e₁Ùe₃Ùe₄ + (Ö7)e₁Ùe₂Ùe₃ .
    A 0-vector is a 0-blade is a scalar. We refer to a k-blade as proper if k>=1.
    A 1-vector is a 1-blade is a conventional vector.
    A 2-vector (aka. bivector) is a sum of scaled 2-blades and need not "reduce" to a single 2-blade for N>3.
    2-blades can be considered geometrically as directed areas (ie. a plane and a signed scalar).
     For N£3, any k-vector is a k-blade. This is true only for k<2 when N > 3.
[ Proof :  ae₁Ùe₃+be₃Ùe₁+ge₁Ùe₂ = (e₁+(a/g)e₃)Ù(ge₂-be₃) if g¹0, (ae₂-be₁)Ùe₃ else . All 3D 3-vectors are multiples of e₁Ùe₂Ùe₃    .]
    This makes N=3 a fundamentally simpler case than N³4 to the extent that "geometric intuitions" founded in 3D can be actively misleading for N³4.
    Because of this, the reader is here advised to initally consider multivectors as algebraic rather than geometric entitiesto be manipulated symbolicsllyand numerically using grammar rules rather than pertaining to geoemetric constructions.

    Consider the set Â_N (aka. G_N and Cl_N in the literature) of all linear combinations (with real "coefficients" or "wieghtings") of k-vectors for 0£k£N, our 1-vectors being taken from the N-dimensional vector space Â^N. Clearly Â^N Ì Â_N (or, more properly, is represented within Â_N) and infact Â_N has dimension å_k=0^N NC_k = 2^k, there being ^NC_k º N! (N-k)!^-1 k!^-1 distinct k-blades in N-dimensions.

    We refer to the elements of Â_N as multivectors An example Â₃ multivector, albeit one unlikely to arise in practice, is 3+e₁-4e₂+½e₁Ùe₂+(Ö7)e₂Ùe₃ + pe₁Ùe₂Ùe₃ . We can thus think of a general N-D multivector as an arbitary real-weighted combination of 2^N distinct basis blades. [  Mathematically, one can take the "blade coefficients" from any field or "number space". It is the utilisation of  - essentially identifying the blade "coefficients" or "coordinates" with "scalars" (0-vectors) - that distinguishes "geometric algebra" from more general Clifford algebras of only passing concern here. If we allow "complex number" blade coeeficients, for example,  we obtain a space C_N of dimension 2^N+1 . ]
    Multivectors can thus be represented with 2^N dimensional 1-vectors, with respect to the "extended basis" generated by a given set of N linearly independant N-D basis 1-vectors e₁,e₂,...,e_N and this tells us how to add and subtract multivectors, but not how to multiply and divide them. For that we will need the "geometric product" and its associated "subproducts".

Conflicting Terminologies
    Some authors such as Pavsik use the term "k-vector" for what we will call a k-blade and "multivector" for our k-vector (ie. a single-graded multivector), adopting the term polyvector for our multivector.
    The term k-vector is sometimes used in the literature to refer to a k-dimensional 1-vector. However it is more common to spell the number so that, for example, four-vector typically denotes a 1-vector in a 4D spacetime; three-vector denotes a 1-vector in ³ and so on, and we will adopt this convention here.  

Notations and Coordinates
    A general N-D multivector is thus the sum of a scalar, a 1-vector, a 2-vector,..., and an N-vector but we will frequently be interested in pure k-blades or k-vectors and will benefit from a notation that that distinguishes such from general multivectors.  We will use the following fonts to denote geometric algebra elements

Font	Represents	Also known as
a	Proper blade
a	General multivector
a	0-vector	Scalar
a	1-vector	Vector, 1-blade
a	2-blade
a	2-vector	Bivector
a	3-blade
a	3-vector	Trivector
a	(N-1)-vector	Hyperblade, Pseudovector
a	N-vector	Pseudoscalar
a	<0;N>-vector	Scalar-pseudoscalar pair.

Blades
    Given an orthonormal basis (ie. a Cartesian coordinate frame) consisting of N orthonormal N-D 1-vectors {e₁,e₂,...,e_N} for Â^N, we can consider multivectors as elements of Â^2N, ie. as 2^N-dimensional 1-vectors,
    We use the notation e_ij..m º e_iÙe_j....Ùe_m     where i,j,..m are distinct indices. [  If we relax the distinctness restriction in the case of N indices we have e_ij..m = e_12..Ne_ij..m where scalar e_ij..m is the traditional "alternating tensor" returning ±1 or 0 according as whether ordered set {i,j,..,m } is an even/odd, or not a, permutation of ordered set { 1,2, .. N } ]
    If all 2^N coefficients of the basis blades arising from a given orthonormal basis are integers ( eg. 5 + 3e₁ - 7e₂Ùe₃) then we will say the multivector is integer-coefficiented with regard to that basis. If the coeffients are further restricted to { ±1 , 0 } then the resulting 3^2N "discrete" multivectors are said to be Mantheyian with regard to the basis.

Specific subcomponents
    We will write a^ij..m or a_[ij..m] for the coefficient of e_ij..m in multivector a. . [  We will seldom have to raise individual components to integer powers and this raised suffix notation has definite advantages ]
    We will refer to the integer ½(2^j+2^k+...+2^m) as the bitwise enumerator for the extended basis blade e_jk..m. The ½ arises because we labelled our first frame vector e₁ rather than e₀. We will denote coordinates or blades indexed by a particular bitwise enumerator n by _[.n.] . Thus eg. e_[.0.]=1 ; e_[.1.]=e₁ ; e_[.6.]=e₂₃ and so on and in general we have a = å_k=0^2N-1 a_[.k.] e_[.k.] .
    We will use the notation a_k to indicate a k-vector, a_k a k-blade, and the notation a_<k> to indicate the k-vector component of a general multivector a (with a_<k>=0_<k>=0 for k>N) . The notation a_<i,j,..m>º a_<i>+ a_<j>+ .. + a_<m> will prove useful.
    We say a is pure or k-pure if a_<k> = a. By an impure k-vector we mean a multivector having a nonzero k-vector component and possible nonzero j-vector components for j<k.
    If we wish to specify a multivector as having only components of grades 2,3 and 5, for example, we will refer to a <2;3;5>-vector. A <k>-vector is thus a pure k-vector. A <0;1>-vector is sometimes known as a paravector, particularly when N=3.
    We will use the notations a_<+> º a_{<0,2,..,2*[N/2|>} to denote the multivector obtained by taking only the even grade components of a, and a_<-> º a_{<1,3,..,2*[N/2|+1>} for the odd grade components.  

Zeroes
    The degenerate (zero magnitude) k-blade is denoted by 0_k.
    All degenerate blades are considered equivalent ( 0 = 0₁ = 0₂ = ....0_N = 0 ) , and this zero is the sole equivalence. Note that within a computational context (finite precision arithmetic) we will typically be checking for proximity rather than equivalence with zero. It is often worth keeping track of the "grades" of expected zeros.

