In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a geometric algebra (also known as a
real Clifford algebra) is an extension of
elementary algebra
Elementary algebra encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces variables (quantities without fixed values).
This use of variables entail ...
to work with geometrical objects such as
vector
Vector most often refers to:
*Euclidean vector, a quantity with a magnitude and a direction
*Vector (epidemiology), an agent that carries and transmits an infectious pathogen into another living organism
Vector may also refer to:
Mathematic ...
s. Geometric algebra is built out of two fundamental operations, addition and the geometric product. Multiplication of vectors results in higher-dimensional objects called
multivectors. Compared to other formalisms for manipulating geometric objects, geometric algebra is noteworthy for supporting vector division and addition of objects of different dimensions.
The geometric product was first briefly mentioned by
Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
, who was chiefly interested in developing the closely related
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
. In 1878,
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
greatly 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 the
Grassmann algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is a ...
and Hamilton's
quaternion algebra
In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
. Adding the
dual of the Grassmann exterior product (the "meet") allows the use of the
Grassmann–Cayley algebra, and a
conformal version of the latter together with a conformal Clifford algebra yields a
conformal geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an -dimensional base space to null vectors in . This allows operations on the base space, including reflections, rotations an ...
providing a framework for
classical geometries. In practice, these and several derived operations allow a correspondence of elements,
subspaces and operations of the algebra with geometric interpretations. For several decades, geometric algebras went somewhat ignored, greatly eclipsed by the
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by
Hestenes, who advocated its importance to relativistic physics.
The scalars and vectors have their usual interpretation, and make up distinct subspaces of a geometric algebra.
Bivector In mathematics, a bivector or 2-vector is a quantity in exterior algebra or geometric algebra that extends the idea of scalars and vectors. If a scalar is considered a degree-zero quantity, and a vector is a degree-one quantity, then a bivector ca ...
s provide a more natural representation of the pseudovector quantities in
vector algebra In mathematics, vector algebra may mean:
* Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
* The algebraic operations in vector calculus, namely the specific additional stru ...
such as oriented area, oriented angle of rotation, torque, angular momentum and the electromagnetic field. A
trivector
In multilinear algebra, a multivector, sometimes called Clifford number, is an element of the exterior algebra of a vector space . This algebra is graded, associative and alternating, and consists of linear combinations of simple -vectors (a ...
can represent an oriented volume, and so on. An element called a
blade
A blade is the portion of a tool, weapon, or machine with an edge that is designed to puncture, chop, slice or scrape surfaces or materials. Blades are typically made from materials that are harder than those they are to be used on. Historic ...
may be used to represent a subspace of
and
orthogonal projection
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it wer ...
s onto 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 in
relativity.
Examples of geometric algebras applied in physics include the
spacetime algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of speci ...
(and the less common
algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
) and the
conformal geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an -dimensional base space to null vectors in . This allows operations on the base space, including reflections, rotations an ...
.
Geometric calculus
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential ...
, an extension of GA that incorporates
differentiation and
integration
Integration may refer to:
Biology
*Multisensory integration
*Path integration
* Pre-integration complex, viral genetic material used to insert a viral genome into a host genome
*DNA integration, by means of site-specific recombinase technology, ...
, can be used to formulate other theories such as
complex analysis
Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates Function (mathematics), functions of complex numbers. It is helpful in many branches of mathemati ...
and
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
, e.g. by using the Clifford algebra instead of
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s. Geometric algebra has been advocated, most notably by
David Hestenes
David Orlin Hestenes (born May 21, 1933) is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics, and as founder of Modelling Instructio ...
and
Chris Doran
Christopher Doran (born 22 November 1979) is an Irish singer from Waterford. He was the winner of Ireland's '' You're a Star'' 2003-2004 talent search competition to find Ireland's Eurovision Song Contest entry. The competition held auditions t ...
, as the preferred mathematical framework for
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
. Proponents claim that it provides compact and intuitive descriptions in many areas including
classical and
quantum mechanics
Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, ...
,
electromagnetic theory
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
and
relativity. GA has also found use as a computational tool in
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and
robotics
Robotics is an interdisciplinary branch of computer science and engineering. Robotics involves design, construction, operation, and use of robots. The goal of robotics is to design machines that can help and assist humans. Robotics integrat ...
.
Definition and notation
There are a number of different ways to define a geometric algebra. Hestenes's original approach was axiomatic, "full of geometric significance" and equivalent to the universal Clifford algebra.
Given a finite-dimensional
quadratic space
In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example,
:4x^2 + 2xy - 3y^2
is a quadratic form in the variables and . The coefficients usually belong to ...
over a
field
Field may refer to:
Expanses of open ground
* Field (agriculture), an area of land used for agricultural purposes
* Airfield, an aerodrome that lacks the infrastructure of an airport
* Battlefield
* Lawn, an area of mowed grass
* Meadow, a grass ...
with a symmetric bilinear form (the ''inner product'', e.g. the Euclidean or
Lorentzian metric
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the r ...
)
, the geometric algebra for this quadratic space is the
Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
. As usual in this domain, for the remainder of this article, only the
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
case,
, will be considered. The notation
(respectively
) will be used to denote a geometric algebra for which the bilinear form
has the
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
(respectively
).
