Geometric algebra
Inmathematics,ageometric algebra(also known as aClifford algebra) is an extension ofelementary algebrato work with geometrical objects such asvectors.Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects calledmultivectors.Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division (though generally not for all elements) and addition of objects of different dimensions.
The geometric product was first briefly mentioned byHermann Grassmann,[1]who was chiefly interested in developing the closely relatedexterior algebra.In 1878,William Kingdon Cliffordgreatly expanded on Grassmann's work to form what are now usually called Clifford algebras in his honor (although Clifford himself chose to call them "geometric algebras" ). Clifford defined the Clifford algebra and its product as a unification of theGrassmann algebraand Hamilton'squaternion algebra.Adding thedualof the Grassmann exterior product (the "meet" ) allows the use of theGrassmann–Cayley algebra,and aconformal versionof the latter together with a conformal Clifford algebra yields aconformal geometric algebra(CGA) providing a framework forclassical geometries.[2]In practice, these and several derived operations allow a correspondence of elements,subspacesand operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by thevector calculusthen newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s byDavid Hestenes,who advocated its importance to relativistic physics.[3]
The scalars and vectors have their usual interpretation and make up distinct subspaces of a geometric algebra.Bivectorsprovide a more natural representation of thepseudovectorquantities of 3D vector calculus that are derived as across product,such as oriented area, oriented angle of rotation, torque, angular momentum and themagnetic field.Atrivectorcan represent an oriented volume, and so on. An element called ablademay be used to represent a subspace andorthogonal projectionsonto that subspace. Rotations and reflections are represented as elements. Unlike a vector algebra, a geometric algebra naturally accommodates any number of dimensions and any quadratic form such as inrelativity.
Examples of geometric algebras applied in physics include thespacetime algebra(and the less commonalgebra of physical space) and theconformal geometric algebra.Geometric calculus,an extension of GA that incorporatesdifferentiationandintegration,can be used to formulate other theories such ascomplex analysisanddifferential geometry,e.g. by using the Clifford algebra instead ofdifferential forms.Geometric algebra has been advocated, most notably byDavid Hestenes[4]andChris Doran,[5]as the preferred mathematical framework forphysics.Proponents claim that it provides compact and intuitive descriptions in many areas includingclassicalandquantum mechanics,electromagnetic theory,andrelativity.[6]GA has also found use as a computational tool incomputer graphics[7]androbotics.
Definition and notation
[edit]There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic,[8]"full of geometric significance" and equivalent to the universal[a]Clifford algebra.[9] Given a finite-dimensional vector spaceover afieldwith a symmetric bilinear form (theinner product,[b]e.g., the Euclidean orLorentzian metric),thegeometric algebraof thequadratic spaceis theClifford algebra,an element of which is called a multivector. The Clifford algebra is commonly defined as aquotient algebraof thetensor algebra,though this definition is abstract, so the following definition is presented without requiringabstract algebra.
- Definition
- A unital associative algebrawith anondegeneratesymmetric bilinear formis the Clifford algebra of the quadratic spaceif[10]
- it containsandas distinct subspaces
- for
- generatesas an algebra
- is not generated by any proper subspace of.
To cover degenerate symmetric bilinear forms, the last condition must be modified.[c]It can be shown that these conditions uniquely characterize the geometric product.
For the remainder of this article, only therealcase,,will be considered. The notation(respectively) will be used to denote a geometric algebra for which the bilinear formhas thesignature(respectively).
The product in the algebra is called thegeometric product,and the product in the contained exterior algebra is called theexterior product(frequently called thewedge productor theouter product[d]). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol.
The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectors:
- (closure)
- ,whereis the identity element (existence of anidentity element)
- (associativity)
- and(distributivity)
- for.
The exterior product has the same properties, except that the last property above is replaced byfor.
Note that in the last property above, the real numberneed not be nonnegative ifis not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vector,ifthenexists and is equal to.A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, ifis a vector insuch that,the elementis both a nontrivialidempotent elementand a nonzerozero divisor,and thus has no inverse.[e]
It is usual to identifyandwith their images under the naturalembeddingsand.In this article, this identification is assumed. Throughout, the termsscalarandvectorrefer to elements ofandrespectively (and of their images under this embedding).
Geometric product
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/7/79/GA_parallel_and_perpendicular_vectors.svg/200px-GA_parallel_and_perpendicular_vectors.svg.png)
For vectorsand,we may write the geometric product of any two vectorsandas the sum of a symmetric product and an antisymmetric product:
Thus we can define theinner productof vectors as
so that the symmetric product can be written as
Conversely,is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the containedexterior algebra:
Then by simple addition:
- the ungeneralized or vector form of the geometric product.
The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically,andareparallelif their geometric product is equal to their inner product, whereasandareperpendicularif their geometric product is equal to their exterior product. In a geometric algebra for which the square of any nonzero vector is positive, the inner product of two vectors can be identified with thedot productof standard vector algebra. The exterior product of two vectors can be identified with thesigned areaenclosed by aparallelogramthe sides of which are the vectors. Thecross productof two vectors indimensions with positive-definite quadratic form is closely related to their exterior product.
Most instances of geometric algebras of interest have a nondegenerate quadratic form. If the quadratic form is fullydegenerate,the inner product of any two vectors is always zero, and the geometric algebra is then simply an exterior algebra. Unless otherwise stated, this article will treat only nondegenerate geometric algebras.
The exterior product is naturally extended as an associative bilinear binary operator between any two elements of the algebra, satisfying the identities
where the sum is over all permutations of the indices, withthesign of the permutation,andare vectors (not general elements of the algebra). Since every element of the algebra can be expressed as the sum of products of this form, this defines the exterior product for every pair of elements of the algebra. It follows from the definition that the exterior product forms analternating algebra.
The equivalent structure equation for Clifford algebra is[16][17]
whereis thePfaffianofandprovidescombinations,,ofindices divided intoandparts andis theparityof thecombination.
The Pfaffian provides a metric for the exterior algebra and, as pointed out by Claude Chevalley, Clifford algebra reduces to the exterior algebra with a zero quadratic form.[18]The role the Pfaffian plays can be understood from a geometric viewpoint by developing Clifford algebra fromsimplices.[19]This derivation provides a better connection betweenPascal's triangleandsimplicesbecause it provides an interpretation of the first column of ones.
Blades, grades, and basis
[edit]A multivector that is the exterior product oflinearly independent vectors is called ablade,and is said to be of grade.[f]A multivector that is the sum of blades of gradeis called a (homogeneous) multivector of grade.From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.