    We are freuwently interested in issues such as whether the square of a multivector is a pure scalar but in practice this may mean checking that any residual nonzero coordinates are a "negligable" proportion of the scalar part, which can be problematic if the scalar part is also small.
    We define the sparsity of a multivector with respect to a given basis as the number of zero coefficients (coordinates) in its representation in that basis.

Inverse frames
    Suppose we have a possibly non-orthonormal linearly independant basis of N N-D 1-vectors providing a coordinate frame (ie. a set of axies) E=(e₁,e₂,...,e_N) for U^N. We can construct a reciprocal or inverse frame (e¹,e²,...,e^N) so that eⁱ.e_j = 1 when i=j and 0 else where . is the traditional scalar ("dot") product of two 1-vectors.
    If E is orthogonal then provided no e_i²=0 we can set e^k º e_k^-2e_k . More generally we require e^k º (-1)^k-1(e₁Ù..e_k-1Ùe_k+1Ù...e_N) i^-1 where i=e_12..N , though this may not make much sense to the reader till he is more familiar with Ù and the pseudoscalar i discussed later.
    We define a notation e^ij..m º eⁱÙe^jÙ...e^m .

    If E is orthogonal (ie. e_i.e_j=0 for i¹j) then (using the geometric product defined below) we have e^ke_k = 1 (provided e_k²¹0) ; but in general e^ke_k has a non-zero 2-blade component because e^kÙe_k ¹ 0 .
    E induces both the coordinate expression x = å_i=1^N xⁱe_i     [ with xⁱ º eⁱ.x ], and the reciprocal coordinate expression x = å_i=1^N x_ieⁱ     [ with x_i º x.e_i ]. eⁱ is the 1-vector geometric multiplier that "seperates"  1-vector x into xⁱ + å_{j ¹ i} x^jeⁱÙe_j .

    In Â^N : (i) an orthonormal frame is self-inverse ( e_i = eⁱ ; x_i = xⁱ ) ; (ii) a general frame, expressed as an N×N matrix E with respect to a fixed orthonormal frame F=(f₁,f₂,...f_N) in conventional manner via ( e_i = å_i=1^N E_jif_j ) has as its reciprocal frame the frame having matrix (E^-1)^T = (E^T)^-1   with respect to F, ie. the inverse transpose matrix.

    Letting E_ij º e_i¿e_j  and E^ij º eⁱ¿e^j ,  we have x_j = å_i=1^N E_ijxⁱ     ;      x^j = å_i=1^N E^ijx_i .
    The N×N symmetric matrices {E_ij} and {E^ij} are related by {E^ij} = {E_ij}^-1   where ^-1 is  the conventional matrix inverse.

Inverse Frame Units
    We discuss the mathematically ticklish issue of inverse frames here because we are discussing coordinate representations.
    If frame vectors are assigned units, e₁ having length 5 m say, then e¹ must be regarded as having "length" 5^-1 m^-1 so that e¹¿e₁ = 1 m⁰ is dimensionless. Coordinates xⁱ =eⁱ¿x are then unitless while reciprocal coordinates x_i = e_i¿x have units m².

Extended inverse frames
    Given an extended basis {e_[.i.] : 0£i<2^N } we can construct an extended pureblade inverse frame {e^[.i.]} which satisfies e^[.i.]_* e_[.j.] º (e^[.i.]e_[.j.])_<0> = 1 when i=j and 0 else .

The Geometric Product
"He who can propery define and divide is to be considered a god." --- Plato

    To make Â_N a linear space we require a "multiplication" with the following properties:
    a(bc) = (ab)c     (associativity)
    a(b+c) = ab + ac ; (b+c)a = ba + ca     (distributivity)
    aa = Sig(a) where Sig(a) is a scalar for all 1-vector a     (contraction).
    Of principle interest here is the contraction Sig(a) = e|a|². where e (the signature of a) is either ±1 or 0 and |a| is the conventional magnitude ("length") of 1-vector a. A vector is null if a²=0.   
    We write Â^p,q,r for a vector space having orthogonal basis {e₁,e₂,...e_N} where N=p+q+r and Sig(e_i) = 1 for 1£i£p ; -1 for p<i£p+q ; 0 for p+q<i£N .
    We write Â_p,q,r for the associated geometric algebra. We write Â_p,q (aka. Cl_p,q) as an abbreviation for Â_p,q,0 and Â_N as an abbreviation for Â_N,0,0 .

    We define the geometric product of any a by a scalar b in the obvious "coordinatewise" commutative manner (ba)_[ij..m] =    (ab)_[ij..m] º b(a_[ij...m]) .
     We define the geometric product of two 1-vectors by ab º a.b + aÙb where a.b is the conventional Â^N (or U^N) vector "dot" product. We can then extend this definiton by means of the associativity and contraction rules to higher grade blades and hence (by distributivity) to multivectors generally. The geometric product is noncommutative (ab¹ba in general), but this is actually an assett; the "degree" of noncommutativity of the geometric product of two multivectors being a measure of their orthogonality. A unit multivector is a multivector satisfying (aa)_<0>=±1.
    More generally we have ab= a¿b + aÙb where a is a 1-vector and b is a general multivector.
    abc=a¿((b¿c)+bÙc) + aÙ((b¿c)+bÙc) = a¿(bÙc) + a(b¿c) +aÙbÙc)
    abcd= = a¿(b¿(cÙd) + b(c¿d) +bÙcÙd)) + aÙ(b¿(cÙd) + b(c¿d) +bÙcÙd)) = aÙ(b¿(cÙd) + (aÙb)(c¿d) + aÙ(bÙcÙd))

Â₂
    We can tabulate the geometric product for Â₂ with respect to a basis for Â₂ derived from an orthonormal basis {e₁,e₂} for ².

ab for Â2		a
		1	e1	e2	e12
	1	1	e1	e2	e12
b	e1	e1	1	e12	-e2
	e2	e2	-e12	1	e1
	e12	e12	e2	-e1	-1

[ Proof :  e₁₂e₁ = (e₁Ùe₂)e₁ = -(e₂Ùe₁)e₁ = -(e₂e₁)e₁ = -e₂(e₁e₁) = -e₂ . Other products likewise.  .]
    We note in passing that the subspace Â_{2 +} consisting of all 2D multivectors having no 1-vector component, ie. the space of multivectors of the form a + be₁₂   is closed under the geometric product and is isomorphic to the complex number space C,  as is Â_0,1.