The essential product in the algebra is called the ''geometric product'', and the product in the contained exterior algebra is called the ''exterior product'' (frequently called the ''wedge product'' and less often the ''outer product''). 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 abstract, so we summarize the properties of the geometric product by the following set of axioms. The geometric product has the following properties, for
:
*
(
closure)
*
, where
is the identity element (existence of an
identity element
In mathematics, an identity element, or neutral element, of a binary operation operating on a set is an element of the set that leaves unchanged every element of the set when the operation is applied. This concept is used in algebraic structures su ...
)
*
(
associativity
In mathematics, the associative property is a property of some binary operations, which means that rearranging the parentheses in an expression will not change the result. In propositional logic, associativity is a valid rule of replacement f ...
)
*
and
(
distributivity
In mathematics, the distributive property of binary operations generalizes the distributive law, which asserts that the equality
x \cdot (y + z) = x \cdot y + x \cdot z
is always true in elementary algebra.
For example, in elementary arithmeti ...
)
*
, where
is any element of the subspace
of the algebra.
The exterior product has the same properties, except that the last property above is replaced by
for
.
Note that in the last property above, the real number
need not be nonnegative if
is not positive-definite. An important property of the geometric product is the existence of elements having a multiplicative inverse. For a vector
, if
then
exists and is equal to
. A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if
is a vector in
such that
, the element
is both a nontrivial
idempotent element
Idempotence (, ) is the property of certain operations in mathematics and computer science whereby they can be applied multiple times without changing the result beyond the initial application. The concept of idempotence arises in a number of pl ...
and a nonzero
zero divisor
In abstract algebra, an element of a ring is called a left zero divisor if there exists a nonzero in such that , or equivalently if the map from to that sends to is not injective. Similarly, an element of a ring is called a right zero ...
, and thus has no inverse.
It is usual to identify
and
with their images under the natural
embedding
In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.
When some object X is said to be embedded in another object Y, the embedding is gi ...
s
and
. In this article, this identification is assumed. Throughout, the terms ''scalar'' and ''vector'' refer to elements of
and
respectively (and of their images under this embedding).
The geometric product
For vectors
and
, we may write the geometric product of any two vectors
and
as the sum of a symmetric product and an antisymmetric product:
:
Thus we can define the ''inner product'' of 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 contained
exterior algebra
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is ...
:
:
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,
and
are
parallel
Parallel is a geometric term of location which may refer to:
Computing
* Parallel algorithm
* Parallel computing
* Parallel metaheuristic
* Parallel (software), a UNIX utility for running programs in parallel
* Parallel Sysplex, a cluster of ...
if their geometric product is equal to their inner product, whereas
and
are
perpendicular
In elementary geometry, two geometric objects are perpendicular if they intersect at a right angle (90 degrees or π/2 radians). The condition of perpendicularity may be represented graphically using the ''perpendicular symbol'', ⟂. It can ...
if 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 the
dot product
In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an algebra ...
of standard vector algebra. The exterior product of two vectors can be identified with the
signed area
In mathematics, an integral assigns numbers to Function (mathematics), functions in a way that describes Displacement (geometry), displacement, area, volume, and other concepts that arise by combining infinitesimal data. The process of finding ...
enclosed by a
parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equa ...
the sides of which are the vectors. The
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of two vectors in
dimensions 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 fully
degenerate
Degeneracy, degenerate, or degeneration may refer to:
Arts and entertainment
* Degenerate (album), ''Degenerate'' (album), a 2010 album by the British band Trigger the Bloodshed
* Degenerate art, a term adopted in the 1920s by the Nazi Party i ...
, 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, with
the
sign of the permutation, and
are 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 an
alternating algebra
In mathematics, an alternating algebra is a -graded algebra for which for all nonzero homogeneous elements and (i.e. it is an anticommutative algebra) and has the further property that for every homogeneous element of odd degree.
Examples ...
.
Blades, grades, and canonical basis
A multivector that is the exterior product of
linearly independent vectors is called a ''blade'', and is said to be of grade
. A multivector that is the sum of blades of grade
is 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 of
linearly independent vectors
spanning an
-dimensional subspace of the vector space. With these, we can define a real
symmetric matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally,
Because equal matrices have equal dimensions, only square matrices can be symmetric.
The entries of a symmetric matrix are symmetric with re ...
(in the same way as a
Gramian matrix
In linear algebra, the Gram matrix (or Gramian matrix, Gramian) of a set of vectors v_1,\dots, v_n in an inner product space is the Hermitian matrix of inner products, whose entries are given by the inner product G_ = \left\langle v_i, v_j \right\r ...
)
:
By the
spectral theorem
In mathematics, particularly linear algebra and functional analysis, a spectral theorem is a result about when a linear operator or matrix (mathematics), matrix can be Diagonalizable matrix, diagonalized (that is, represented as a diagonal matrix i ...
,
can be diagonalized to
diagonal matrix
In linear algebra, a diagonal matrix is a matrix in which the entries outside the main diagonal are all zero; the term usually refers to square matrices. Elements of the main diagonal can either be zero or nonzero. An example of a 2×2 diagonal ma ...
by an
orthogonal matrix
In linear algebra, an orthogonal matrix, or orthonormal matrix, is a real square matrix whose columns and rows are orthonormal vectors.
