Geometric algebra

In mathematics, a geometric algebra (also known as a Clifford algebra) is an algebra that can represent and manipulate geometrical objects such as vectors. 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 (though generally not by all elements) and addition of objects of different dimensions.

The geometric product was first briefly mentioned by Hermann Grassmann,^[1] who was chiefly interested in developing the closely related exterior algebra. In 1878, William Kingdon Clifford 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 and Hamilton's quaternion algebra. 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 (CGA) providing a framework for classical geometries.^[2] 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 then newly developed to describe electromagnetism. The term "geometric algebra" was repopularized in the 1960s by David 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. Bivectors provide a more natural representation of the pseudovector quantities of 3D vector calculus that are derived as a cross product, such as oriented area, oriented angle of rotation, torque, angular momentum and the magnetic field. A trivector can represent an oriented volume, and so on. An element called a blade may be used to represent a subspace and orthogonal projections 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 (and the less common algebra of physical space) and the conformal geometric algebra. Geometric calculus, an extension of GA that incorporates differentiation and integration, can be used to formulate other theories such as complex analysis and differential geometry, e.g. by using the Clifford algebra instead of differential forms. Geometric algebra has been advocated, most notably by David Hestenes^[4] and Chris Doran,^[5] as the preferred mathematical framework for physics. Proponents claim that it provides compact and intuitive descriptions in many areas including classical and quantum mechanics, electromagnetic theory, and relativity.^[6] GA has also found use as a computational tool in computer graphics^[7] and robotics.

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 space ⁠${\displaystyle V}$⁠ over a field ⁠${\displaystyle F}$⁠ with a symmetric bilinear form (the inner product,^[b] e.g., the Euclidean or Lorentzian metric) ⁠${\displaystyle g:V\times V\to F}$⁠, the geometric algebra of the quadratic space ⁠${\displaystyle (V,g)}$⁠ is the Clifford algebra ⁠${\displaystyle \operatorname {Cl} (V,g)}$⁠, an element of which is called a multivector. The Clifford algebra is commonly defined as a quotient algebra of the tensor algebra, though this definition is abstract, so the following definition is presented without requiring abstract algebra.

Definition

A unital associative algebra ⁠${\displaystyle \operatorname {Cl} (V,g)}$⁠ with a nondegenerate symmetric bilinear form ⁠${\displaystyle g:V\times V\to F}$⁠ is the Clifford algebra of the quadratic space ⁠${\displaystyle (V,g)}$⁠ if^[10]

• it contains ⁠${\displaystyle F}$⁠ and ⁠${\displaystyle V}$⁠ as distinct subspaces
• ⁠${\displaystyle a^{2}=g(a,a)1}$⁠ for ⁠${\displaystyle a\in V}$⁠
• ⁠${\displaystyle V}$⁠ generates ⁠${\displaystyle \operatorname {Cl} (V,g)}$⁠ as an algebra
• ⁠${\displaystyle \operatorname {Cl} (V,g)}$⁠ is not generated by any proper subspace of ⁠${\displaystyle V}$⁠.

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 the real case, ⁠${\displaystyle F=\mathbb {R} }$⁠, will be considered. The notation ⁠${\displaystyle {\mathcal {G}}(p,q)}$⁠ (respectively ⁠${\displaystyle {\mathcal {G}}(p,q,r)}$⁠) will be used to denote a geometric algebra for which the bilinear form ⁠${\displaystyle g}$⁠ has the signature ⁠${\displaystyle (p,q)}$⁠ (respectively ⁠${\displaystyle (p,q,r)}$⁠).

The 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 or the outer product^[d]). It is standard to denote these respectively by juxtaposition (i.e., suppressing any explicit multiplication symbol) and the symbol ⁠${\displaystyle \wedge }$⁠.

The above definition of the geometric algebra is still somewhat abstract, so we summarize the properties of the geometric product here. For multivectors ⁠${\displaystyle A,B,C\in {\mathcal {G}}(p,q)}$⁠:

• ⁠${\displaystyle AB\in {\mathcal {G}}(p,q)}$⁠ (closure)
• ⁠${\displaystyle 1A=A1=A}$⁠, where ⁠${\displaystyle 1}$⁠ is the identity element (existence of an identity element)
• ⁠${\displaystyle A(BC)=(AB)C}$⁠ (associativity)
• ⁠${\displaystyle A(B+C)=AB+AC}$⁠ and ⁠${\displaystyle (B+C)A=BA+CA}$⁠ (distributivity)
• ⁠${\displaystyle a^{2}=g(a,a)1}$⁠ for ⁠${\displaystyle a\in V}$⁠.