Â_1.1
    If e₁²=1 and e₂²=-1 then (e₁Ùe₂)² = 1 and the even subalgebra Â_{1,1 +} generated by 1 and (e₁Ùe₂) is isomorphic to the hyperbolic numbers x+yh where h²=1.
      These are less well studied than complex numbers and rarely implimented in code libraries. However it is easy to generalise a complex number class to allow for a plusquare or null i. The hyperbolic number x+yh has inverse (x-yh)(x²-y²)^-1 provided x¹±y . Though we can express x+yh in "hyperbolic polar form" r(qh)^↑ where r=|x²-y²|^½ and q =  tanh^-1(xy^-1) if |x|<|y| ; or  tanh^-1(yx^-1) if |y|<|x| , the inverse hyperbolic tangent providing q must be carefully defined since hyperbolic numbers divide naturally into four "regions" or "quadrants" seperated by the lines y=x and y=-x according as to which of |x| and |y| is larger and the sign of the larger coordinate. We will refer to hyperbolic numbers with x>0 and x>|y| as being in the (1)-quadrant since this is the quadrant containing 1. Hyperbolic numbers in this quadrant remain in it when raised to any power or exponentiated.
    We can express ay (1)-quadrant hypercomplex as r(qh)^↑ for real r³0 and q Î (-¥,¥) . Whereas the complex polar angle parameter can be restricted to (-p,p] and maps all 2-D directions, the hyperbolic polar angle parameter is unbounded and yet only maps a quarter of the 2-D directions. We cannot recover x and y from r and q, as we can for complex numbers, without also knowing which quadrant we are in. Multiplication with h flips x and y and so corresonds to reflection in x=y , mapping the (1)-quadrant into the (h)-quadrant, the (h)-quadrant into the (-1)-quadrant, and so on.   Computationally, we must specify the quadrant "mapped" by q as well as r and q, which requires an additional two bits.

Â₃
    We can tabulate the geometric product for Â₃ with respect to a basis derived from a standard orthonormal basis {e₁,e₂,e₃} for ³.

ab for Â3		a
		1	e1	e2	e3	e23	e31	e12	e123
	1	1	e1	e2	e3	e23	e31	e12	e123
	e1	e1	1	-e12	e31	e123	e3	-e2	e23
	e2	e2	e12	1	-e23	-e3	e123	e1	e31
b	e3	e3	-e31	e23	1	e2	-e1	e123	e12
	e23	e23	e123	e3	-e2	-1	e12	-e31	-e1
	e31	e31	-e3	e123	e1	-e12	-1	e23	-e2
	e12	e12	e2	-e1	e123	e31	-e23	-1	-e3
	e123	e123	e23	e31	e12	-e1	-e2	-e3	-1

and similar geometric multiplication tables can be constructed for higher N.
    What is happening here is e_[.j.]e_[.k.] = ± e_{[.j  ^  k.]} where  ^  denotes bitwise XOR ("exclusive or") and the sign is determined both by how many commutations we have to do to bring the common e_i together, and the signatures of the common e_i.

    Writing i º e₁₂₃ º e₁Ùe₂Ùe₃ = e₁e₂e₃ we see from the above table that i commutes with all multivectors and satisfies i² = -1.
    We also observe that aÙb = i(a×b) = (a×b)i where a×b = (aÙb)e₁₂₃^-1 is the conventional ³ vector "cross" product and that ia spans the plane normal to a. We also have aÙbÙc = (a.(b×c))i.
    We note in passing that the subspace Â_{3 +} consisting of all 3D multivectors having no odd grade component, ie. the space of multivectors of the form a + be₂₃ + ge₃₁ + de₁₂ is closed under the geometric product and is isomorphic to the quaternion space Q, as is Â_0,2 . a + bi + gj + dk.

Biquaternions
    We further note that a general Â₃ multivector can be uniquely expressed as (a + a₂) + (b + b₂)e₁₂₃ where a,b are scalars and a₂,b₂ are pure Â₃ bivectors. An alternative biquaternion (aka. complex four-vector aka. Pauli spinor) representation of Â₃ sets i=i=e₁₂₃ , s₁=e₁ , s₂=e₂ , s₃=e₃ (satisfying s_i s_j=e_ijki s_k). The biquaternion (a₀+b₀i) +(a₁+b₁i)e₁ +(a₂+b₂i)e₂ +(a₃+b₃i)e₃ is equivalent to the Â₃ multivector a₀ + a₁e₁+a₂e₂+a₃e₃ + b₁e₂₃+b₂e₃₁+b₃e₁₂ + b₀e₁₂₃ . The product of two 3D 1-vectors is usually taken in this context to be ab º a.b + i(a×b) which is equivalent to the Â₃ geometric product since i(a×b)=aÙb.

Overview
    Given an orthonormal 1-vector basis e₁,e₂,..,e_N we can regard multivectors as complex weighted sums of the 2^N basis blades and such a "coordinates form" makes addition and multiplication particularly straightforward. As an example computation consider (1+2e₁₃)(_3e₁+4e₁₂₃₄)   = _3e₁+4e₁₂₃₄   + 6e₁₃e₁ + 8e₁₃e₁₂₃₄   = _3e₁+4e₁₂₃₄   + 6e₁₃e₁ + 8e₁₃e₁₂₃₄   = _3e₁+4e₁₂₃₄   + 6e₁e₃e₁ + 8e₁e₃e₁e₂e₃e₄   = _3e₁+4e₁₂₃₄   - 6(e₁)²e₃ - 8e₃(e₁)²e₂e₃e₄   = _3e₁+4e₁₂₃₄   - 6e₃ - 8e₃e₂e₃e₄   = _3e₁+4e₁₂₃₄   - 6e₃ + 8(e₃)²e₂e₄   = _3e₁+4e₁₂₃₄   - 6e₃ + 8e₂₄ where we have assumed e₁²=e₃²=+1.
    The "geometric product" can thus be viewed as a purely computational construct in which we can think of sliding basis 1-vectors across eachother, introducing a sign flip for every basis 1-vector crossed, until another instance of the same base 1-vector is encountered whereupon the two 1-vectors "condense" into their ±1 scalar signature. The grade of the product of an orthonormal basis k-blade with an l-blade from the same basis is thus £k+l.
    Though we constructed our extended basis and defined the geometric product using the outter product Ù, we could have assumed only the traditional scalar vector dot product to define basis orthonormality condition e_i.e_j = e_i d_ij where d_ij=1 i=j and 0 else ;   defined the extended basis elemnets by incorporating 1 and unordered pairs, triples, ... ,and N-tuples of the e_i into the basis written as e_ij..m; and finally defined the geometric product of two basis elements e_ij..me_no..r logically in this "index hoping with sign tracking" manner.

Pseudoscalars
    The N-vectors from a given U^N are all equivalent apart from magnitude ("scale","volume") and sign ("handedness"). They are accordingly known as pseudoscalars. Conversely a nonzero pseudoscalar "spans" (and can be thought of as representing) U^N .
    We will use the font a to denote a blade viewed as a pseudoscalar.   
    Let i º i_N be the unit pseudoscalar e_12..N for Â_N. i satisifes i ² = (-1)^½(N-1)N and commutes with all multivectors if N is odd. For even N we have ia_k = (-1)^ka_ki so that i (anti)commutes with (odd)even blades.
    We say a multivector is central if it commutes with all other multivectors. For even N, only scalars are central but for odd N any <0;N>-multivector (scalar plus pseudoscalar) is central.
    An (N-1)-vector is sometimes refered to as a pseudovector but we favour the term hyperblade here.
    Taking the geometric product of a multivector with i maps k-blades to (N-k)-blades and vice versa. In particular it maps scalars to pseudoscalars (and vice versa) and vectors to pseudovectors (and vice versa).
    Note that a k-blade acts as a pseudoscalar when acting upon multivectors wholly contained within the space it spans. The geometric product of a pseudoscalar i with a blade a_k contained in the subspace spanned by i spans the subspace of i complimentary (orthogonal) to subspace a_k. In particular, the geometric product of any blade with itself is a scalar.
    The signature   of a blade b_k, e_bk , is the sign of b_k² (or zero if b_k² = 0 in which case the blade is said to be null).