One way to express this is
Q^\mathrm Q = Q Q^\mathrm = I,
where is the transpose of and is the identity ma ...
via
:
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 that
and thus the
are perpendicular. In other words, the geometric product of two distinct vectors
is completely specified by their exterior product, or more generally
:
Therefore, every blade of grade
can be written as a geometric product of
vectors. More generally, if a degenerate geometric algebra is allowed, then the orthogonal matrix is replaced by a
block matrix that 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 are
normalized according to
:
then these normalized vectors must square to
or
. By
Sylvester's law of inertia
Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real number, real quadratic form that remain invariant (mathematics), invariant under a change of basis. Namely, if ''A'' is the symme ...
, the total number of
s and the total number of
s along the diagonal matrix is invariant. By extension, the total number
of these vectors that square to
and the total number
that square to
is 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-dimensional
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
,
relativistic
spacetime
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why differen ...
and
a
conformal geometric algebra
Conformal geometric algebra (CGA) is the geometric algebra constructed over the resultant space of a map from points in an -dimensional base space to null vectors in . This allows operations on the base space, including reflections, rotations an ...
of a three-dimensional space.
The set of all possible products of
orthogonal basis vectors with indices in increasing order, including
as the empty product, forms a basis for the entire geometric algebra (an analogue of the
PBW theorem PBW may refer to:
* Philadelphia-Baltimore-Washington Stock Exchange
* Peanut Butter Wolf, American hip hop record producer
* Proton beam writing, a lithography process
* Play by Web, Play-by-post role-playing game
* Prosopography of the Byzantine ...
). For example, the following is a basis for the geometric algebra
:
:
A basis formed this way is called a canonical basis for the geometric algebra, and any other orthogonal basis for
will produce another canonical basis. Each canonical basis consists of
elements. Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. If the canonical basis elements are
with
being 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 of
vectors). By way of example,
in
cannot 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.
Grade projection
Using an orthogonal basis, a
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
structure can be established. Elements of the geometric algebra that are scalar multiples of
are grade-
blades and are called ''scalars''. Multivectors that are in the span of
are grade-
blades and are the ordinary vectors. Multivectors in the span of
are grade-
blades and are the bivectors. This terminology continues through to the last grade of
-vectors. Alternatively, grade-
blades are called
pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not.
Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The ...
s, grade-
blades 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 of ''mixed 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 of
through
-blades, the geometric algebra is a
filtered algebra In mathematics, a filtered algebra is a generalization of the notion of a graded algebra. Examples appear in many branches of mathematics, especially in homological algebra and representation theory.
A filtered algebra over the field k is an alge ...
.
A multivector
may be decomposed with the grade-projection operator
, which outputs the grade-
portion of
. As a result:
:
As an example, the geometric product of two vectors
since
and
and
, for
other than
and
.
The decomposition of a multivector
may also be split into those components that are even and those that are odd:
:
:
This is the result of forgetting structure from a
-
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
to
-
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
. The geometric product respects this coarser grading. Thus in addition to being a
-
graded vector space
In mathematics, a graded vector space is a vector space that has the extra structure of a '' grading'' or a ''gradation'', which is a decomposition of the vector space into a direct sum of vector subspaces.
Integer gradation
Let \mathbb be th ...
, the geometric algebra is a
-
graded algebra
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups R_i such that R_i R_j \subseteq R_. The index set is usually the set of nonnegative integers or the se ...
or
superalgebra
In mathematics and theoretical physics, a superalgebra is a Z2-graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
Th ...
.
Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines an ''
even subalgebra
In mathematics and theoretical physics, a superalgebra is a Z2- graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
...
''. The even subalgebra of an
-dimensional geometric algebra is
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word is ...
(without preserving either filtration or grading) to a full geometric algebra of
dimensions. Examples include
and
.
Representation of subspaces
Geometric algebra represents subspaces of
as blades, and so they coexist in the same algebra with vectors from
. A
-dimensional subspace
of
is represented by taking an orthogonal basis
and using the geometric product to form the
blade
A blade is the portion of a tool, weapon, or machine with an edge that is designed to puncture, chop, slice or scrape surfaces or materials. Blades are typically made from materials that are harder than those they are to be used on. Historic ...
. There are multiple blades representing
; all those representing
are scalar multiples of
. These blades can be separated into two sets: positive multiples of
and negative multiples of
. The positive multiples of
are said to have ''the same
orientation
Orientation may refer to:
Positioning in physical space
* Map orientation, the relationship between directions on a map and compass directions
* Orientation (housing), the position of a building with respect to the sun, a concept in building de ...
'' as
, and the negative multiples the ''opposite 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
Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non-degenerate subspace
of
is a blade that is the product of the members of an orthonormal basis for
. It can be shown that if
and
are both unit pseudoscalars for
, then
and
. If one doesn't choose an orthonormal basis for
, then the
Plücker embedding In mathematics, the Plücker map embeds the Grassmannian \mathbf(k,V), whose elements are ''k''-dimensional subspaces of an ''n''-dimensional vector space ''V'', in a projective space, thereby realizing it as an algebraic variety.
More precisely ...
gives 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 of
for all
. Orthonormality gets rid of this ambiguity except for the signs above.
Suppose the geometric algebra
with the familiar positive definite inner product on
is formed. Given a plane (two-dimensional subspace) of
, one can find an orthonormal basis
spanning the plane, and thus find a unit pseudoscalar
representing this plane. The geometric product of any two vectors in the span of
and
lies in
, that is, it is the sum of a
-vector and a
-vector.