Consider a set oflinearly independent vectorsspanning an-dimensional subspace of the vector space. With these, we can define a realsymmetric matrix(in the same way as aGramian matrix)
By thespectral theorem,can be diagonalized todiagonal matrixby anorthogonal matrixvia
Define a new set of vectors,known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:
Since orthogonal transformations preserve inner products, it follows thatand thus theare perpendicular. In other words, the geometric product of two distinct vectorsis completely specified by their exterior product, or more generally
Therefore, every blade of gradecan be written as the exterior product ofvectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by ablock matrixthat is orthogonal in the nondegenerate block, and the diagonal matrix has zero-valued entries along the degenerate dimensions. If the new vectors of the nondegenerate subspace arenormalizedaccording to
then these normalized vectors must square toor.BySylvester's law of inertia,the total number ofand the total number ofs along the diagonal matrix is invariant. By extension, the total numberof these vectors that square toand the total numberthat square tois invariant. (The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed.) We denote this algebra.For example,models three-dimensionalEuclidean space,relativisticspacetimeandaconformal geometric algebraof a three-dimensional space.
The set of all possible products oforthogonal basis vectors with indices in increasing order, includingas the empty product, forms a basis for the entire geometric algebra (an analogue of thePBW theorem). For example, the following is a basis for the geometric algebra:
A basis formed this way is called astandard basisfor the geometric algebra, and any other orthogonal basis forwill produce another standard basis. Each standard basis consists ofelements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements arewithbeing an index set, then the geometric product of any two multivectors is
The terminology "-vector "is often encountered to describe multivectors containing elements of only one grade. In higher dimensional space, some such multivectors are not blades (cannot be factored into the exterior product ofvectors). By way of example,incannot be factored; typically, however, such elements of the algebra do not yield to geometric interpretation as objects, although they may represent geometric quantities such as rotations. Only-,-,- and-vectors are always blades in-space.
Versor
[edit]A-versor is a multivector that can be expressed as the geometric product ofinvertible vectors.[g][21]Unit quaternions (originally called versors by Hamilton) may be identified with rotors in 3D space in much the same way as real 2D rotors subsume complex numbers; for the details refer to Dorst.[22]
Some authors use the term "versor product" to refer to the frequently occurring case where an operand is "sandwiched" between operators. The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. These outermorphisms have a particularly simple algebraic form.[h]Specifically, a mapping of vectors of the form
- extends to the outermorphism
Since both operators and operand are versors there is potential for alternative examples such as rotating a rotor or reflecting a spinor always provided that some geometrical or physical significance can be attached to such operations.
By theCartan–Dieudonné theoremwe have that every isometry can be given as reflections in hyperplanes and since composed reflections provide rotations then we have that orthogonal transformations are versors.
In group terms, for a real, non-degenerate,having identified the groupas the group of all invertible elements of,Lundholm gives a proof that the "versor group"(the set of invertible versors) is equal to the Lipschitz group(a.k.a.Clifford group, although Lundholm deprecates this usage).[23]
Subgroups of the Lipschitz group
[edit]We denote the grade involution asand reversion as.
Although the Lipschitz group (defined as) and the versor group (defined as) have divergent definitions, they are the same group. Lundholm defines the,,andsubgroups of the Lipschitz group.[24]
Subgroup | Definition | GA term |
---|---|---|
versors | ||
unit versors | ||
even unit versors | ||
rotors |
Multiple analyses of spinors use GA as a representation.[25]
Grade projection
[edit]A-graded vector spacestructure can be established on a geometric algebra by use of the exterior product that is naturally induced by the geometric product.
Since the geometric product and the exterior product are equal on orthogonal vectors, this grading can be conveniently constructed by using an orthogonal basis.
Elements of the geometric algebra that are scalar multiples ofare of gradeand are calledscalars.Elements that are in the span ofare of gradeand are the ordinary vectors. Elements in the span ofare of gradeand are the bivectors. This terminology continues through to the last grade of-vectors. Alternatively,-vectors are calledpseudoscalars,-vectors are called pseudovectors, etc. Many of the elements of the algebra are not graded by this scheme since they are sums of elements of differing grade. Such elements are said to be ofmixed grade.The grading of multivectors is independent of the basis chosen originally.
This is a grading as a vector space, but not as an algebra. Because the product of an-blade and an-blade is contained in the span ofthrough-blades, the geometric algebra is afiltered algebra.
A multivectormay be decomposed with thegrade-projection operator,which outputs the grade-portion of.As a result:
As an example, the geometric product of two vectorssinceandand,forother thanand.
A multivectormay also be decomposed into even and odd components, which may respectively be expressed as the sum of the even and the sum of the odd grade components above:
This is the result of forgetting structure from a-graded vector spaceto-graded vector space.The geometric product respects this coarser grading. Thus in addition to being a-graded vector space,the geometric algebra is a-graded algebra,a.k.a.asuperalgebra.
Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines aneven subalgebra.The even subalgebra of an-dimensional geometric algebra isalgebra-isomorphic(without preserving either filtration or grading) to a full geometric algebra ofdimensions. Examples includeand.
Representation of subspaces
[edit]Geometric algebra represents subspaces ofas blades, and so they coexist in the same algebra with vectors from.A-dimensional subspaceofis represented by taking an orthogonal basisand using the geometric product to form theblade.There are multiple blades representing;all those representingare scalar multiples of.These blades can be separated into two sets: positive multiples ofand negative multiples of.The positive multiples ofare said to havethe sameorientationas,and the negative multiples theopposite orientation.
Blades are important since geometric operations such as projections, rotations and reflections depend on the factorability via the exterior product that (the restricted class of)-blades provide but that (the generalized class of) grade-multivectors do not when.
Unit pseudoscalars
[edit]Unit pseudoscalars are blades that play important roles in GA. Aunit pseudoscalarfor a non-degenerate subspaceofis a blade that is the product of the members of an orthonormal basis for.It can be shown that ifandare both unit pseudoscalars for,thenand.If one doesn't choose an orthonormal basis for,then thePlücker embeddinggives a vector in the exterior algebra but only up to scaling. Using the vector space isomorphism between the geometric algebra and exterior algebra, this gives the equivalence class offor all.Orthonormality gets rid of this ambiguity except for the signs above.
Suppose the geometric algebrawith the familiar positive definite inner product onis formed. Given a plane (two-dimensional subspace) of,one can find an orthonormal basisspanning the plane, and thus find a unit pseudoscalarrepresenting this plane. The geometric product of any two vectors in the span ofandlies in,that is, it is the sum of a-vector and a-vector.
By the properties of the geometric product,.The resemblance to theimaginary unitis not incidental: the subspaceis-algebra isomorphic to thecomplex numbers.In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace ofon which the quadratic form is definite.
It is sometimes possible to identify the presence of an imaginary unit in a physical equation. Such units arise from one of the many quantities in the real algebra that square to,and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.