Duality

    We define the dual of a multivector a with respect to a pseudoscalar i spanning a space containing a by
    a^* º ai^-1 = a¿i^-1
[  Where ¿ is the contractive inner product defined below. Some authors favour ai, but if i is a unit pseudoscalar the difference is only one of sign. ]
    a^* spans the subspace of i "perpendicular" to pureblade a.
    If b is an unmixed (ie. odd or even) multivector, it will either commute or anticommute with i. For odd N, the pseudoscalar commutes with everything and we have (a^*)b = a(b^*) = (ab)^* .
    For even N, i (anti)commutes with (odd) even multivectors and we have (a^*)b = a(b^#*) = (a(b^#))^* where ^# is the grade involution conjugation defined below.

    In the presence of a standard basis for Â_N, computing a^* for the unit pseudoscalar i=e_123..N is a computationally trivial "shuffling" of coordinates requiring no numeric computations. For the bitwise ordering we have (ai^-1)_[.i.] = ± a_{[.(i XOR (2^N-1)).]} where the actual sign depends upon N and the bitwise parity of i.

    The inverse dual or undual is defined by a^-* º a^*i² = ai so that (a^-*)^* = a.

Centrality

    Any multivector a defines an algebra _Cent(a) known as the centralizer of a consisting of all multivectors that commute with a.

Bivectors
    For N£4 the squares of bivectors commute with even multivectors while for N=4 the pseudoscalar part of w² is negated on commutaion with odd multivectors.
    Left multiplication by a bivector ( a ® b₂a ) casts scalars into b₂. If b₂ is a 2-blade It rotates directions within b₂ (by b₂), while casting directions perpendicular to b₂ into trivectors. For N=3, it casts bivectors in b₂ to scalars while bivectors normal to b₂ are rotated by b₂ and the pseudoscalar is cast to 1-vector b₂^*.
    Right multiplication by a bivector has similar effects. Indeed a 2-blade commutes with all multivectors in its dual space , which is not the case for a 1-vector. Consequently, it is occasionally advantageous to represent 3D 1-vectors by their bivector duals.
    Let b be a multivector having only bivector and scalar components. a¿b = a.b while b¿a = b.a + b₀a
    b₂×a sends ^(a,b₂) to 0 and rotates ¯(a,b₂) in b₂ by ½p, scaling it by |b₂|.
    The operation a ® (ba).a for nonnull b is interesting, casting ¯(a,b) to 0 and ^(a,b) into
|^(a,b)|²b = |aÙb|²(b^-2)b = |aÙb^~|²b .

Matrix representations
    Multivectors can be represented with matrices (physicists have done so, largely unknowingly, for decades) with the geometric product corresponding to the traditonal matrix product) but matrices are seldom the best way to impliment multivectors  computationally and tend to obscure the underlying geometries. Nonetheless, we will describe some matrix representations here: partly to convince the more skeptical reader that multivectors do actually "exist", and also to provide an alternate model for those who have philosophical difficulties with the concept of adding "different grade" blades to form a composite "entity". Geometric (multivector) algebra becomes the algebra of a particular "form" of matrix, requiring only standard matrix multiply and inversion techniques. Such an approach is computationally profligate, but can sometimes provide alternative insights as well as quick-and-dirty programming applications exploiting existing matrix suites.

Â_p.q.r in Â_2^N×2^N
    With regard to a particular extended basis, an N-D multivector a can be expressed as a 2^N dimensional real 1-vector. But as a function mapping multivectors to multivectors a(x) = ax , ie.. a transform of 2^N-D 1-vectors, a can also be represented as a 2^N×2^N matrix.
    Taking a=a⁰+a¹e₁+a²e₂+a¹²e₁₂ in Â_p,q,r with p+q+r=2, for example, we have
    (a⁰+a¹e₁+a²e₂+a¹²e₁₂)(x⁰+x¹e₁+x²e₂+x¹²e₁₂)
        = (a⁰x⁰+e₊₁a¹x¹+e₊₂a²x²-e₊₁e₊₂a¹²x¹²) + (a¹x⁰+a⁰x¹+e₊₂a¹²x²-e₊₂a²x¹²)e₁ + (a²x⁰+a⁰x²-e₊₁a¹²x¹+e₊₁a¹x¹²)e₂ + (a⁰x¹²+a¹x²-a²x¹+a¹²)e₁₂       
which we can express as

æ	a0	e+1a1	e+2a2	-e+1e+2a12	ö		æ	x0	ö
ç	a1	a0	e+2a12	-a2	÷		ç	x1	÷
ç	a2	-e+1a12	a0	e+1a1	÷		ç	x2	÷
è	a12	-a2	a1	a0	ø		è	x12	ø

    Note that the leftmost column of the matrix representation of a contains the 1-vector representation of a, and that the matrix is symmetric apart from occasional sign changes determined by both the position in the matrix and the signatures of the basis 1-vectors.
    Note also that the matrix trace (sum of leading diagonal elements) of the matrix representatrion of a is the frame-invariant scalar 2^Na⁰ = 2^Na_<0> .
    More generally we have a = (a,ae₁,ae₂,...,ai) reagradablde as 2^N column vectors.  
    The geometric product ab corresponds to conventional matrix multiplication of the matrix representations of a and b. Multivector 1 has as its matrix representation the 2^N×2^N identity matrix.
    If the pseudoscalar is central (all commuting) then we can express each multivector as a 2^N-1 complex 1-vector and obtain a C_2^N-1×2^N-1 complex matrix, which requires only half as many reals as a Â_2^N×2^N matrix.

    Far more compact matrix representors for multivectors are typically available. The following are all "maximally compact" in that they require precisely 2^N real scalar parameters to hold a general N-D multivector.

Â₂ in Â_2×2

1	=	æ	1	0	ö	e1	=	æ	1	0	ö	e2	=	æ	0	1	ö	e12	=	æ	0	1	ö
		è	0	1	ø			è	0	-1	ø			è	1	0	ø			è	-1	0	ø

satisfy the required e₁²=e₂²=1 ; e₁₂=e₁e₂=-e₂e₁ .

Â_1.1 in Â_2×2

1	=	æ	1	0	ö	e1	=	æ	0	1	ö	e2	=	æ	0	-1	ö	e12	=	æ	1	0	ö
		è	0	1	ø			è	1	0	ø			è	1	0	ø			è	0	-1	ø

satisfy the required e₁²=1 ; e₂²=-1 ; e₁₂=e₁e₂=-e₂e₁ .