By the properties of the geometric product,
. The resemblance to the
imaginary unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition an ...
is not incidental: the subspace
is
-algebra isomorphic to the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace of
on 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 canonical basis consisting of orthonormal vectors
of
, the set of ''all''
-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 the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s, another important algebraic system.
Extensions of the inner and exterior products
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 mentioned
grade projection operator:
:
(the ''exterior product'')
This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:
:
(the ''commutator product'')
The regressive product (usually referred to as the "meet") is the dual of the exterior product (or "join" in this context). The dual specification of elements permits, for blades
and
, the intersection (or meet) where the duality is to be taken relative to the smallest grade blade containing both
and
(the join).
:
with
the unit pseudoscalar of the algebra. The regressive product, like the exterior product, is associative.
The inner product on vectors can also be generalized, but in more than one non-equivalent way. The paper 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:
:
(the ''left contraction'')
:
(the ''right contraction'')
:
(the ''scalar product'')
:
(the "(fat) dot" product)
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 identity
is extended to
for any vector
and multivector
, and that the
projection
Projection, projections or projective may refer to:
Physics
* Projection (physics), the action/process of light, heat, or sound reflecting from a surface to another in a different direction
* The display of images by a projector
Optics, graphic ...
operation
is extended to
for any blade
and any multivector
(with a minor modification to accommodate null
, given
below
Below may refer to:
*Earth
*Ground (disambiguation)
*Soil
*Floor
*Bottom (disambiguation)
Bottom may refer to:
Anatomy and sex
* Bottom (BDSM), the partner in a BDSM who takes the passive, receiving, or obedient role, to that of the top or ...
).
Dual basis
Let
be a basis of
, i.e. a set of
linearly independent vectors that span the
-dimensional vector space
. The basis that is dual to
is the set of elements of the
dual vector space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
that forms a
biorthogonal system In mathematics, a biorthogonal system is a pair of indexed families of vectors
\tilde v_i \text E \text \tilde u_i \text F
such that
\left\langle\tilde v_i , \tilde u_j\right\rangle = \delta_,
where E and F form a pair of topological vector spaces t ...
with this basis, thus being the elements denoted
satisfying
:
where
is the
Kronecker delta
In mathematics, the Kronecker delta (named after Leopold Kronecker) is a function of two variables, usually just non-negative integers. The function is 1 if the variables are equal, and 0 otherwise:
\delta_ = \begin
0 &\text i \neq j, \\
1 &\ ...
.
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 the
denotes that the
th basis vector is omitted from the product.
A dual basis is also known as a
reciprocal basis or reciprocal frame.
A major usage of a dual basis is to separate vectors into components. Given a vector
, scalar components
can be defined as
:
in terms of which
can be separated into vector components as
:
We can also define scalar components
as
:
in terms of which
can 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. 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
:
where
is the scalar product.
With
a multivector, we can now define scalar components as
:
in terms of which
can be separated into component blades as
:
We can alternatively define scalar components
as
:
in terms of which
can be separated into component blades as
:
Linear functions
Although a versor is easier to work with because it can be directly represented in the algebra as a multivector, versors are a subgroup of
linear function
In mathematics, the term linear function refers to two distinct but related notions:
* In calculus and related areas, a linear function is a function (mathematics), function whose graph of a function, graph is a straight line, that is, a polynomia ...
s on multivectors, which can still be used when necessary. The geometric algebra of an
-dimensional vector space is spanned by a basis of
elements. If a multivector is represented by a
real
column matrix of coefficients of a basis of the algebra, then all linear transformations of the multivector can be expressed as the
matrix multiplication
In mathematics, particularly in linear algebra, matrix multiplication is a binary operation that produces a matrix from two matrices. For matrix multiplication, the number of columns in the first matrix must be equal to the number of rows in the s ...
by a
real 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, the ''
outermorphism'' of the linear transformation is the unique extension of the versor. If
is 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
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.
Vector space model
may be considered as an extension or completion of
vector algebra In mathematics, vector algebra may mean:
* Linear algebra, specifically the basic algebraic operations of vector addition and scalar multiplication; see vector space.
* The algebraic operations in vector calculus, namely the specific additional stru ...
. ''From Vectors to Geometric Algebra'' covers basic analytic geometry and gives an introduction to stereographic projection.
The
even subalgebra
In mathematics and theoretical physics, a superalgebra is a Z2- graded algebra. That is, it is an algebra over a commutative ring or field with a decomposition into "even" and "odd" pieces and a multiplication operator that respects the grading.
...
of
is isomorphic to the
complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form ...
s, as may be seen by writing a vector
in terms of its components in an orthonormal basis and left multiplying by the basis vector
, yielding
:
where we identify
since
:
Similarly, the even subalgebra of
with basis
is isomorphic to the
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
s as may be seen by identifying
,
and
.
Every
associative algebra
In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative), and a scalar multiplication by elements in some field ''K''. The addition and multiplic ...
has a matrix representation; replacing the three Cartesian basis vectors by the
Pauli matrices
In mathematical physics and mathematics, the Pauli matrices are a set of three complex matrices which are Hermitian, involutory and unitary. Usually indicated by the Greek letter sigma (), they are occasionally denoted by tau () when used in ...
gives a representation of
:
:
Dotting the "
Pauli vector" (a
dyad
Dyad or dyade may refer to:
Arts and entertainment
* Dyad (music), a set of two notes or pitches
* ''Dyad'' (novel), by Michael Brodsky, 1989
* ''Dyad'' (video game), 2012
* ''Dyad 1909'' and ''Dyad 1929'', ballets by Wayne McGregor
Other uses ...