In,a further familiar case occurs. Given a standard basis consisting of orthonormal vectorsof,the set ofall-vectors is spanned by
Labelling these,and(momentarily deviating from our uppercase convention), the subspace generated by-vectors and-vectors is exactly.This set is seen to be the even subalgebra of,and furthermore is isomorphic as an-algebra to thequaternions,another important algebraic system.
Extensions of the inner and exterior products
[edit]It is common practice to extend the exterior product on vectors to the entire algebra. This may be done through the use of the above mentionedgrade projectionoperator:
- (theexterior product)
This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
- (thecommutator product)
The regressive product is the dual of the exterior product (respectively corresponding to the "meet" and "join" in this context).[i]The dual specification of elements permits, for bladesand,the intersection (or meet) where the duality is to be taken relative to the a blade containing bothand(the smallest such blade being the join).[27]
withthe unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.[28]
The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper (Dorst 2002) gives a full treatment of several different inner products developed for geometric algebras and their interrelationships, and the notation is taken from there. Many authors use the same symbol as for the inner product of vectors for their chosen extension (e.g. Hestenes and Perwass). No consistent notation has emerged.
Among these several different generalizations of the inner product on vectors are:
- (theleft contraction)
- (theright contraction)
- (thescalar product)
- (the "(fat) dot" product)[j]
Dorst (2002)makes an argument for the use of contractions in preference to Hestenes's inner product; they are algebraically more regular and have cleaner geometric interpretations. A number of identities incorporating the contractions are valid without restriction of their inputs. For example,
Benefits of using the left contraction as an extension of the inner product on vectors include that the identityis extended tofor any vectorand multivector,and that theprojectionoperationis extended tofor any bladeand any multivector(with a minor modification to accommodate null,givenbelow).
Dual basis
[edit]Letbe a basis of,i.e. a set oflinearly independent vectors that span the-dimensional vector space.The basis that is dual tois the set of elements of thedual vector spacethat forms abiorthogonal systemwith this basis, thus being the elements denotedsatisfying
whereis theKronecker delta.
Given a nondegenerate quadratic form on,becomes naturally identified with,and the dual basis may be regarded as elements of,but are not in general the same set as the original basis.
Given further a GA of,let
be the pseudoscalar (which does not necessarily square to) formed from the basis.The dual basis vectors may be constructed as
where thedenotes that theth basis vector is omitted from the product.
A dual basis is also known as areciprocal basisor reciprocal frame.
A major usage of a dual basis is to separate vectors into components. Given a vector,scalar componentscan be defined as
in terms of whichcan be separated into vector components as
We can also define scalar componentsas
in terms of whichcan be separated into vector components in terms of the dual basis as
A dual basis as defined above for the vector subspace of a geometric algebra can be extended to cover the entire algebra.[29]For compactness, we'll use a single capital letter to represent an ordered set of vector indices. I.e., writing
where, we can write a basis blade as
The corresponding reciprocal blade has the indices in opposite order:
Similar to the case above with vectors, it can be shown that
whereis the scalar product.
Witha multivector, we can define scalar components as[30]
in terms of whichcan be separated into component blades as
We can alternatively define scalar components
in terms of whichcan be separated into component blades as
Linear functions
[edit]Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup oflinear functionson multivectors, which can still be used when necessary. The geometric algebra of an-dimensional vector space is spanned by a basis ofelements. If a multivector is represented by arealcolumn matrixof coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as thematrix multiplicationby areal matrix. However, such a general linear transformation allows arbitrary exchanges among grades, such as a "rotation" of a scalar into a vector, which has no evident geometric interpretation.
A general linear transformation from vectors to vectors is of interest. With the natural restriction to preserving the induced exterior algebra, theoutermorphismof the linear transformation is the unique[k]extension of the versor. Ifis a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule
for a blade, extended to the whole algebra through linearity.
Modeling geometries
[edit]Although a lot of attention has been placed on CGA, it is to be noted that GA is not just one algebra, it is one of a family of algebras with the same essential structure.[31]
Vector space model
[edit]Theeven subalgebraofis isomorphic to thecomplex numbers,as may be seen by writing a vectorin terms of its components in an orthonormal basis and left multiplying by the basis vector,yielding
where we identifysince
Similarly, the even subalgebra ofwith basisis isomorphic to thequaternionsas may be seen by identifying,and.
Everyassociative algebrahas a matrix representation; replacing the three Cartesian basis vectors by thePauli matricesgives a representation of:
Dotting the "Pauli vector"(adyad):
- with arbitrary vectorsandand multiplying through gives:
- (Equivalently, by inspection,)
Spacetime model
[edit]In physics, the main applications are the geometric algebra ofMinkowski 3+1 spacetime,,calledspacetime algebra(STA),[3]or less commonly,,interpreted thealgebra of physical space(APS).
While in STA, points of spacetime are represented simply by vectors, in APS, points of-dimensional spacetime are instead represented byparavectors,a three-dimensional vector (space) plus a one-dimensional scalar (time).
In spacetime algebra the electromagnetic field tensor has a bivector representation.[32]Here, theis the unit pseudoscalar (or four-dimensional volume element),is the unit vector in time direction, andandare the classic electric and magnetic field vectors (with a zero time component). Using thefour-current,Maxwell's equationsthen become
Formulation Homogeneous equations Non-homogeneous equations Fields Potentials (any gauge) Potentials (Lorenz gauge)
In geometric calculus, juxtaposition of vectors such as inindicate the geometric product and can be decomposed into parts as.Hereis the covector derivative in any spacetime and reduces toin flat spacetime. Whereplays a role in Minkowski-spacetime which is synonymous to the role ofin Euclidean-space and is related to thed'Alembertianby.Indeed, given an observer represented by a future pointing timelike vectorwe have
Boostsin this Lorentzian metric space have the same expressionas rotation in Euclidean space, whereis the bivector generated by the time and the space directions involved, whereas in the Euclidean case it is the bivector generated by the two space directions, strengthening the "analogy" to almost identity.
TheDirac matricesare a representation of,showing the equivalence with matrix representations used by physicists.
Homogeneous models
[edit]Homogeneous models generally refer to a projective representation in which the elements of the one-dimensional subspaces of a vector space represent points of a geometry.
In a geometric algebra of a space ofdimensions, the rotors represent a set of transformations withdegrees of freedom, corresponding to rotations – for example,whenandwhen.Geometric algebra is often used to model aprojective space,i.e. as ahomogeneous model:a point, line, plane, etc. is represented by an equivalence class of elements of the algebra that differ by an invertible scalar factor.
The rotors in a space of dimensionhavedegrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an-dimensional space.