Â₃ in C_2×2

    Â₃ has a "biquaternian" representation with Hermitian 2×2 complex matrices

1=1

=

æ

1

0

ö

; e1= s1

=

æ

0

1

ö

; e2= s2

=

æ

0

-i

ö

; e3= s3

=

æ

1

0

ö

è

0

1

ø

è

1

0

ø

è

i

0

ø

è

0

-1

ø

which provide the Pauli algebra if we set i=e₁₂₃=i (so that s₁ s₂ s₃=i s₃² = i 1 )

ab		a
		1	s1	s2	s3
b	1	1	s1	s2	s3
	s1	s1	1	-i s3	+i s2
	s2	s2	+i s3	1	-i s1
	s3	s3	-i s2	+i s1	1

Note that s_i^D=- s_i while 1^D=1 , traditional 2×2 matrix adjoint ^D corresponding to a biquaternian "vector conjugation".

    The s_i and 1 act as a basis for the full   C_2×2 algebra of complex 2×2 matrices since

æ	a	b	ö	= ½(a+d)1 + ½(b+c) s1   + ½i(b-c) s2 + ½(a-d) s3
è	c	d	ø

    Each element of the special unitary group SU(2) of all C_2×2 unitary (AA^†ºA(A^{^T})=1) matrices having unit positive determinant can be expressed via U=(½iå_j=1³ q_je_j)^↑ for three real scalar parameters q_j. e₁,e₂, and e₃ are then refered to as the generators of SU(2) so we can view SU(2) geometrically as the space of exponentiated Â₃ 1-vectors.
    SU(2) is isomorphic to the group SO(3) of all orthogonal (AA^T=1) Â_3×3 matrices, with U (U)

æ z	x-iy	ö	U-1
è x+iy	z	ø

equivalent to Ax where x=(x,y,z)_transp, yielding the quaternian representation Â_{3 +} @ SO(3).

Â_3.1 in Â_4×4

1=1	=	æ	1	0	0	0	ö	; e1	=	æ	1	0	0	0	ö	; e2	=	æ	0	1	0	0	ö	; e3	=	æ	0	0	1	0	ö	; e4	=	æ	0	0	-1	0	ö
		ç	0	1	0	0	÷			ç	0	-1	0	0	÷			ç	1	0	0	0	÷			ç	0	0	0	-1	÷			ç	0	0	0	1	÷
		ç	0	0	1	0	÷			ç	0	0	-1	0	÷			ç	0	0	0	1	÷			ç	1	0	0	0	÷			ç	1	0	0	0	÷
		è	0	0	0	1	ø			è	0	0	0	1	ø			è	0	0	1	0	ø			è	0	-1	0	0	ø			è	0	-1	0	0	ø

satisfy the required e₁²=e₂²= e₃²=-e₄²=1 ; e_ie_j=-e_je_i for i¹j . [ Due to Lounesto.] Note that the 1-vector representors are traceless and have determinant ±1. Transpose negates only e₄ and coresponds to geometric Hermitian conjugation.

Â_4.1 in C_4×4

1=	æ	1	0	0	0	ö    ;	e1=	æ	0	0	0	i	ö    ;	e2=	æ	0	0	0	1	ö ;	e3=	æ	0	0	i	0	ö ;	e4=	æ	0	0	1	0	ö ;	e5=	æ	-i	0	0	0	ö
	ç	0	1	0	0	÷		ç	0	0	i	0	÷		ç	0	0	-1	0	÷		ç	0	0	0	-i	÷		ç	0	0	0	1	÷		ç	0	-i	0	0	÷
	ç	0	0	1	0	÷		ç	0	-i	0	0	÷		ç	0	-1	0	0	÷		ç	-i	0	0	0	÷		ç	1	0	0	0	÷		ç	0	0	i	0	÷
	è	0	0	0	1	ø		è	-i	0	0	0	ø		è	1	0	0	0	ø		è	0	i	0	0	ø		è	0	1	0	0	ø		è	0	0	0	i	ø

satisfy the required e₁²=e₂²=e₃²=e₄²= -e₅²=1 ; e_ie_j=-e_je_i  for  i¹j . Once again, the 1-vector representors are traceless and have determinant ±1. Note that e₁₂₃₄₅ = e₁e₂e₃e₄e₅ = i corresponds to i1 and with e₁₂₃₄=-ie₅ .
    We cannot simply multiply e₅ by i to obtain a basis for Â₅ because we would then have e₁e₂e₃e₄e₅=-1 rather than i1. We can only change the basis signatures in pairs, such as by multiplying e₁,..,e₄ by i to obtain a matrix basis for Â_0,5.
    Setting g⁰=ie₅ , g¹=-ie₁ , g²=-ie₂ , g³=-ie₃ provides a "Dirac matrix" basis for Â_1,3 , inefficient in that it requires 16 complex values to represent a 4D multivector having 16 real coordinates.
    Note that matrix transpose coresponds to negation of e₁ and e₃ whereas matrix complex conjugation negates e₁,e₃, and e₅ so that reversing matrix Hermitian conjugation A^{T^} negates only negative signatured e₅ and so coresponds to geometric Hermitian conjugation defined below. Negation of the on-lead-diagonal blocks is provided by e₅₌ and negates 1 and e₅.

    We can construct matrices having 1 in the first to fourth entries of the first column (and zeroes elsewhere) respectively as

• u   =   -(i-e₅)(i+e₁₂)   =   (-1+ie₅)(-1+ie₁₂)   =   (1-ie₅)(1-ie₁₂)   =   (1+e₁₂₃₄)(1+e₃₄₅) [ an absorber or emmitter of e₃₄₅, e₁₂₃₄ and e₁₂₅] ;
• e₁₃u = -e₁₄₅u     ;
• ie₃u   =   e₁₂₄₅u   =   -e₃₅u     ; and
• ie₁u   =   e₂u=e₄u=-e₂₃₄₅u =e₁₂₄₅ (-e₁₄₅u)   =   e₁₅u

    We can keep repeating this trick to create C_16×16 representations of an orthogonal 1-vector basis for Â_8.1@Â_6,3@...@Â_0,7 ; C_32×32 representations for Â₁₁@Â_9.2@...@Â_1,10 ; C_64×64 representations for Â_12,1@...@Â_0,13 and so on.

Other matrix representations
    It can be shown that Â_0,4 @ Â_4,0 @ Q_2×2  ; Â_0,6 @ Â_8×8  ; Â_6,0 @ Q_4×4  ; Â_7,0 @ Â_5,2 @ C_8×8 ;    Â_0,8 @ Â_8,0 @ Â_16×16     [ where Q_2×2 denotes the space of 2×2 quaternion matrices etc. ] but we do not provide example basies here. More generally, for even p+q, Â_p,q is isomorphic to either a real, complex, or quaternion matrix algbera. For odd p+q we sometimes have a sum (ordered pair) of two such matrix algebras.