):
:
with arbitrary vectors
and
and multiplying through gives:
:
(Equivalently, by inspection,
)
Spacetime model
In physics, the main applications are the geometric algebra of
Minkowski 3+1 spacetime, , called
spacetime algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of speci ...
(STA), or less commonly, , interpreted the
algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
(APS).
While in STA, points of spacetime are represented simply by vectors, in APS, points of
-dimensional spacetime are instead represented by
paravector
The name paravector is used for the sum of a scalar and a vector in any Clifford algebra, known as geometric algebra among physicists.
This name was given by J. G. Maks in a doctoral dissertation at Technische Universiteit Delft, Netherlands, in ...
s, a three-dimensional vector (space) plus a one-dimensional scalar (time).
In spacetime algebra the electromagnetic field tensor has a bivector representation
. Here, the
is the unit pseudoscalar (or four-dimensional volume element),
is the unit vector in time direction, and
and
are the classic electric and magnetic field vectors (with a zero time component). Using the
four-current
In special and general relativity, the four-current (technically the four-current density) is the four-dimensional analogue of the electric current density. Also known as vector current, it is used in the geometric context of ''four-dimensional spa ...
,
Maxwell's equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.
...
then become
:
In geometric calculus, juxtaposition of vectors such as in
indicate the geometric product and can be decomposed into parts as
. Here
is the covector derivative in any spacetime and reduces to
in flat spacetime. Where
plays a role in Minkowski
-spacetime which is synonymous to the role of
in Euclidean
-space and is related to the
d'Alembertian
In special relativity, electromagnetism and wave theory, the d'Alembert operator (denoted by a box: \Box), also called the d'Alembertian, wave operator, box operator or sometimes quabla operator (''cf''. nabla symbol) is the Laplace operator of ...
by
. Indeed, given an observer represented by a future pointing timelike vector
we have
:
:
Boosts in this Lorentzian metric space have the same expression
as rotation in Euclidean space, where
is 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.
The
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
are a representation of
, showing the equivalence with matrix representations used by physicists.
Homogeneous model
Projective geometric algebra (PGA), also known as the homogeneous model, provides a complete algebra containing representations of all Euclidean isometries and the linear subspaces on which they operate. In this model, a single degenerate dimension is added to the ''n'' ordinary dimensions of space to form the algebra
. A comprehensive treatment of
in particular is given by
Lengyel. In the three-dimensional PGA, vectors correspond to points, bivectors correspond to lines, and trivectors correspond to planes. Proper Euclidean isometries, which are always
screw motions in 3D space, are represented by ''motors'' consisting of eight even-graded components, and improper Euclidean isometries, which contain a reflection, are represented by ''flectors'' consisting of eight odd-graded components. The motor algebra is the correct generalization of dual quaternions to the full set of objects that appear in PGA.
A central concept in PGA is the symmetry that arises through the inversion of the basis vectors that are present and absent in each of an object's components. This gives rise not only to a fundamental duality in the geometric objects of the algebra, but also the operations on those objects. Every product in the algebra (wedge product, inner product, geometric product, interior product) has a matching "antiproduct" that performs the same operation on the duals of its operands. The inclusion of both types of products is essential for completing the algebra. Join and meet operations between various geometries are performed by the wedge product and its dual, the antiwedge product, respectively. Two different norms, called the bulk norm and weight norm, are given by the inner product and its dual, and these produce scalar and antiscalar quantities, respectively. Euclidean isometries are performed by the geometric antiproduct.
All objects in projective geometric algebra are homogeneous, including magnitudes that would simply be scalars in nonprojective settings. Multiplying any point, line, plane, motor, or flector by a nonzero scalar has no effect on its meaning. An object is projected into 3D space through a process called ''unitization'', which makes the collective size of the components extending into the degenerate dimension unity. Concrete distance measurements arise as homogeneous magnitudes given by the geometric norm, which is a sum of the bulk norm and weight norm. When the geometric norm is unitized, its scalar component represents an actual distance.
Conformal model
Working within GA, Euclidean space
(along with a conformal point at infinity) is embedded projectively in the CGA
via 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 is
covariant, extending incidence relations of projective geometry to circles and spheres.
Specifically, we add orthogonal basis vectors
and
such that
and
to the basis of the vector space that generates
and identify
null vectors
:
as a conformal point at infinity (see ''
Compactification
Compactification may refer to:
* Compactification (mathematics), making a topological space compact
* Compactification (physics), the "curling up" of extra dimensions in string theory
See also
* Compaction (disambiguation)
Compaction may refer t ...
'') and
:
as the point at the origin, giving
:
.
This procedure has some similarities to the procedure for working with
homogeneous coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. T ...
in projective geometry and in this case allows the modeling of
Euclidean transformation
In mathematics, a rigid transformation (also called Euclidean transformation or Euclidean isometry) is a geometric transformation of a Euclidean space that preserves the Euclidean distance between every pair of points.