This is the case inProjective Geometric Algebra(PGA), which is used[33][34][35]to representEuclidean isometriesin Euclidean geometry (thereby covering the large majority of engineering applications of geometry). In this model, a degenerate dimension is added to the three Euclidean dimensions to form the algebra.With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are alwaysscrew motionsin 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections.
PGA combineswith acomplementoperator to obtain join, meet, distance, and angle formulas.[36]In effect, the complement switches basis vectors that are present and absent in the expression of each term of the algebraic representation. For example, in the PGA or 3-dimensional space, the complement of the lineis the line,becauseandare basis elements that arenotcontained inbutarecontained in.In the PGA of 2-dimensional space, the complement ofis,since there is noelement.
PGA is a widely used system that combines geometric algebra with homogeneous representations in geometry, but there exist several other such systems. The conformal model discussed below is homogeneous, as is "Conic Geometric Algebra",[37]and seePlane-based geometric algebrafor discussion of homogeneous models of elliptic and hyperbolic geometry compared with the Euclidean geometry derived from PGA.
Conformal model
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/8/8d/Horosphere-3d.svg/300px-Horosphere-3d.svg.png)
Working within GA, Euclidean space(along with a conformal point at infinity) is embedded projectively in the CGAvia the identification of Euclidean points with 1D subspaces in the 4D null cone of the 5D CGA vector subspace. This allows all conformal transformations to be performed as rotations and reflections and iscovariant,extending incidence relations of projective geometry to rounds objects such as circles and spheres.
Specifically, we add orthogonal basis vectorsandsuch thatandto the basis of the vector space that generatesand identifynull vectors
- as the point at the origin and
- as a conformal point at infinity (seeCompactification), giving
(Some authors setand.[36]) This procedure has some similarities to the procedure for working withhomogeneous coordinatesin projective geometry, and in this case allows the modeling ofEuclidean transformationsofasorthogonal transformationsof a subset of.
A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.
Table of models
[edit]Note in this list thatandcan be swapped and the same name applies; for example, withrelativelylittle change occurring, seesign convention.For example,andarebothreferred to as Spacetime Algebra.[38]
Name | Signature | Blades, e.g., oriented geometric objects that algebra can represent | Rotors, e.g.,orientation-preserving transformations that the algebra can represent | Notes |
---|---|---|---|---|
Vectorspace GA, VGA | Planes and lines through the origin | Rotations, eg | First GA to be discovered[by whom?] | |
Projective GA, PGA, Rigid GA, RGA,Plane-based GA | Planes, lines, and points anywhere in space | Rotations and translations, e.g.,rigid motions,aka | Slight modifications to the signature allow for the modelling of hyperbolic and elliptic space, see main article. Cannot model the entire "projective" group. | |
Spacetime Algebra,STA | Volumes, planes and lines through the origin in spacetime | Rotations and spacetime boosts, e.g.,theLorentz group | Basis forGauge Theory Gravity. | |
Spacetime Algebra Projectivized,[39]STAP | Volumes, planes, lines, and points (events) in spacetime | Rotations, translations, and spacetime boosts (Poincare group) | ||
Conformal GA,CGA | Spheres, circles, point pairs (or dipoles), round points, flat points, lines, and planes anywhere in space | Transformations of space that preserve angles (Conformal group) | ||
Conformal Spacetime Algebra,[40]CSTA | Spheres, circles, planes, lines, light-cones, trajectories of objects with constant acceleration, all in spacetime | Conformal transformations of spacetime, e.g. transformations that preserverapidityalong arclengths through spacetime | Related toTwistor theory. | |
Mother Algebra[41] | Unknown | Projective group | ||
GA for Conics, GAC | Points, point pair/triple/quadruple, Conic, Pencil of up to 6 independent conics. | Reflections, translations, rotations, dilations, others | Conics can be created from control points and pencils of conics. | |
Quadric Conformal GA, QCGA[43] | Points, tuples of up to 8 points, quadric surfaces, conics, conics on quadratic surfaces (such asSpherical conic), pencils of up to 9 quadric surfaces. | Reflections, translations, rotations, dilations, others | Quadric surfaces can be created from control points and their surface normals can be determined. | |
Double Conformal Geometric Algebra (DCGA)[44] | Points, Darboux Cyclides, quadrics surfaces | Reflections, translations, rotations, dilations, others | Uses bivectors of two independent CGA basis to represents 5×5 symmetric "matrices" of 15 unique coefficients. This is at the cost of the ability to perform intersections and construction by points. |
Geometric interpretation in the vector space model
[edit]Projection and rejection
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/c/c4/GA_plane_subspace_and_projection.svg/300px-GA_plane_subspace_and_projection.svg.png)
For any vectorand any invertible vector,
where theprojectionofonto(or the parallel part) is
and therejectionoffrom(or the orthogonal part) is
Using the concept of a-bladeas representing a subspace ofand every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible-bladeas[l]
with the rejection being defined as
The projection and rejection generalize to null bladesby replacing the inversewith the pseudoinversewith respect to the contractive product.[m]The outcome of the projection coincides in both cases for non-null blades.[45][46]For null blades,the definition of the projection given here with the first contraction rather than the second being onto the pseudoinverse should be used,[n]as only then is the result necessarily in the subspace represented by.[45] The projection generalizes through linearity to general multivectors.[o]The projection is not linear inand does not generalize to objectsthat are not blades.
Reflection
[edit]Simple reflectionsin a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of generalrotoreflectionsandrotations.
![](https://upload.wikimedia.org/wikipedia/commons/thumb/b/b4/GA_reflection_along_vector.svg/200px-GA_reflection_along_vector.svg.png)
The reflectionof a vectoralong a vector,or equivalently in the hyperplane orthogonal to,is the same as negating the component of a vector parallel to.The result of the reflection will be
This is not the most general operation that may be regarded as a reflection when the dimension.A general reflection may be expressed as the composite of any odd number of single-axis reflections. Thus, a general reflectionof a vectormay be written
where
- and
If we define the reflection along a non-null vectorof the product of vectors as the reflection of every vector in the product along the same vector, we get for any product of an odd number of vectors that, by way of example,
and for the product of an even number of vectors that
Using the concept of every multivector ultimately being expressed in terms of vectors, the reflection of a general multivectorusing any reflection versormay be written
whereis theautomorphismofreflection through the originof the vector space () extended through linearity to the whole algebra.
Rotations
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/e/e9/GA_planar_rotations.svg/200px-GA_planar_rotations.svg.png)
If we have a product of vectorsthen we denote the reverse as
As an example, assume thatwe get
Scalingso thatthen
soleaves the length ofunchanged. We can also show that
so the transformationpreserves both length and angle. It therefore can be identified as a rotation or rotoreflection;is called arotorif it is aproper rotation(as it is if it can be expressed as a product of an even number of vectors) and is an instance of what is known in GA as aversor.