Adding Blades
    Expressing multivectors in coordinate or matrix forms induces a natural addition of multivectors and provides a potent computational technique, but regarding multivector a as being the real weighted sum of the 2^N outter product progency of N elemental "generators" e₁,e₂,...e_N or a particular real matrix risks loosing sight of duality. All the "information" in a blade b is also "encoded" in its dual bi^-1 and we might equally naturally regard a as bb +gc + .. where b = b₁ + b₂i etc. as a sum of "complex"-weighted blades, particularly for odd N when i is central and (when i²=-1) we have Â_p,q @ C_p-1,q @ C_p,q-1. Because we can express a in terms of an extended basis we know that any multivector a can be expressed as a complex-weighted sum of 2^N-1 blades but it is the minimal number of blades necessasry that best "classifies" the multivector. We define the positive integer spread of a multivector a to be the number of blades in its "irreducible" real weighted summation.   The spreads of 1 and of e₁₃ + e₂₃ = (e₁+e₂)Ùe₃  are 1, for example. However the spread does not fully categorise the "complexity" of a because we don;t know how many blades appear with their duals. We will refer to the Expression of a multivector a suach as bb +gc  of reduced spread 2 in coordinate form "smears" b and c together. The spread and reduced spreads are not apparent.
    Though the addition of multivectors is well defined and frame-independant, it is in many cases of somewhat dubious merit, particularly when adding high glade blades. Though the coordinate approach facilitates and encourages multivector additions and we will do so freely throughout this work, the strict mathmatical purist is nonetheless correct to maintain an element of unease. Informally, multivectors are made for multiplying, not adding.

Multivector Products


Restricted products
    Given the geometric product we can define a large variety of partial-geometric or restricted products by evaluating ab and then throwing away (zeroing the "coordinate" of) one or more particular blades. We might zero all the odd blades, for example, but more useful restricted products arise when the blades we zero are determined by the blades we multiply. The outter product is an example, with a_kÙb_l = (a_kb_l)_<k+l> . Though all these products can in principle be implimented using a geometric product primative, this is frequently grossly computationally inefficient and we will later see that one should generalise our encoded multivector product to "streamline" the evaluation of the more useful restricted products.
    There are two "outer" products and a plethora of potential "inner" products, most of which have something to commend them. All are restrictions of the geometric product in that their result for two extended basis blades is either the same as that of the geometric product or zero. From a programmer's perspective, the contractive inner product ¿ (defined below) is the most fundamental inner product. The reader should accordingly familiarise himself with ¿ , and also the Hestenes (.) product since it is ubiquitous in much of the literature, often required in this work, and (unlike ¿) is "dual" to the outter product . The other inner produts (·, ë, ¿₊) are not used here and seldom seen elsewhere. They may have their place in a particular application, however, and are defined here for completeness.

The outer product
      Given the geometric product, we can fully define the outer or exterior product (aka. wedge product)   by defining
    a_kÙb_l º (a_kb_l)_<k+l>     where a_k,b_l are blades of any grade;
and thence extending over multivectors by insisting on associativity and multilinearity.
    From this definition it follows that:
    aÙb = bÙa = ab ;
    a_kÙb_l = (-1)^klb_lÙa_k    ;    
    aÙb = ½(ab - ba) º a×b    ;     aÙb_k = ½(ab_k + (-1)^kb_ka) = ½(ab_k + b_k^#a)    ;     b_kÙa = (-1)^kaÙb_k
    a₁Ùa₂Ù...a_k = k!^-1 å_ij..me_ij..m a_i a_j... a_m     summing over all k! permutations of {1,2,..k} .
    (aÙb)²   =   (a¿b)² - (a²)(b²)   =   -a²b² sin(q)     where q is angle subtended by a and b
[ Proof :  (aÙb)² = -(aÙb)(bÙa) = -(ab-a.b)(ba-b.a) = -abba+(a.b)(ab+ba)-(a.b)² = -a²b² + 2(a.b)² - (a.b)²  .]
    (a₁Ùa₂Ù...a_k)² = (-1)^½k(k-1) (k! Volume(k-simplex of points 0,a₁,a₂,...a_k))² = (-1)^½k(k-1) (Volume(parallelepiped with edge vectors a₁,a₂,...a_k))²
    a_kÙa_k=0 is true for single proper blade a_k but not for general k-vectors when k>1. Note that 0-blades (scalars) have aÙa=a² .

aÙb for Â3		a
		1	e1	e2	e3	e23	e31	e12	e123
	1	1	e1	e2	e3	e23	e31	e12	e123
	e1	e1	0	-e12	e31	e123	0	0	0
	e2	e2	e12	0	-e23	0	e123	0	0
b	e3	e3	-e31	e23	0	0	0	e123	0
	e23	e23	e123	0	0	0	0	0	0
	e31	e31	0	e123	0	0	0	0	0
	e12	e12	0	0	-e123	0	0	0	0
	e123	e123	0	0	0	0	0	0	0

The "thin" outer product
    The thin outter product is here defined and denoted by by
    a^b º aÙb - a_<0> b - b_<0> a so that a^b = b^a = 0.
    It has the same multiplication table as Ù but with the scalar row and column zeroed. We make no use of it in this work.

The contractive inner product

    We define the contractive inner product (aka. Lounesto inner product) (¿ or û ) by
    a_k¿b_l º ( a_kb_l)_<l-k>       for l³k    ,    0  else   
where a_k,b_l are blades of any grade; and thence extending over multivectors by insisting on bilinearity.
    ¿ is neither associative nor commutative (symmetric). In particular, (a¿a)=a² so  (a¿a)¿b=a² b but a¿(a¿b) = 0 .
    For k-vectors with k>1, contraction with a 1-vector a corresponds to orthogonal projection into subspace a^* so
a¿b is the "component factor" of blade b perpendicular to 1-vector a   .

    We have the following properties:

• a¿b = ab    ;     b¿a = ab_<0>
• a¿b = ½(ab+ba)
• ab = a¿b + aÙb
• a¿b_k = ½(ab_k - (-1)^kb_ka) = ½(ab_k - b_k^#a)     k³0
• a¿(bÙc) = (a¿b)Ùc + b^*Ù(a¿c)
• a¿(b¿c) = (aÙb)¿c         [ the ¿ for Ù swapping rule ].     Þ a¿(a¿b)   = 0
• a¿(b¿j) = (aÙb)¿j     where j is any pseudoscalar spanning a subspace containing b.
• a¿(b₁Ù...Ùb_k) = å_i=1^k (-1)ⁱ⁺¹(a¿b_i) (b₁Ù..Ùb_i-1Ùb_i+1Ù...Ùb_k)     [ the expanded k-blade contraction rule]. Examples include:
      a¿(bÙc) = (a¿b)c - (a¿c)b       [ the expanded bivector contraction rule] giving a¿(bÙc) = - a×(b×c) in Â₃ .
      a¿(bÙcÙd) =(a¿b)(cÙd) - (a¿c)(bÙd) + (a¿d)(bÙc)
• (a₁Ù...Ùa_r)¿(b₁Ù...Ùb_s) = (a₁Ù...Ùa_r-1)¿(a_r¿(b₁Ù...Ùb_s)) = å_i=1^s (-1)ⁱ⁺¹(a_r¿b_i)(a₁Ù...Ùa_r-1) ¿(b₁Ù...Ùb_i-1Ùb_i+1Ù...b_s)
  [  note ¿ rather than . and that expression is vanishingly true for r>s ]
      Whence (a_kÙ...Ùa₂Ùa₁)¿(b₁Ùb₂Ù...Ùb_k) = Det([a_{i j}]) where a_{i j} = a_i¿b_j .
      In particular,     (a₁Ùa₂)¿(b₁Ùb₂) = a₁¿(a₂¿(b₁Ùb₂)) = (a₂¿b₁)(a₁¿b₂) - (a₁¿b₁)(a₂¿b₂)
• a¿(b₂¿c₃) = (aÙb₂)¿c₃     where b₂ is a 2-vector and c₃ is a 3-vector.
• (aÙb)(a+b) = (a²+a¿b)(a-b)     when a²=b²
• a¿(b_kÙc) = (a¿b_k)Ùc + (-1)^kb_kÙ(a¿c).     called the expanded contraction rule with special case a¿(bÙc) = (a¿b)c - bÙ(a¿c).