The exterior product has the same properties, except that the last property above is replaced by ⁠${\displaystyle a\wedge a=0}$⁠ for ⁠${\displaystyle a\in V}$⁠.

Note that in the last property above, the real number ⁠${\displaystyle g(a,a)}$⁠ need not be nonnegative if ⁠${\displaystyle g}$⁠ is not positive-definite. An important property of the geometric product is the existence of elements that have a multiplicative inverse. For a vector ⁠${\displaystyle a}$⁠, if ${\displaystyle a^{2}\neq 0}$ then ${\displaystyle a^{-1}}$ exists and is equal to ⁠${\displaystyle g(a,a)^{-1}a}$⁠. A nonzero element of the algebra does not necessarily have a multiplicative inverse. For example, if ${\displaystyle u}$ is a vector in ${\displaystyle V}$ such that ⁠${\displaystyle u^{2}=1}$⁠, the element ${\displaystyle \textstyle {\frac {1}{2}}(1+u)}$ is both a nontrivial idempotent element and a nonzero zero divisor, and thus has no inverse.^[e]

It is usual to identify ${\displaystyle \mathbb {R} }$ and ${\displaystyle V}$ with their images under the natural embeddings ${\displaystyle \mathbb {R} \to {\mathcal {G}}(p,q)}$ and ⁠${\displaystyle V\to {\mathcal {G}}(p,q)}$⁠. In this article, this identification is assumed. Throughout, the terms scalar and vector refer to elements of ${\displaystyle \mathbb {R} }$ and ${\displaystyle V}$ respectively (and of their images under this embedding).

Orientation defined by an ordered set of vectors.

Reversed orientation corresponds to negating the exterior product.

Geometric interpretation of grade-${\displaystyle n}$ elements in a real exterior algebra for ${\displaystyle n=0}$ (signed point), ${\displaystyle 1}$ (directed line segment, or vector), ${\displaystyle 2}$ (oriented plane element), ${\displaystyle 3}$ (oriented volume). The exterior product of ${\displaystyle n}$ vectors can be visualized as any ⁠${\displaystyle n}$⁠-dimensional shape (e.g. ⁠${\displaystyle n}$⁠-parallelotope, ⁠${\displaystyle n}$⁠-ellipsoid); with magnitude (hypervolume), and orientation defined by that on its ⁠${\displaystyle (n-1)}$⁠-dimensional boundary and on which side the interior is.^[14][15]

For vectors ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠, we may write the geometric product of any two vectors ⁠${\displaystyle a}$⁠ and ⁠${\displaystyle b}$⁠ as the sum of a symmetric product and an antisymmetric product:

${\displaystyle ab={\frac {1}{2}}(ab+ba)+{\frac {1}{2}}(ab-ba).}$

Thus we can define the inner product of vectors as

${\displaystyle a\cdot b:=g(a,b),}$

so that the symmetric product can be written as

${\displaystyle {\frac {1}{2}}(ab+ba)={\frac {1}{2}}\left((a+b)^{2}-a^{2}-b^{2}\right)=a\cdot b.}$

Conversely, ⁠${\displaystyle g}$⁠ is completely determined by the algebra. The antisymmetric part is the exterior product of the two vectors, the product of the contained exterior algebra:

${\displaystyle a\wedge b:={\frac {1}{2}}(ab-ba)=-(b\wedge a).}$

Then by simple addition:

${\displaystyle ab=a\cdot b+a\wedge b}$ the ungeneralized or vector form of the geometric product.

The inner and exterior products are associated with familiar concepts from standard vector algebra. Geometrically, ${\displaystyle a}$ and ${\displaystyle b}$ are parallel if their geometric product is equal to their inner product, whereas ${\displaystyle a}$ and ${\displaystyle b}$ are perpendicular 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 of standard vector algebra. The exterior product of two vectors can be identified with the signed area enclosed by a parallelogram the sides of which are the vectors. The cross product of two vectors in ${\displaystyle 3}$ 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, 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

${\displaystyle {\begin{aligned}1\wedge a_{i}&=a_{i}\wedge 1=a_{i}\\a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}&={\frac {1}{r!}}\sum _{\sigma \in {\mathfrak {S}}_{r}}\operatorname {sgn} (\sigma )a_{\sigma (1)}a_{\sigma (2)}\cdots a_{\sigma (r)},\end{aligned}}}$

where the sum is over all permutations of the indices, with ${\displaystyle \operatorname {sgn} (\sigma )}$ the sign of the permutation, and ${\displaystyle a_{i}}$ 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.