The rigid transformations ...
s of
as
orthogonal transformation In linear algebra, an orthogonal transformation is a linear transformation ''T'' : ''V'' → ''V'' on a real inner product space ''V'', that preserves the inner product. That is, for each pair of elements of ''V'', we h ...
s of a subset of
.
A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.
Models for projective transformation
Two potential candidates are currently under investigation as the foundation for affine and projective geometry in three dimensions
and
which includes representations for shears and non-uniform scaling, as well as
quadric surfaces and
conic section
In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a specia ...
s.
A new research model, Quadric Conformal Geometric Algebra (QCGA)
is an extension of CGA, dedicated to quadric surfaces. The idea is to represent the objects in low dimensional subspaces of the algebra. QCGA is capable of constructing quadric surfaces either using control points or implicit equations. Moreover, QCGA can compute the intersection of quadric surfaces as well as the surface tangent and normal vectors at a point that lies in the quadric surface.
Geometric interpretation
Projection and rejection
For any vector
and any invertible vector
,
:
where the projection of
onto
(or the parallel part) is
:
and the rejection of
from
(or the orthogonal part) is
:
Using the concept of a
-blade
as representing a subspace of
and every multivector ultimately being expressed in terms of vectors, this generalizes to projection of a general multivector onto any invertible
-blade
as
:
with the rejection being defined as
:
The projection and rejection generalize to null blades
by replacing the inverse
with the pseudoinverse
with respect to the contractive product. The outcome of the projection coincides in both cases for non-null blades. 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, as only then is the result necessarily in the subspace represented by
.
The projection generalizes through linearity to general multivectors
. The projection is not linear in
and does not generalize to objects
that are not blades.
Reflection
Simple reflections in a hyperplane are readily expressed in the algebra through conjugation with a single vector. These serve to generate the group of general
rotoreflection
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
s and
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
s.
The reflection
of a vector
along 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 reflection
of a vector
may be written
:
where
:
and
If we define the reflection along a non-null vector
of 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 multivector
using any reflection versor
may be written
:
where
is the
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms ...
of
reflection through the origin
In geometry, a point reflection (point inversion, central inversion, or inversion through a point) is a type of isometry of Euclidean space. An object that is invariant under a point reflection is said to possess point symmetry; if it is invari ...
of the vector space (
) extended through linearity to the whole algebra.
Rotations
If we have a product of vectors
then we denote the reverse as
:
As an example, assume that
we get
:
Scaling
so that
then
:
so
leaves the length of
unchanged. We can also show that
:
so the transformation
preserves both length and angle. It therefore can be identified as a rotation or rotoreflection;
is called a
rotor
Rotor may refer to:
Science and technology
Engineering
*Rotor (electric), the non-stationary part of an alternator or electric motor, operating with a stationary element so called the stator
* Helicopter rotor, the rotary wing(s) of a rotorcraft ...
if it is a
proper rotation
In geometry, an improper rotation,. also called rotation-reflection, rotoreflection, rotary reflection,. or rotoinversion is an isometry in Euclidean space that is a combination of a rotation about an axis and a reflection in a plane perpendicul ...
(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 a ''
versor
In mathematics, a versor is a quaternion of norm one (a ''unit quaternion''). The word is derived from Latin ''versare'' = "to turn" with the suffix ''-or'' forming a noun from the verb (i.e. ''versor'' = "the turner"). It was introduced by Will ...
''.
There is a general method for rotating a vector involving the formation of a multivector of the form
that produces a rotation
in the
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* ''Planes' ...
and with the orientation defined by a
-blade
.
Rotors are a generalization of quaternions to
-dimensional spaces.
Versor
A
-versor is a multivector that can be expressed as the geometric product of
invertible vectors. 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.
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. 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 the
Cartan–Dieudonné theorem
In mathematics, the Cartan–Dieudonné theorem, named after Élie Cartan and Jean Dieudonné, establishes that every orthogonal transformation in an ''n''-dimensional symmetric bilinear space can be described as the composition of at most ''n'' r ...
we 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 group
as 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
( Clifford group, although Lundholm deprecates this usage).
Subgroups of
Lundholm defines the
,
, and
subgroups, generated by unit vectors, and in the case of
and
, only an even number of such vector factors can be present.
Spinors are defined as elements of the even subalgebra of a real GA; an analysis of the GA approach to spinors is given by Francis and Kosowsky.
Examples and applications
Hypervolume of a parallelotope spanned by vectors
For vectors
and
spanning a parallelogram we have
:
with the result that
is 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
-dimensional
parallelotope; 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
We may define the line parametrically by
where
and
are position vectors for points P and T and
is the direction vector for the line.
Then
:
and
so
:
and
:
Rotating systems
The mathematical description of rotational forces such as
torque
In physics and mechanics, torque is the rotational equivalent of linear force. It is also referred to as the moment of force (also abbreviated to moment). It represents the capability of a force to produce change in the rotational motion of th ...
and
angular momentum
In physics, angular momentum (rarely, moment of momentum or rotational momentum) is the rotational analog of linear momentum. It is an important physical quantity because it is a conserved quantity—the total angular momentum of a closed syst ...
often makes use of the
cross product
In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and is ...
of
vector calculus
Vector calculus, or vector analysis, is concerned with differentiation and integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for the broader subject ...
in three dimensions with a convention of orientation (handedness).