    ¿ is sometimes known as left-contraction or onto-contraction as opposed to the right- or by-contraction defined by
    a_k ë b_m º ( a_kb_m)_<k-m>       for k³m ; 0 else
or equivalently by aëb = (a^§¿b^§)^§ where ^§ is the reverse operator defined below.

The semi-commutative inner product
    Some authors favour the semi-symmetric or semi-commutative inner product (aka. Hestenes inner product) (.) defined by
    a.b_k º b_k.a º 0 where b_k is any blade  ;  a_k.b_m º ( a_kb_m)_<|k-m|> where a_k,b_m are proper (nonscalar) blades;
and thence extending over multivectors by inisiting on bilinearity. The result is neither associative nor commutative ("symmetric").  It is "semi-symmetric" in that a_j.b_k = (-1)^j(k-j)b_k.a_j for pure j and k-vectors a_j,b_k with j£k.
    Note that scalars (0-blades) a,b satisfy aÙb = ab ; a¿b = a.b = 0 so we can think of scalars as "self-orthogonal".
    We obtain the identities

• a.b_k = (-1)^k+1b_k.a .
• b_ka = b_k.a + b_kÙa     for kÎ{1,2}
• a.b_k = ½(ab_k - (-1)^kb_ka)    ;     b_k.a = (-1)^k+1a.b_k .
• a.(b_kÙc) = (a.b_k)Ùc + (-1)^kb_kÙ(a.c).     called the expanded inner product rule with special case a.(bÙc) = (a.b)c - bÙ(a.c).
• (aÙb_k)i =  a.(b_ki)
• c₂.(aÙb) = (b.c₂).a = b.(c₂.a) =_( )= b . c₂ . a

    We will use ¿ in preference to . whenever possible trhoughout this work. a¿b = a.b " 1-vector a if b_<0>=0 so in many equations ¿ and . are interchangeable. However, only . is dual to Ù in the sense that a.(bi)   =   (aÙb)i    ;     ie. aÙ(bi)   =   (a.b)i ; or equivalently a.(b^*) = (aÙb)^* ; aÙ(b^*) = (a.b)^* ;
    a.b = (aÙ(bi^-1))i    ;     aÙb = (a.(bi^-1))i    for any 1-vector a and multivector b .
    In particular aÙ(b.c) = (a.(bÙ(ci^-1)))i , ie. (aÙ(b.c))^* = a.(bÙ(c^*)) .
    We cannot replace . by ¿ in these fundamental duality equations.

    In Â₃ we have a×b = (aÙb)i^-1 ; a×(b×c) = (aÙ((bÙc)i^-1))i^-1 = (a.c)b - (a.b)c ; and aÙbÙc = (a.(b×c))i where × denotes the traditional ³ vector product.

The "fatdot" inner product
    A variation of the sem-commutative inner product defined by   a_k·b_m º ( a_kb_m)_<|k-m|>     k,m ³ 0.
    This is sometimes known as the modified Hestenes or dot product but we will call it the fatdot product here to avoid confusion with the traditional (Hestenes) inner product .
    The multiplication table for · is the same as that for . except with regard to scalars. The "scalar" row and column are filled as for Ù , ie. according to 1·a  = a·1 = a    .
    We can think of · as "abrogating" the "scalar handling" from Ù, reducing it to the "thin" outter product ^. Thus · and ^ provide an alternate "decomposition" of the geometric product to . and Ù which may be fruitful in some contexts but will not be exploited here.

The forced Euclidean contractive inner product
    The forced Euclidean contractive inner product ¿₊ "overrides" the signatures of the vectors on which it operates. In a Euclidean space Â_N , ¿₊ = ¿ .   In Â_p,q , ¿₊ is defined with regard to an orthonormal basis {e_i} . We take 1¿₊a º a and  e_i¿₊e_j º |e_i¿e_j| and extend ¿₊ bilinearly.
    The tabulation of ¿₊ with regard to the extended (multivector) basis for Â_p,q is thus the tabulation for ¿ for Â_p+q.

    Although frame dependant, ¿₊ is a useful product computationally, since one can sometimes simply "lift" a problem (such as computing meets and joins) in a nonEuclidean space into Euclidean space and solve it there.
    We define the frame-dependant forced Euclidean geometric product by extending a¨₊b º a¿₊b + aÙb over Â_p,q linearly and asscoiatively so that the tabulation of ¨₊ for a given basis for Â_p,q is the tabulation for ¨ for Â_p+q.

    We will later consider higher dimensional embeddings of U^N obtained by adding "extension" dimensions e₊ and e_- to a basis and in such cases will often wish to retain the negative signature of the extender while forcing U^N Euclidean. We represent the unextended forced Euclidean contractive product by ¿₍₊₎ and the unextended forced Euclidean geometeric product by ¨₍₊₎ .

Commutator product
    The antisymmetric commutator or product is defined by a×b º ½(ab-ba). It is our first non-partial-geometric product, potentially containing blades not present in ab. [  aka. , without the ½ factor, as the Lie product ]
    × is nonassociative, and often represents the appropriate generalisation of the ³ vector product ×.
    We have

• a×b = - b×a .
• (a×b)² = ¼((ab)² + (ba)² -2a²b²) procided a² commutes with b and b² commutes with a
• (a×b)×c ¹ a×(b×c)     in general. We say the commutator product is "nonassociative".
• a×(b×c) + b×(c×a) + c×(a×b) = 0     . This is known as the Jacobi Identity.
• ba = ab - 2a×b
• a(a×b) = - (a×b)a provided a²b = ba²
• a×b_k = aÙb_k  if k is odd ; a¿b_k else.
• a×b = 0
• a×b = aÙb
• a₂×b = a₂.b = - b¿a₂.
• a₂b_k = a₂¿b_k + a₂Ùb_k + a₂×b_k     for k ³ 0.
      Þ b_ka₂ = 2 (a₂¿b_k + a₂Ùb_k - ½a₂b_k) = 2(b_këa₂ + b_kÙa₂) - a₂b_k = b_këa₂ + b_kÙa₂ + b_k×a₂     [  Derives from the pure product rule below. Fails for k=1 if ¿ replaced by . .   ]
• a₂×b_k = (a₂b_k)_<k>     In particular a₂×b₂ is a pure bivector.
• a₂×(b_<k>) = (a₂×b)_<k>
• (aÙb)×c = a(b.c)-(c.a)b
• (aÙb)×c₂ = aÙ(b.c₂) - bÙ(a.c₂)
• a×b_k = (a×b_k)_{<k>     [ GAfp 4.63 ]}

    If a and b have a^{^}=a and b^{^}=b or a^{^}-a and b^{^}=-b for reversing conjugation ^{^} then (a×b)^{^} = -a×b . Thus the commutator product of two same grade blades has grade <2;6;10;...>.