The equivalent structure equation for Clifford algebra is ^[16][17]

${\displaystyle a_{1}a_{2}a_{3}\dots a_{n}=\sum _{i=0}^{[{\frac {n}{2}}]}\sum _{\mu \in {}{\mathcal {C}}}(-1)^{k}\operatorname {Pf} (a_{\mu _{1}}\cdot a_{\mu _{2}},\dots ,a_{\mu _{2i-1}}\cdot a_{\mu _{2i}})a_{\mu _{2i+1}}\land \dots \land a_{\mu _{n}}}$

where ${\displaystyle \operatorname {Pf} (A)}$ is the Pfaffian of ⁠${\displaystyle A}$⁠ and ${\textstyle {\mathcal {C}}={\binom {n}{2i}}}$ provides combinations, ⁠${\displaystyle \mu }$⁠, of ⁠${\displaystyle n}$⁠ indices divided into ⁠${\displaystyle 2i}$⁠ and ⁠${\displaystyle n-2i}$⁠ parts and ⁠${\displaystyle k}$⁠ is the parity of the combination.

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 from simplices.^[19] This derivation provides a better connection between Pascal's triangle and simplices because it provides an interpretation of the first column of ones.

A multivector that is the exterior product of ${\displaystyle r}$ linearly independent vectors is called a blade, and is said to be of grade ⁠${\displaystyle r}$⁠.^[f] A multivector that is the sum of blades of grade ${\displaystyle r}$ is called a (homogeneous) multivector of grade ⁠${\displaystyle r}$⁠. From the axioms, with closure, every multivector of the geometric algebra is a sum of blades.

Consider a set of ${\displaystyle r}$ linearly independent vectors ${\displaystyle \{a_{1},\ldots ,a_{r}\}}$ spanning an ⁠${\displaystyle r}$⁠-dimensional subspace of the vector space. With these, we can define a real symmetric matrix (in the same way as a Gramian matrix)

${\displaystyle [\mathbf {A} ]_{ij}=a_{i}\cdot a_{j}}$

By the spectral theorem, ${\displaystyle \mathbf {A} }$ can be diagonalized to diagonal matrix ${\displaystyle \mathbf {D} }$ by an orthogonal matrix ${\displaystyle \mathbf {O} }$ via

${\displaystyle \sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {A} ]_{kl}[\mathbf {O} ^{\mathrm {T} }]_{lj}=\sum _{k,l}[\mathbf {O} ]_{ik}[\mathbf {O} ]_{jl}[\mathbf {A} ]_{kl}=[\mathbf {D} ]_{ij}}$

Define a new set of vectors ⁠${\displaystyle \{e_{1},\ldots ,e_{r}\}}$⁠, known as orthogonal basis vectors, to be those transformed by the orthogonal matrix:

${\displaystyle e_{i}=\sum _{j}[\mathbf {O} ]_{ij}a_{j}}$

Since orthogonal transformations preserve inner products, it follows that ${\displaystyle e_{i}\cdot e_{j}=[\mathbf {D} ]_{ij}}$ and thus the ${\displaystyle \{e_{1},\ldots ,e_{r}\}}$ are perpendicular. In other words, the geometric product of two distinct vectors ${\displaystyle e_{i}\neq e_{j}}$ is completely specified by their exterior product, or more generally

${\displaystyle {\begin{array}{rl}e_{1}e_{2}\cdots e_{r}&=e_{1}\wedge e_{2}\wedge \cdots \wedge e_{r}\\&=\left(\sum _{j}[\mathbf {O} ]_{1j}a_{j}\right)\wedge \left(\sum _{j}[\mathbf {O} ]_{2j}a_{j}\right)\wedge \cdots \wedge \left(\sum _{j}[\mathbf {O} ]_{rj}a_{j}\right)\\&=(\det \mathbf {O} )a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r}\end{array}}}$

Therefore, every blade of grade ${\displaystyle r}$ can be written as the exterior product of ${\displaystyle r}$ 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