There is a general method for rotating a vector involving the formation of a multivector of the formthat produces a rotationin theplaneand with the orientation defined by a-blade.
Rotors are a generalization of quaternions to-dimensional spaces.
Examples and applications
[edit]Hypervolume of a parallelotope spanned by vectors
[edit]For vectorsandspanning a parallelogram we have
with the result thatis linear in the product of the "altitude" and the "base" of the parallelogram, that is, its area.
Similar interpretations are true for any number of vectors spanning an-dimensionalparallelotope;the exterior product of vectors,that is,has a magnitude equal to the volume of the-parallelotope. An-vector does not necessarily have a shape of a parallelotope – this is a convenient visualization. It could be any shape, although the volume equals that of the parallelotope.
Intersection of a line and a plane
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/f/f1/LinePlaneIntersect.png/220px-LinePlaneIntersect.png)
We may define the line parametrically by,whereandare position vectors for points P and T andis the direction vector for the line.
Then
- and
so
and
Rotating systems
[edit]A rotational quantity such astorqueorangular momentumis described in geometric algebra as a bivector. Suppose a circular path in an arbitrary plane containing orthonormal vectorsandis parameterized by angle.
By designating the unit bivector of this plane as the imaginary number
this path vector can be conveniently written in complex exponential form
and the derivative with respect to angle is
![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/d8/Exterior_calc_cross_product.svg/220px-Exterior_calc_cross_product.svg.png)
For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle. Thus the torque, the rate of change of workwith respect to angle, due to a force,is
Rotational quantities are represented invector calculusin three dimensions using thecross product.Together with a choice of an oriented volume form,these can be related to the exterior product with its more natural geometric interpretation of such quantities as a bivectors by using thedualrelationship
Unlike the cross product description of torque,,the geometric algebra description does not introduce a vector in the normal direction; a vector that does not exist in two and that is not unique in greater than three dimensions. The unit bivector describes the plane and the orientation of the rotation, and the sense of the rotation is relative to the angle between the vectorsand.
Geometric calculus
[edit]Geometric calculus extends the formalism to include differentiation and integration including differential geometry anddifferential forms.[47]
Essentially, the vector derivative is defined so that the GA version ofGreen's theoremis true,
and then one can write
as a geometric product, effectively generalizingStokes' theorem(including the differential form version of it).
In 1D whenis a curve with endpointsand,then
reduces to
or the fundamental theorem of integral calculus.
Also developed are the concept ofvector manifoldand geometric integration theory (which generalizes differential forms).
History
[edit]Before the 20th century
[edit]Although the connection of geometry with algebra dates as far back at least toEuclid'sElementsin the third century B.C. (seeGreek geometric algebra), GA in the sense used in this article was not developed until 1844, when it was used in asystematic wayto describe the geometrical properties andtransformationsof a space. In that year,Hermann Grassmannintroduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to thepropositional calculus) that encoded all of the geometrical information of a space.[48]Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them beingEuclidean space,affine space,andprojective space.Following Grassmann, in 1878William Kingdon Cliffordexamined Grassmann's algebraic system alongside thequaternionsofWilliam Rowan Hamiltonin (Clifford 1878). From his point of view, the quaternions described certaintransformations(which he calledrotors), whereas Grassmann's algebra described certainproperties(orStreckensuch as length, area, and volume). His contribution was to define a new product – thegeometric product– on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently,Rudolf Lipschitzin 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations indimensions. Later these developments would lead other 20th-century mathematicians to formalize and explore the properties of the Clifford algebra.
Nevertheless, another revolutionary development of the 19th-century would completely overshadow the geometric algebras: that ofvector analysis,developed independently byJosiah Willard GibbsandOliver Heaviside.Vector analysis was motivated byJames Clerk Maxwell's studies ofelectromagnetism,and specifically the need to express and manipulate conveniently certaindifferential equations.Vector analysis had a certain intuitive appeal compared to the rigors of the new algebras. Physicists and mathematicians alike readily adopted it as their geometrical toolkit of choice, particularly following the influential 1901 textbookVector AnalysisbyEdwin Bidwell Wilson,following lectures of Gibbs.
In more detail, there have been three approaches to geometric algebra:quaternionicanalysis, initiated by Hamilton in 1843 and geometrized as rotors by Clifford in 1878; geometric algebra, initiated by Grassmann in 1844; and vector analysis, developed out of quaternionic analysis in the late 19th century by Gibbs and Heaviside. The legacy of quaternionic analysis in vector analysis can be seen in the use of,,to indicate the basis vectors of:it is being thought of as the purely imaginary quaternions. From the perspective of geometric algebra, the even subalgebra of the Space Time Algebra is isomorphic to the GA of 3D Euclidean space and quaternions are isomorphic to the even subalgebra of the GA of 3D Euclidean space, which unifies the three approaches.
20th century and present
[edit]Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work ofabstract algebraistssuch asÉlie Cartan,Hermann WeylandClaude Chevalley.Thegeometricalapproach to geometric algebras has seen a number of 20th-century revivals. In mathematics,Emil Artin'sGeometric Algebra[49]discusses the algebra associated with each of a number of geometries, includingaffine geometry,projective geometry,symplectic geometry,andorthogonal geometry.In physics, geometric algebras have been revived as a "new" way to do classical mechanics and electromagnetism, together with more advanced topics such as quantum mechanics and gauge theory.[5]David Hestenesreinterpreted thePauliandDiracmatrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.
Incomputer graphicsand robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (screw theory,kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).
See also
[edit]- Comparison of vector algebra and geometric algebra
- Clifford algebra
- Grassmann–Cayley algebra
- Spacetime algebra
- Spinor
- Quaternion
- Algebra of physical space
- Universal geometric algebra
Notes
[edit]- ^A 'universal' algebra is the most "complete" or least degenerate algebra that satisfies all the defining equations. In this article, by 'Clifford algebra' we mean the universal Clifford algebra.
- ^The terminner productas used in geometric algebra refers to the symmetric bilinear form on the-vector subspace, and is a synonym for thescalar productof apseudo-Euclidean vector space,not theinner producton a normed vector space. Some authors may extend the meaning ofinner productto the entire algebra, but there is little consensus on this. Even in texts on geometric algebras, the term is not universally used.
- ^It may be replaced by the condition that[11]the product of any set of linearly independent vectors inmust not be inor that[12]the dimension of the algebra must be.
- ^The termouter productused in geometric algebra conflicts with the meaning ofouter productelsewhere in mathematics
- ^Given,we have that,showing thatis idempotent, and that,showing that it is a nonzero zero divisor.