    We have 2^3N frame-dependant commutation coefficients C^ij..._kl... lm... º e^ij.. _* (e_kl..×e_lm...) which can be regarded as measuring the nonorthogonality of the frame.

    With (a×) denoting the operator v ® a×v, we have (a×)(b×)v = ½(a×)(bv-vb) = ¼(abv +vba - avb - bva) .
    Whence ((a×)×(b×))v = ¼((a×b)v + v(b×a)) =  ½(a×b)×v .

a×b for Â3		a
		1	e1	e2	e3	e23	e31	e12	e123
	1	0	0	0	0	0	0	0	0
	e1	0	0	-e12	e31	0	e3	-e2	0
	e2	0	e12	0	-e23	-e3	0	e1	0
b	e3	0	-e31	e23	0	e2	-e1	0	0
	e23	0	0	e3	-e2	0	e12	-e31	0
	e31	0	-e3	0	e1	-e12	0	e23	0
	e12	0	e2	-e1	0	e31	-e23	0	0
	e123	0	0	0	0	0	0	0	0

    We extend this definiton of × to give commutator "products" of operators and functions. If ¦(a) and g(a) are multivector-valued functions of multivectors we define (¦×g)(a) º ½(¦(g(a)) - g(¦(a)) º ½(¦g(a) - g¦(a)) .

    We also define ¦^×(a,b) º ¦(a)צ(b) .

    We define the k-fold commutator product by
    a×^kb º a×(a×(....(a×(a×b))..)) = (-1)^k ((...(b×a)×a)×...)a     where there are k a's and ×'s in the sequence.
    a^×0b = b ;     a^×1b = ½(ab-ba) ;     a^×2b = ¼(a²b-2aba+ba²) ;     ... ;     a^×kb = 2^-kå_i=0^k  (-1)^{k k}C_i a^k-ibaⁱ .

AntiCommutator product
    The symmetric anticommutator product is defined by a~b º ½(ab+ba).

    The geometric product decomposes naturally as ab = a×b + a~b . Provided a² commutes with b and b² commutes with a, then a and b both commute with a~b and anticommute with a×b. Further a×b commutes with a~b.
    We have

• a~b =  b~a .
• (a~b)² = ¼((ab)² + (ba)² +2a²b²) = (a×b)² + a²b² provided a² commutes with b and b² commutes with a

    The anticommutator product of two same grade blades has grade <4;8;12;...>.

Scalar product
    The commutative (symmetric) scalar product is defined by a_*b º (ab)_<0> = (ab)¿1 .
    It satisfies a_{k *}b_k = a_k¿b_k .

    Taking the scalar product with 1 corresponds to taking the scalar part a_*1 = a_<0> = a¿1 .
    We have the powerful cyclic scalar rule
    (abc)_<0> = (cab)_<0>     for all multivectors a,b,c

    From a programmers' perspective the scalar product can be regarded both as a multivector-valued product returning a 0-grade (ie. scalar valued) multivector, and as a real-valued bilinear function of two multivectors. For code efficiency it is frequently preferable to impliment it returning a real rather than a multivector type.

Scalar-Pseudoscalar product
    The commutative scalar-pseudoscalar product is defined by a_*b º (ab)_<0,N> = (ab)¿(1+i) .
    For N odd we will also refer to _* as the central product since it is central-valued and we have the vastly powerful cyclic scalar-pseudoscalar rule (abc)_<0;N> = (cab)_<0;N> .
    For even N, the "central product" coincides with _* rather than _*.

Inversive product
    a*b º (a¿b) / (|a||b|)    .
    Thus a*b = cos(q_Ð(a , b))     where the magnitude |a| is as defined below and q_Ð(a,b) denotes the "angle subtended by" 1-vectors a and b.

Delta products
    The greater delta product D (aka. disjoint) is a nonbilinear restrictive product returning the highest grade nonvanishing component of the geometric product, or zero if the geometric product is zero. This can be computed, albeit somewhat inefficiently, by calculating the geometric product keeping track of, or subsequently seeking, the highest grade attained by the blades in the product, and then restricting the product to only this grade. It is most commonly applied to blades and spans the "difference" between them, so for example ((e₁+e₂)Ùe₃₄) D e₂₃₅ = -e₁₂₄₅.   It is profoundly useful in constructing meets and joins.

    Of course, calculating a delta product can present us with dilemas when small coordinate values are encountered. If a product ab has a large scalar part and some comparatively tiny 2-blade coordinates, is the highest "nonvanishing" grade zero or two? Are the small 2-vector coordinates a genuine geometrical artifact, or just "failed zeroes" arising from finite precision computations? What magnitudes may we discard? Our choice will substantively effect the grade and magnitude of aDb and our computations risk becoming more art than science. Nonetheless, the delta product is profoundly useful since in many cases in can be computed unambiguously and is then fundamental in constructing meets,  joints, and disjoints as described later.

    We also have the lesser delta product returning the nonzero component of minimal grade adb = (ab)_Min .

Conjugative Products
    For every multivector conjugation ^{^} there are two associated restricted geometric products. One restricting to the "^{^}-real" blades (b^{^} = b) , and one to the "^{^}-imaginary" blades with b^{^}=-b.

Rescaled Product
    The recaled product is defined by a§b º (a_rsccb_rscc)_rscc where x_rscc is an arbitary, potentially frame dependant, computationally convenient rescaling normalisation of x such as division by the maximal absolute coordinate, or the sum of absolute coordinates, of x.in a particvular frame.. a§b is equivalent to a¨b apart from scale.
    The recalsed product is particularly useful when evaluating sandwich rotor products such as (½qa)^↑ x (-½qa)^↑ for large q when intermediary coordinates can aquire collosal magnitudes and rounding errors become significant. The evaluation of (½qa)^↑ §  x § (-½qa)^↑.is usually far better behaved ; and the correct scale can easily be recovered for non null x from , ((½qa)^↑ x (-½qa)^↑)² =  x² whenver x² commutes with a .

Pure Product Rule
    a_kb_l = (a_kb_l)_<|k-l|> + (a_kb_l)_<|k-l|+2> + .. + (a_kb_l)_<k+l>   =   å_m=0^{½(k+l-|k-l|)}    (a_kb_l)_<|k-l|+2m>
hence the geometric product of two pure multivectors is either odd or even (as defined under Involution below) according as integer k-l is odd or even.
[ Proof :  True for k=1 and then by induction on k for single k-blade a_k. Result follows by distributivity.  .]

The Intersective Product
    The intersective product is a not a restricted geometric product and is frame dependant. It picks out the "shared axies" eg. e_3Çe₁₂₃=e₃. In programming terms, while the geometeric product is based on the exclusive or (XOR) of the bitwise ennumeration indices, the intersective product is based on their logical AND. We bring in the sign changes due to commutations ( e_2Çe₁₂ = -e₂) but not the signatures.