The cross product can be viewed in terms of the exterior product allowing a more natural geometric interpretation of the cross product as a bivector using the
dual relationship
:
For example, torque is generally defined as the magnitude of the perpendicular force component times distance, or work per unit angle.
Suppose a circular path in an arbitrary plane containing orthonormal vectors
and
is 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
:
So the torque, the rate of change of work
, due to a force
, is
:
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 vectors
and
.
Geometric calculus
Geometric calculus extends the formalism to include differentiation and integration including differential geometry and
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s.
Essentially, the vector derivative is defined so that the GA version of
Green's theorem
In vector calculus, Green's theorem relates a line integral around a simple closed curve to a double integral over the plane region bounded by . It is the two-dimensional special case of Stokes' theorem.
Theorem
Let be a positively orient ...
is true,
:
and then one can write
:
as a geometric product, effectively generalizing
Stokes' theorem
Stokes's theorem, also known as the Kelvin–Stokes theorem Nagayoshi Iwahori, et al.:"Bi-Bun-Seki-Bun-Gaku" Sho-Ka-Bou(jp) 1983/12Written in Japanese)Atsuo Fujimoto;"Vector-Kai-Seki Gendai su-gaku rekucha zu. C(1)" :ja:培風館, Bai-Fu-Kan( ...
(including the differential form version of it).
In
when
is a curve with endpoints
and
, then
:
reduces to
:
or the fundamental theorem of integral calculus.
Also developed are the concept of
vector manifold and geometric integration theory (which generalizes differential forms).
History
Before the 20th century
Although the connection of geometry with algebra dates as far back at least to
Euclid
Euclid (; grc-gre, Wikt:Εὐκλείδης, Εὐκλείδης; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the ''Euclid's Elements, Elements'' trea ...
's ''
Elements'' in the third century B.C. (see
Greek geometric algebra
Algebra can essentially be considered as doing computations similar to those of arithmetic but with non-numerical mathematical objects. However, until the 19th century, algebra consisted essentially of the theory of equations. For example, the fu ...
), GA in the sense used in this article was not developed until 1844, when it was used in a ''systematic way'' to describe the geometrical properties and ''transformations'' of a space. In that year,
Hermann Grassmann
Hermann Günther Grassmann (german: link=no, Graßmann, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mat ...
introduced the idea of a geometrical algebra in full generality as a certain calculus (analogous to the
propositional calculus
Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations b ...
) that encoded all of the geometrical information of a space. Grassmann's algebraic system could be applied to a number of different kinds of spaces, the chief among them being
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
,
affine space
In mathematics, an affine space is a geometric structure that generalizes some of the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties relate ...
, and
projective space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally ...
. Following Grassmann, in 1878
William Kingdon Clifford
William Kingdon Clifford (4 May 18453 March 1879) was an English mathematician and philosopher. Building on the work of Hermann Grassmann, he introduced what is now termed geometric algebra, a special case of the Clifford algebra named in his ...
examined Grassmann's algebraic system alongside the
quaternions
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
of
William Rowan Hamilton
Sir William Rowan Hamilton Doctor of Law, LL.D, Doctor of Civil Law, DCL, Royal Irish Academy, MRIA, Royal Astronomical Society#Fellow, FRAS (3/4 August 1805 – 2 September 1865) was an Irish mathematician, astronomer, and physicist. He was the ...
in . From his point of view, the quaternions described certain ''transformations'' (which he called ''rotors''), whereas Grassmann's algebra described certain ''properties'' (or ''Strecken'' such as length, area, and volume). His contribution was to define a new product — the ''geometric product'' – on an existing Grassmann algebra, which realized the quaternions as living within that algebra. Subsequently,
Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...
in 1886 generalized Clifford's interpretation of the quaternions and applied them to the geometry of rotations in
dimensions. 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 of
vector analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
, developed independently by
Josiah Willard Gibbs
Josiah Willard Gibbs (; February 11, 1839 – April 28, 1903) was an American scientist who made significant theoretical contributions to physics, chemistry, and mathematics. His work on the applications of thermodynamics was instrumental in t ...
and
Oliver Heaviside
Oliver Heaviside FRS (; 18 May 1850 – 3 February 1925) was an English self-taught mathematician and physicist who invented a new technique for solving differential equations (equivalent to the Laplace transform), independently developed vec ...
. Vector analysis was motivated by
James Clerk Maxwell
James Clerk Maxwell (13 June 1831 – 5 November 1879) was a Scottish mathematician and scientist responsible for the classical theory of electromagnetic radiation, which was the first theory to describe electricity, magnetism and ligh ...
's studies of
electromagnetism
In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...
, and specifically the need to express and manipulate conveniently certain
differential equation
In mathematics, a differential equation is an equation that relates one or more unknown functions and their derivatives. In applications, the functions generally represent physical quantities, the derivatives represent their rates of change, an ...
s. 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 textbook ''
Vector Analysis
Vector calculus, or vector analysis, is concerned with derivative, differentiation and integral, integration of vector fields, primarily in 3-dimensional Euclidean space \mathbb^3. The term "vector calculus" is sometimes used as a synonym for ...
'' by
Edwin Bidwell Wilson
Edwin Bidwell Wilson (April 25, 1879 – December 28, 1964) was an American mathematician, statistician, physicist and general polymath. He was the sole protégé of Yale University physicist Josiah Willard Gibbs and was mentor to MIT economist ...