${\displaystyle {\widehat {e_{i}}}={\frac {1}{\sqrt {|e_{i}\cdot e_{i}|}}}e_{i},}$

then these normalized vectors must square to ${\displaystyle +1}$ or ⁠${\displaystyle -1}$⁠. By Sylvester's law of inertia, the total number of ⁠${\displaystyle +1}$⁠ and the total number of ⁠${\displaystyle -1}$⁠s along the diagonal matrix is invariant. By extension, the total number ${\displaystyle p}$ of these vectors that square to ${\displaystyle +1}$ and the total number ${\displaystyle q}$ that square to ${\displaystyle -1}$ 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 ⁠${\displaystyle {\mathcal {G}}(p,q)}$⁠. For example, ${\displaystyle {\mathcal {G}}(3,0)}$ models three-dimensional Euclidean space, ${\displaystyle {\mathcal {G}}(1,3)}$ relativistic spacetime and ${\displaystyle {\mathcal {G}}(4,1)}$ a conformal geometric algebra of a three-dimensional space.

The set of all possible products of ${\displaystyle n}$ orthogonal basis vectors with indices in increasing order, including ${\displaystyle 1}$ as the empty product, forms a basis for the entire geometric algebra (an analogue of the PBW theorem). For example, the following is a basis for the geometric algebra ⁠${\displaystyle {\mathcal {G}}(3,0)}$⁠:

${\displaystyle \{1,e_{1},e_{2},e_{3},e_{1}e_{2},e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}e_{3}\}}$

A basis formed this way is called a standard basis for the geometric algebra, and any other orthogonal basis for ${\displaystyle V}$ will produce another standard basis. Each standard basis consists of ${\displaystyle 2^{n}}$ elements. Every multivector of the geometric algebra can be expressed as a linear combination of the standard basis elements. If the standard basis elements are ${\displaystyle \{B_{i}\mid i\in S\}}$ with ${\displaystyle S}$ being an index set, then the geometric product of any two multivectors is

${\displaystyle \left(\sum _{i}\alpha _{i}B_{i}\right)\left(\sum _{j}\beta _{j}B_{j}\right)=\sum _{i,j}\alpha _{i}\beta _{j}B_{i}B_{j}.}$

The terminology "${\displaystyle k}$-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 ${\displaystyle k}$ vectors). By way of example, ${\displaystyle e_{1}\wedge e_{2}+e_{3}\wedge e_{4}}$ in ${\displaystyle {\mathcal {G}}(4,0)}$ 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 ⁠${\displaystyle 0}$⁠-, ⁠${\displaystyle 1}$⁠-, ⁠${\displaystyle (n-1)}$⁠- and ⁠${\displaystyle n}$⁠-vectors are always blades in ⁠${\displaystyle n}$⁠-space.

A ⁠${\displaystyle k}$⁠-versor is a multivector that can be expressed as the geometric product of ${\displaystyle k}$ invertible 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

${\displaystyle V\to V:a\mapsto RaR^{-1}}$ extends to the outermorphism ${\displaystyle {\mathcal {G}}(V)\to {\mathcal {G}}(V):A\mapsto RAR^{-1}.}$

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 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 ⁠${\displaystyle {\mathcal {G}}(p,q)}$⁠, having identified the group ${\displaystyle {\mathcal {G}}^{\times }}$ as the group of all invertible elements of ⁠${\displaystyle {\mathcal {G}}}$⁠, Lundholm gives a proof that the "versor group" ${\displaystyle \{v_{1}v_{2}\cdots v_{k}\in {\mathcal {G}}\mid v_{i}\in V^{\times }\}}$ (the set of invertible versors) is equal to the Lipschitz group ${\displaystyle \Gamma }$ (a.k.a. Clifford group, although Lundholm deprecates this usage).^[23]

We denote the grade involution as ⁠${\displaystyle {\widehat {S}}}$⁠ and reversion as ⁠${\displaystyle {\widetilde {S}}}$⁠.