- ^Grade is a synonym fordegreeof a homogeneous element under thegrading as an algebrawith the exterior product (a-grading), and not under the geometric product.
- ^"reviving and generalizing somewhat a term from hamilton's quaternion calculus which has fallen into disuse" Hestenes defined a-versor as a multivector which can be factored into a product ofvectors.[20]
- ^Only the outermorphisms of linear transformations that respect the bilinear form fit this description; outermorphisms are not in general expressible in terms of the algebraic operations.
- ^[...] the exterior product operation and the join relation have essentially the same meaning. TheGrassmann–Cayley algebraregards the meet relation as its counterpart and gives a unifying framework in which these two operations have equal footing [...] Grassmann himself defined the meet operation as the dual of the exterior product operation, but later mathematicians defined the meet operator independently of the exterior product through a process calledshuffle,and the meet operation is termed the shuffle product. It is shown that this is an antisymmetric operation that satisfies associativity, defining an algebra in its own right. Thus, the Grassmann–Cayley algebra has two algebraic structures simultaneously: one based on the exterior product (or join), the other based on the shuffle product (or meet). Hence, the name "double algebra", and the two are shown to be dual to each other.[26]
- ^ This should not be confused with Hestenes's irregular generalization,where the distinguishing notation is fromDorst, Fontijne & Mann (2007),p. 590, §B.1, which makes the point that scalar components must be handled separately with this product.
- ^The condition thatis usually added to ensure that thezero mapis unique.
- ^This definition followsDorst, Fontijne & Mann (2007)andPerwass (2009)– the left contraction used by Dorst replaces the ( "fat dot" ) inner product that Perwass uses, consistent with Perwass's constraint that grade ofmay not exceed that of.
- ^Dorst appears to merely assumesuch that,whereasPerwass (2009)defines,whereis the conjugate of,equivalent to the reverse ofup to a sign.
- ^That is to say, the projection must be defined asand not as,though the two are equivalent for non-null blades.
- ^This generalization to allis apparently not considered by Perwass or Dorst.
Citations
[edit]- ^Hestenes 1986,p. 6.
- ^Li 2008,p. 411.
- ^abHestenes 1966.
- ^Hestenes 2003.
- ^abDoran 1994.
- ^Lasenby, Lasenby & Doran 2000.
- ^Hildenbrand et al. 2004.
- ^Hestenes & Sobczyk 1984,p. 3–5.
- ^Aragón, Aragón & Rodríguez 1997,p. 101.
- ^Lounesto 2001,p. 190.
- ^Lounesto 2001,p. 191.
- ^Vaz & da Rocha 2016,p. 58, Theorem 3.1.
- ^abHestenes 2005.
- ^Penrose 2007.
- ^Wheeler, Misner & Thorne 1973,p. 83.
- ^Wilmot 1988a,p. 2338.
- ^Wilmot 1988b,p. 2346.
- ^Chevalley 1991.
- ^Wilmot 2023.
- ^Hestenes & Sobczyk 1984,p. 103.
- ^Dorst, Fontijne & Mann 2007,p. 204.
- ^Dorst, Fontijne & Mann 2007,pp. 177–182.
- ^Lundholm & Svensson 2009,pp. 58et seq.
- ^Lundholm & Svensson 2009,p. 58.
- ^Francis & Kosowsky 2008.
- ^Kanatani 2015,pp. 112–113.
- ^Dorst & Lasenby 2011,p. 443.
- ^Vaz & da Rocha 2016,§2.8.
- ^Hestenes & Sobczyk 1984,p. 31.
- ^Doran & Lasenby 2003,p. 102.
- ^Dorst & Lasenby 2011,p. vi.
- ^"Electromagnetism using Geometric Algebra versus Components".Retrieved2013-03-19.
- ^Selig 2005.
- ^Hadfield & Lasenby 2020.
- ^"Projective Geometric Algebra".projectivegeometricalgebra.org.Retrieved2023-10-03.
- ^abLengyel 2024.
- ^abHrdina, Návrat & Vašík 2018.
- ^Wu 2022.
- ^Sokolov 2013.
- ^Lasenby 2004.
- ^Dorst 2016.
- ^Perwass 2009.
- ^Breuils et al. 2019.
- ^Easter & Hitzer 2017.
- ^abDorst, Fontijne & Mann 2007,§3.6 p. 85.
- ^Perwass 2009,§3.2.10.2 p. 83.
- ^Hestenes & Sobczyk 1984.
- ^Grassmann 1844.
- ^Artin 1988.
References and further reading
[edit]- Arranged chronologically
- Grassmann, Hermann(1844),Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durch Anwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik, Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert,Leipzig: O. Wigand,OCLC20521674
- Clifford, Professor (1878), "Applications of Grassmann's Extensive Algebra",American Journal of Mathematics,1(4): 350–358,doi:10.2307/2369379,JSTOR2369379
- Artin, Emil(1988) [1957],Geometric algebra,Wiley Classics Library, Wiley,doi:10.1002/9781118164518,ISBN978-0-471-60839-4,MR1009557
- Hestenes, David(1966),Space–time Algebra,Gordon and Breach,ISBN978-0-677-01390-9,OCLC996371
- Wheeler, J. A.; Misner, C.; Thorne, K. S. (1973),Gravitation,W.H. Freeman,ISBN978-0-7167-0344-0
- Bourbaki, Nicolas(1980), "Ch. 9" Algèbres de Clifford "",Eléments de Mathématique. Algèbre,Hermann,ISBN9782225655166
- Hestenes, David;Sobczyk, Garret (1984),Clifford Algebra to Geometric Calculus, a Unified Language for Mathematics and Physics,Springer Netherlands,ISBN978-90-277-1673-6
- Hestenes, David(1986), "A Unified Language for Mathematics and Physics", in J.S.R. Chisholm; A.K. Commons (eds.),Clifford Algebras and Their Applications in Mathematical Physics,NATO ASI Series (Series C), vol. 183, Springer, pp. 1–23,doi:10.1007/978-94-009-4728-3_1,ISBN978-94-009-4728-3
- Wilmot, G.P. (1988a),The Structure of Clifford algebra. Journal of Mathematical Physics,vol. 29, pp. 2338–2345
- Wilmot, G.P. (1988b), "Clifford algebra and the Pfaffian expansion",Journal of Mathematical Physics,29:2346–2350,doi:10.1063/1.528118
- Chevalley, Claude (1991),The Algebraic Theory of Spinors and Clifford Algebras, Collected Works,vol. 2, Springer,ISBN3-540-57063-2
- Doran, Chris J. L.(1994),Geometric Algebra and its Application to Mathematical Physics(PhD thesis),University of Cambridge,doi:10.17863/CAM.16148,hdl:1810/251691,OCLC53604228
- Baylis, W. E., ed. (2011) [1996],Clifford (Geometric) Algebra with Applications to Physics, Mathematics, and Engineering,Birkhäuser,ISBN9781461241058
- Aragón, G.; Aragón, J.L.; Rodríguez, M.A. (1997), "Clifford Algebras and Geometric Algebra",Advances in Applied Clifford Algebras,7(2): 91–102,doi:10.1007/BF03041220,S2CID120860757
- Hestenes, David(1999),New Foundations for Classical Mechanics(2nd ed.), Springer Verlag,ISBN978-0-7923-5302-7