, following lectures of Gibbs.
In more detail, there have been three approaches to geometric algebra:
quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
ic analysis, 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
Progress on the study of Clifford algebras quietly advanced through the twentieth century, although largely due to the work of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, and algebras over a field. The term ''a ...
ists such as
Élie Cartan
Élie Joseph Cartan (; 9 April 1869 – 6 May 1951) was an influential French mathematician who did fundamental work in the theory of Lie groups, differential systems (coordinate-free geometric formulation of PDEs), and differential geometry. ...
,
Hermann Weyl
Hermann Klaus Hugo Weyl, (; 9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher. Although much of his working life was spent in Zürich, Switzerland, and then Princeton, New Jersey, he is assoc ...
and
Claude Chevalley
Claude Chevalley (; 11 February 1909 – 28 June 1984) was a French mathematician who made important contributions to number theory, algebraic geometry, class field theory, finite group theory and the theory of algebraic groups. He was a foundin ...
. The ''geometrical'' approach to geometric algebras has seen a number of 20th-century revivals. In mathematics,
Emil Artin
Emil Artin (; March 3, 1898 – December 20, 1962) was an Austrian mathematician of Armenian descent.
Artin was one of the leading mathematicians of the twentieth century. He is best known for his work on algebraic number theory, contributing lar ...
's ''Geometric Algebra'' discusses the algebra associated with each of a number of geometries, including
affine geometry
In mathematics, affine geometry is what remains of Euclidean geometry when ignoring (mathematicians often say "forgetting") the metric notions of distance and angle.
As the notion of ''parallel lines'' is one of the main properties that is inde ...
,
projective geometry
In mathematics, projective geometry is the study of geometric properties that are invariant with respect to projective transformations. This means that, compared to elementary Euclidean geometry, projective geometry has a different setting, pro ...
,
symplectic geometry
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed differential form, closed, nondegenerate form, nondegenerate different ...
, and
orthogonal 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.
David Hestenes
David Orlin Hestenes (born May 21, 1933) is a theoretical physicist and science educator. He is best known as chief architect of geometric algebra as a unified language for mathematics and physics, and as founder of Modelling Instructio ...
reinterpreted the
Pauli and
Dirac matrices as vectors in ordinary space and spacetime, respectively, and has been a primary contemporary advocate for the use of geometric algebra.
In
computer graphics
Computer graphics deals with generating images with the aid of computers. Today, computer graphics is a core technology in digital photography, film, video games, cell phone and computer displays, and many specialized applications. A great de ...
and robotics, geometric algebras have been revived in order to efficiently represent rotations and other transformations. For applications of GA in robotics (
screw theory
Screw theory is the algebraic calculation of pairs of vectors, such as forces and moments or angular and linear velocity, that arise in the kinematics and dynamics of rigid bodies. The mathematical framework was developed by Sir Robert Stawe ...
, kinematics and dynamics using versors), computer vision, control and neural computing (geometric learning) see Bayro (2010).
See also
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Comparison of vector algebra and geometric algebra
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Clifford algebra
In mathematics, a Clifford algebra is an algebra generated by a vector space with a quadratic form, and is a unital associative algebra. As -algebras, they generalize the real numbers, complex numbers, quaternions and several other hyperc ...
*
Grassmann–Cayley algebra
*
Spacetime algebra
In mathematical physics, spacetime algebra (STA) is a name for the Clifford algebra Cl1,3(R), or equivalently the geometric algebra . According to David Hestenes, spacetime algebra can be particularly closely associated with the geometry of speci ...
*
Spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a slight ...
*
Quaternion
In mathematics, the quaternion number system extends the complex numbers. Quaternions were first described by the Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. Hamilton defined a quatern ...
*
Algebra of physical space
In physics, the algebra of physical space (APS) is the use of the Clifford algebra, Clifford or geometric algebra Cl3,0(R) of the three-dimensional Euclidean space as a model for (3+1)-dimensional spacetime, representing a point in spacetime via a ...
*
Universal geometric algebra In mathematics, a universal geometric algebra is a type of geometric algebra generated by real number, real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case.
The universal geometri ...
Notes
Citations
References and further reading
:''Arranged chronologically''
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Chapter 1as PDF
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* Extract online at http://geocalc.clas.asu.edu/html/UAFCG.html #5 New Tools for Computational Geometry and rejuvenation of Screw Theory
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External links
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, from Ian Bell
Ian Ronald Bell (born 11 April 1982) is an English former cricketer who played international cricket in all formats for the England cricket team and county cricket for Warwickshire County Cricket Club. A right-handed higher/middle order batsm ...
.
IMPA Summer School 2010
Fernandes Oliveira Intro and Slides.
E.S.M. Hitzer and Japan GA publications.
Google Group for GA
Geometric Algebra Primer
Introduction to GA, Jaap Suter.
Geometric Algebra Resources
curated wiki, Pablo Bleyer.
Applied Geometric Algebras in Computer Science and Engineering 2018
Early Proceedings
GAME2020
Geometric Algebra Mini Event
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
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
(Arizona State University).
GA-Net blog
an
Geometric Algebra/Clifford Algebra development news.
Geometric Algebra for Perception Action Systems. Geometric Cybernetics Group
(CINVESTAV, Campus Guadalajara, Mexico).
{{DEFAULTSORT:Geometric Algebra