Although the Lipschitz group (defined as ⁠${\displaystyle \{S\in {\mathcal {G}}^{\times }\mid {\widehat {S}}VS^{-1}\subseteq V\}}$⁠) and the versor group (defined as ⁠${\displaystyle \textstyle \{\prod _{i=0}^{k}v_{i}\mid v_{i}\in V^{\times },k\in \mathbb {N} \}}$⁠) have divergent definitions, they are the same group. Lundholm defines the ⁠${\displaystyle \operatorname {Pin} }$⁠, ⁠${\displaystyle \operatorname {Spin} }$⁠, and ⁠${\displaystyle \operatorname {Spin} ^{+}}$⁠ subgroups of the Lipschitz group.^[24]

Subgroup	Definition	GA term
${\displaystyle \Gamma }$	${\displaystyle \{S\in {\mathcal {G}}^{\times }\mid {\widehat {S}}VS^{-1}\subseteq V\}}$	versors
${\displaystyle \operatorname {Pin} }$	${\displaystyle \{S\in \Gamma \mid S{\widetilde {S}}=\pm 1\}}$	unit versors
${\displaystyle \operatorname {Spin} }$	${\displaystyle {\operatorname {Pin} }\cap {\mathcal {G}}^{[0]}}$	even unit versors
${\displaystyle \operatorname {Spin} ^{+}}$	${\displaystyle \{S\in \operatorname {Spin} \mid S{\widetilde {S}}=1\}}$	rotors

Multiple analyses of spinors use GA as a representation.^[25]

A ⁠${\displaystyle \mathbb {Z} }$⁠-graded vector space structure 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 ⁠${\displaystyle \{e_{1},\ldots ,e_{n}\}}$⁠.

Elements of the geometric algebra that are scalar multiples of ${\displaystyle 1}$ are of grade ${\displaystyle 0}$ and are called scalars. Elements that are in the span of ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ are of grade ⁠${\displaystyle 1}$⁠ and are the ordinary vectors. Elements in the span of ${\displaystyle \{e_{i}e_{j}\mid 1\leq i<j\leq n\}}$ are of grade ${\displaystyle 2}$ and are the bivectors. This terminology continues through to the last grade of ⁠${\displaystyle n}$⁠-vectors. Alternatively, ⁠${\displaystyle n}$⁠-vectors are called pseudoscalars, ⁠${\displaystyle (n-1)}$⁠-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 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 ⁠${\displaystyle r}$⁠-blade and an ⁠${\displaystyle s}$⁠-blade is contained in the span of ${\displaystyle 0}$ through ⁠${\displaystyle r+s}$⁠-blades, the geometric algebra is a filtered algebra.

A multivector ${\displaystyle A}$ may be decomposed with the grade-projection operator ⁠${\displaystyle \langle A\rangle _{r}}$⁠, which outputs the grade-⁠${\displaystyle r}$⁠ portion of ⁠${\displaystyle A}$⁠. As a result:

${\displaystyle A=\sum _{r=0}^{n}\langle A\rangle _{r}}$

As an example, the geometric product of two vectors ${\displaystyle ab=a\cdot b+a\wedge b=\langle ab\rangle _{0}+\langle ab\rangle _{2}}$ since ${\displaystyle \langle ab\rangle _{0}=a\cdot b}$ and ${\displaystyle \langle ab\rangle _{2}=a\wedge b}$ and ⁠${\displaystyle \langle ab\rangle _{i}=0}$⁠, for ${\displaystyle i}$ other than ${\displaystyle 0}$ and ⁠${\displaystyle 2}$⁠.

A multivector ${\displaystyle A}$ may 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:

${\displaystyle A^{[0]}=\langle A\rangle _{0}+\langle A\rangle _{2}+\langle A\rangle _{4}+\cdots }$

${\displaystyle A^{[1]}=\langle A\rangle _{1}+\langle A\rangle _{3}+\langle A\rangle _{5}+\cdots }$

This is the result of forgetting structure from a ⁠${\displaystyle \mathrm {Z} }$⁠-graded vector space to ⁠${\displaystyle \mathrm {Z} _{2}}$⁠-graded vector space. The geometric product respects this coarser grading. Thus in addition to being a ⁠${\displaystyle \mathrm {Z} _{2}}$⁠-graded vector space, the geometric algebra is a ⁠${\displaystyle \mathrm {Z} _{2}}$⁠-graded algebra, a.k.a. a superalgebra.

Restricting to the even part, the product of two even elements is also even. This means that the even multivectors defines an even subalgebra. The even subalgebra of an ⁠${\displaystyle n}$⁠-dimensional geometric algebra is algebra-isomorphic (without preserving either filtration or grading) to a full geometric algebra of ${\displaystyle (n-1)}$ dimensions. Examples include ${\displaystyle {\mathcal {G}}^{[0]}(2,0)\cong {\mathcal {G}}(0,1)}$ and ⁠${\displaystyle {\mathcal {G}}^{[0]}(1,3)\cong {\mathcal {G}}(3,0)}$⁠.