- Lasenby, Joan; Lasenby, Anthony N.;Doran, Chris J. L.(2000),"A Unified Mathematical Language for Physics and Engineering in the 21st Century"(PDF),Philosophical Transactions of the Royal Society A,358(1765): 21–39,Bibcode:2000RSPTA.358...21L,doi:10.1098/rsta.2000.0517,S2CID91884543,archived(PDF)from the original on 2015-03-19
- Lounesto, Pertti (2001),Clifford Algebras and Spinors(2nd ed.),Cambridge University Press,ISBN978-0-521-00551-7
- Baylis, W. E. (2002),Electrodynamics: A Modern Geometric Approach(2nd ed.),Birkhäuser,ISBN978-0-8176-4025-5
- Dorst, Leo (2002), "The Inner Products of Geometric Algebra", in Dorst, L.; Doran, C.; Lasenby, J. (eds.),Applications of Geometric Algebra in Computer Science and Engineering,Birkhäuser,pp. 35–46,doi:10.1007/978-1-4612-0089-5_2,ISBN978-1-4612-0089-5
- Doran, Chris J. L.;Lasenby, Anthony N. (2003),Geometric Algebra for Physicists(PDF),Cambridge University Press,ISBN978-0-521-71595-9,archived(PDF)from the original on 2009-01-06
- Hestenes, David(2003),"Oersted Medal Lecture 2002: Reforming the Mathematical Language of Physics"(PDF),Am. J. Phys.,71(2): 104–121,Bibcode:2003AmJPh..71..104H,CiteSeerX10.1.1.649.7506,doi:10.1119/1.1522700
{{citation}}
:CS1 maint: url-status (link) - Hildenbrand, Dietmar; Fontijne, Daniel; Perwass, Christian; Dorst, Leo (2004),"Geometric Algebra and its Application to Computer Graphics"(PDF),Proceedings of Eurographics 2004,doi:10.2312/egt.20041032,archived(PDF)from the original on 2015-09-06
- Lasenby, Anthony (2004), "Conformal Models of de Sitter Space, Initial Conditions for Inflation and the CMB",AIP Conference Proceedings,vol. 736, pp. 53–70,arXiv:astro-ph/0411579,doi:10.1063/1.1835174,S2CID18034896
- Hestenes, David (2005),Introduction to Primer for Geometric Algebra
- Selig, J.M. (2005).Geometric Fundamentals of Robotics.Monographs in Computer Science. New York, NY: Springer New York.doi:10.1007/b138859.ISBN978-0-387-20874-9.
- Bain, J. (2006), "Spacetime structuralism: §5 Manifoldsvs.geometric algebra ", inDennis Dieks(ed.),The ontology of spacetime,Elsevier, p. 54ff,ISBN978-0-444-52768-4
- Dorst, Leo; Fontijne, Daniel; Mann, Stephen (2007),Geometric algebra for computer science: an object-oriented approach to geometry,Elsevier,ISBN978-0-12-369465-2,OCLC132691969
- Penrose, Roger(2007),The Road to Reality,Vintage books,ISBN978-0-679-77631-4
- Francis, Matthew R.; Kosowsky, Arthur (2008), "The Construction of Spinors in Geometric Algebra",Annals of Physics,317(2): 383–409,arXiv:math-ph/0403040v2,Bibcode:2005AnPhy.317..383F,doi:10.1016/j.aop.2004.11.008,S2CID119632876
- Li, Hongbo (2008),Invariant Algebras and Geometric Reasoning,World Scientific,ISBN9789812770110.Chapter 1as PDF
- Vince, John A. (2008),Geometric Algebra for Computer Graphics,Springer,ISBN978-1-84628-996-5
- Lundholm, Douglas; Svensson, Lars (2009), "Clifford Algebra, Geometric Algebra and Applications",arXiv:0907.5356v1[math-ph]
- Perwass, Christian (2009),Geometric Algebra with Applications in Engineering,Geometry and Computing, vol. 4,Bibcode:2009gaae.book.....P,doi:10.1007/978-3-540-89068-3,ISBN978-3-540-89067-6
- Selig, J. M. (2000),"Clifford algebra of points, lines and planes"(PDF),Robotica,18(5): 545–556,doi:10.1017/S0263574799002568,S2CID28929170
- Bayro-Corrochano, Eduardo (2010),Geometric Computing for Wavelet Transforms, Robot Vision, Learning, Control and Action,Springer Verlag,ISBN9781848829299
- Bayro-Corrochano, E.; Scheuermann, Gerik, eds. (2010),Geometric Algebra Computing in Engineering and Computer Science,Springer,ISBN9781849961080Extract online athttps://davidhestenes.net/geocalc/html/UAFCG.html#5 New Tools for Computational Geometry and rejuvenation of Screw Theory
- Goldman, Ron (2010),Rethinking Quaternions: Theory and Computation,Morgan & Claypool, Part III. Rethinking Quaternions and Clifford Algebras,ISBN978-1-60845-420-4
- Dorst, Leo.; Lasenby, Joan (2011),Guide to Geometric Algebra in Practice,Springer,ISBN9780857298119
- Macdonald, Alan (2011),Linear and Geometric Algebra,CreateSpace,ISBN9781453854938,OCLC704377582
- Snygg, John (2011),A New Approach to Differential Geometry using Clifford's Geometric Algebra,Springer,ISBN978-0-8176-8282-8
- Hildenbrand, Dietmar (2012), "Foundations of Geometric Algebra computing",Numerical Analysis and Applied Mathematics Icnaam 2012: International Conference of Numerical Analysis and Applied Mathematics,AIP Conference Proceedings,1479(1): 27–30,Bibcode:2012AIPC.1479...27H,doi:10.1063/1.4756054
- Sokolov, Andrey (2013),Clifford algebra and the projective model of Minkowski (Pseudo-Euclidean) spaces,arXiv:1307.4179
- Bromborsky, Alan (2014),An introduction to Geometric Algebra and Calculus(PDF),archived(PDF)from the original on 2019-10-15
- Klawitter, Daniel (2015),Clifford Algebras,doi:10.1007/978-3-658-07618-4,ISBN978-3-658-07617-7
- Kanatani, Kenichi (2015),Understanding Geometric Algebra: Hamilton, Grassmann, and Clifford for Computer Vision and Graphics,CRC Press,ISBN978-1-4822-5951-3
- Li, Hongbo; Huang, Lei; Shao, Changpeng; Dong, Lei (2015), "Three-Dimensional Projective Geometry with Geometric Algebra",arXiv:1507.06634v1[math.MG]
- Hestenes, David (2017), "The Genesis of Geometric Algebra: A Personal Retrospective",Advances in Applied Clifford Algebras,27:351–379,doi:10.1007/s00006-016-0664-z,S2CID253592888
- Dorst, Leo (2016), "3D Oriented Projective Geometry Through Versors of",Advances in Applied Clifford Algebras,26(4): 1137–1172,doi:10.1007/s00006-015-0625-y
- Vaz, Jayme; da Rocha, Roldão (2016),An Introduction to Clifford Algebras and Spinors,Oxford University Press,Bibcode:2016icas.book.....V,ISBN978-0-19-878292-6
- Easter, Robert Benjamin; Hitzer, Eckhard (2017), "Double Conformal Geometric Algebra",Advances in Applied Clifford Algebras,27(3): 2175–2199,doi:10.1007/s00006-017-0784-0,S2CID253600526
- Du, Juan; Goldman, Ron; Mann, Stephen (2017), "Modeling 3D Geometry in the Clifford Algebra R(4, 4)",Advances in Applied Clifford Algebras,27(4): 3039–3062,doi:10.1007/s00006-017-0798-7,S2CID253587390
- Bayro-Corrochano, Eduardo (2018).Computer Vision, Graphics and Neurocomputing.Geometric Algebra Applications. Vol. I. Springer.ISBN978-3-319-74830-6.