Geometric algebra represents subspaces of ${\displaystyle V}$ as blades, and so they coexist in the same algebra with vectors from ⁠${\displaystyle V}$⁠. A ⁠${\displaystyle k}$⁠-dimensional subspace ${\displaystyle W}$ of ${\displaystyle V}$ is represented by taking an orthogonal basis ${\displaystyle \{b_{1},b_{2},\ldots ,b_{k}\}}$ and using the geometric product to form the blade ⁠${\displaystyle D=b_{1}b_{2}\cdots b_{k}}$⁠. There are multiple blades representing ⁠${\displaystyle W}$⁠; all those representing ${\displaystyle W}$ are scalar multiples of ⁠${\displaystyle D}$⁠. These blades can be separated into two sets: positive multiples of ${\displaystyle D}$ and negative multiples of ⁠${\displaystyle D}$⁠. The positive multiples of ${\displaystyle D}$ are said to have the same orientation as ⁠${\displaystyle D}$⁠, 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) ⁠${\displaystyle n}$⁠-blades provide but that (the generalized class of) grade-⁠${\displaystyle n}$⁠ multivectors do not when ⁠${\displaystyle n\geq 4}$⁠.

Unit pseudoscalars are blades that play important roles in GA. A unit pseudoscalar for a non-degenerate subspace ${\displaystyle W}$ of ${\displaystyle V}$ is a blade that is the product of the members of an orthonormal basis for ⁠${\displaystyle W}$⁠. It can be shown that if ${\displaystyle I}$ and ${\displaystyle I'}$ are both unit pseudoscalars for ⁠${\displaystyle W}$⁠, then ${\displaystyle I=\pm I'}$ and ⁠${\displaystyle I^{2}=\pm 1}$⁠. If one doesn't choose an orthonormal basis for ⁠${\displaystyle W}$⁠, then the Plücker embedding 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 ${\displaystyle \alpha I}$ for all ⁠${\displaystyle \alpha \neq 0}$⁠. Orthonormality gets rid of this ambiguity except for the signs above.

Suppose the geometric algebra ${\displaystyle {\mathcal {G}}(n,0)}$ with the familiar positive definite inner product on ${\displaystyle \mathbb {R} ^{n}}$ is formed. Given a plane (two-dimensional subspace) of ⁠${\displaystyle \mathbb {R} ^{n}}$⁠, one can find an orthonormal basis ${\displaystyle \{b_{1},b_{2}\}}$ spanning the plane, and thus find a unit pseudoscalar ${\displaystyle I=b_{1}b_{2}}$ representing this plane. The geometric product of any two vectors in the span of ${\displaystyle b_{1}}$ and ${\displaystyle b_{2}}$ lies in ⁠${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$⁠, that is, it is the sum of a ⁠${\displaystyle 0}$⁠-vector and a ⁠${\displaystyle 2}$⁠-vector.

By the properties of the geometric product, ⁠${\displaystyle I^{2}=b_{1}b_{2}b_{1}b_{2}=-b_{1}b_{2}b_{2}b_{1}=-1}$⁠. The resemblance to the imaginary unit is not incidental: the subspace ${\displaystyle \{\alpha _{0}+\alpha _{1}I\mid \alpha _{i}\in \mathbb {R} \}}$ is ⁠${\displaystyle \mathbb {R} }$⁠-algebra isomorphic to the complex numbers. In this way, a copy of the complex numbers is embedded in the geometric algebra for each two-dimensional subspace of ${\displaystyle V}$ 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 ⁠${\displaystyle -1}$⁠, and these have geometric significance because of the properties of the algebra and the interaction of its various subspaces.