- Breuils, Stéphane (2018).Structure algorithmique pour les opérateurs d'Algèbres Géométriques et application aux surfaces quadriques(PDF)(PHD). université-paris-est.Archived(PDF)from the original on 2019-07-14.
- Lavor, Carlile; Xambó-Descamps, Sebastià; Zaplana, Isiah (2018).A Geometric Algebra Invitation to Space-Time Physics, Robotics and Molecular Geometry.Springer. pp. 1–.ISBN978-3-319-90665-2.
- Hrdina, Jaroslav; Návrat, Aleš; Vašík, Petr (2018), "Geometric Algebra for Conics",Advances in Applied Clifford Algebras,28(3),doi:10.1007/s00006-018-0879-2,S2CID125649450
- Breuils, Stéphane; Fuchs, Laurent; Hitzer, Eckhard; Nozick, Vincent; Sugimoto, Akihiro (2019),"Three-dimensional quadrics in extended conformal geometric algebras of higher dimensions from control points, implicit equations and axis alignment"(PDF),Advances in Applied Clifford Algebras,29(3),doi:10.1007/s00006-019-0974-z,S2CID253597480
- Josipović, Miroslav (2019).Geometric Multiplication of Vectors: An Introduction to Geometric Algebra in Physics.Springer International Publishing;Birkhäuser. p. 256.ISBN978-3-030-01756-9.
- Hadfield, Hugo; Lasenby, Joan (2020),"Constrained Dynamics in Conformal and Projective Geometric Algebra",Advances in Computer Graphics,Lecture Notes in Computer Science, vol. 12221, pp. 459–471,doi:10.1007/978-3-030-61864-3_39,ISBN978-3-030-61863-6,S2CID224820480
- Wu, Bofeng (2022), "A signature invariant geometric algebra framework for spacetime physics and its applications in relativistic dynamics of a massive particle and gyroscopic precession",Scientific Reports,12(1): 3981,arXiv:2111.07353,Bibcode:2022NatSR..12.3981W,doi:10.1038/s41598-022-06895-0,PMC8901677,PMID35256628
- Wilmot, G.P. (2023)."The Algebra Of Geometry".GitHub.
- Lengyel, Eric (2024).Projective Geometric Algebra Illuminated.Lincoln, California: Terathon Software LLC.ISBN979-8-9853582-5-4.
External links
[edit]![](https://upload.wikimedia.org/wikipedia/commons/thumb/d/df/Wikibooks-logo-en-noslogan.svg/40px-Wikibooks-logo-en-noslogan.svg.png)
![](https://upload.wikimedia.org/wikipedia/commons/thumb/0/0b/Wikiversity_logo_2017.svg/40px-Wikiversity_logo_2017.svg.png)
- A Survey of Geometric Algebra and Geometric CalculusAlan Macdonald,Luther College, Iowa.
- Imaginary Numbers are not Real – the Geometric Algebra of Spacetime.Introduction (Cambridge GA group).
- Geometric Algebra 2015, Masters Course in Scientific Computing,from Dr. Chris Doran (Cambridge).
- Maths for (Games) Programmers: 5 – Multivector methods.Comprehensive introduction and reference for programmers, fromIan Bell.
- IMPA Summer School 2010Fernandes Oliveira Intro and Slides.
- University of FukuiE.S.M. Hitzer and Japan GA publications.
- Google Group for GA
- Geometric Algebra PrimerIntroduction to GA, Jaap Suter.
- Geometric Algebra Resourcescurated wiki, Pablo Bleyer.
- Applied Geometric Algebras in Computer Science and Engineering 2018Early Proceedings
- bivector.netGeometric Algebra for CGI, Vision and Engineering community website
- AGACSE 2021 Videos
English translations of early books and papers
- G. Combebiac, "calculus of tri-quaternions"(Doctoral dissertation)
- M. Markic, "Transformants: A new mathematical vehicle. A synthesis of Combebiac's tri-quaternions and Grassmann's geometric system. The calculus of quadri-quaternions"
- C. Burali-Forti, "The Grassmann method in projective geometry"A compilation of three notes on the application of exterior algebra to projective geometry
- C. Burali-Forti, "Introduction to Differential Geometry, following the method of H. Grassmann"Early book on the application of Grassmann algebra
- H. Grassmann, "Mechanics, according to the principles of the theory of extension"One of his papers on the applications of exterior algebra.
Research groups
- Geometric Calculus International.Links to Research groups, Software, and Conferences, worldwide.
- Cambridge Geometric Algebra group.Full-text online publications, and other material.
- University of Amsterdam group
- Geometric Calculus research & development(archive of Hestenes's website at Arizona State University).
- GA-Net blogandnewsletter archive.Geometric Algebra/Clifford Algebra development news.
- Geometric Algebra for Perception Action Systems. Geometric Cybernetics Group(CINVESTAV, Campus Guadalajara, Mexico).