In ⁠${\displaystyle {\mathcal {G}}(3,0)}$⁠, a further familiar case occurs. Given a standard basis consisting of orthonormal vectors ${\displaystyle e_{i}}$ of ⁠${\displaystyle V}$⁠, the set of all ⁠${\displaystyle 2}$⁠-vectors is spanned by

${\displaystyle \{e_{3}e_{2},e_{1}e_{3},e_{2}e_{1}\}.}$

Labelling these ⁠${\displaystyle i}$⁠, ${\displaystyle j}$ and ${\displaystyle k}$ (momentarily deviating from our uppercase convention), the subspace generated by ⁠${\displaystyle 0}$⁠-vectors and ⁠${\displaystyle 2}$⁠-vectors is exactly ⁠${\displaystyle \{\alpha _{0}+i\alpha _{1}+j\alpha _{2}+k\alpha _{3}\mid \alpha _{i}\in \mathbb {R} \}}$⁠. This set is seen to be the even subalgebra of ⁠${\displaystyle {\mathcal {G}}(3,0)}$⁠, and furthermore is isomorphic as an ⁠${\displaystyle \mathbb {R} }$⁠-algebra to the quaternions, another important algebraic system.

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:

${\displaystyle C\wedge D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r+s}}$     (the exterior product)

This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is the commutator product:

${\displaystyle C\times D:={\tfrac {1}{2}}(CD-DC)}$     (the commutator 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 blades ⁠${\displaystyle C}$⁠ and ⁠${\displaystyle D}$⁠, the intersection (or meet) where the duality is to be taken relative to the a blade containing both ⁠${\displaystyle C}$⁠ and ⁠${\displaystyle D}$⁠ (the smallest such blade being the join).^[27]

${\displaystyle C\vee D:=((CI^{-1})\wedge (DI^{-1}))I}$

with ⁠${\displaystyle I}$⁠ the 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:

${\displaystyle C\;\rfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{s-r}}$   (the left contraction)

${\displaystyle C\;\lfloor \;D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{r-s}}$   (the right contraction)

${\displaystyle C*D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{0}}$   (the scalar product)

${\displaystyle C\bullet D:=\sum _{r,s}\langle \langle C\rangle _{r}\langle D\rangle _{s}\rangle _{|s-r|}}$   (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,

${\displaystyle C\;\rfloor \;D=(C\wedge (DI^{-1}))I}$

${\displaystyle C\;\lfloor \;D=I((I^{-1}C)\wedge D)}$

${\displaystyle (A\wedge B)*C=A*(B\;\rfloor \;C)}$

${\displaystyle C*(B\wedge A)=(C\;\lfloor \;B)*A}$

${\displaystyle A\;\rfloor \;(B\;\rfloor \;C)=(A\wedge B)\;\rfloor \;C}$

${\displaystyle (A\;\rfloor \;B)\;\lfloor \;C=A\;\rfloor \;(B\;\lfloor \;C).}$

Benefits of using the left contraction as an extension of the inner product on vectors include that the identity ${\displaystyle ab=a\cdot b+a\wedge b}$ is extended to ${\displaystyle aB=a\;\rfloor \;B+a\wedge B}$ for any vector ${\displaystyle a}$ and multivector ⁠${\displaystyle B}$⁠, and that the projection operation ${\displaystyle {\mathcal {P}}_{b}(a)=(a\cdot b^{-1})b}$ is extended to ${\displaystyle {\mathcal {P}}_{B}(A)=(A\;\rfloor \;B^{-1})\;\rfloor \;B}$ for any blade ${\displaystyle B}$ and any multivector ${\displaystyle A}$ (with a minor modification to accommodate null ⁠${\displaystyle B}$⁠, given below).

Let ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ be a basis of ⁠${\displaystyle V}$⁠, i.e. a set of ${\displaystyle n}$ linearly independent vectors that span the ⁠${\displaystyle n}$⁠-dimensional vector space ⁠${\displaystyle V}$⁠. The basis that is dual to ${\displaystyle \{e_{1},\ldots ,e_{n}\}}$ is the set of elements of the dual vector space ${\displaystyle V^{*}}$ that forms a biorthogonal system with this basis, thus being the elements denoted ${\displaystyle \{e^{1},\ldots ,e^{n}\}}$ satisfying

${\displaystyle e^{i}\cdot e_{j}=\delta ^{i}{}_{j},}$

where ${\displaystyle \delta }$ is the Kronecker delta.

Given a nondegenerate quadratic form on ⁠${\displaystyle V}$⁠, ${\displaystyle V^{*}}$ becomes naturally identified with ⁠${\displaystyle V}$⁠, and the dual basis may be regarded as elements of ⁠${\displaystyle V}$⁠, but are not in general the same set as the original basis.

Given further a GA of ⁠${\displaystyle V}$⁠, let

${\displaystyle I=e_{1}\wedge \cdots \wedge e_{n}}$

be the pseudoscalar (which does not necessarily square to ⁠${\displaystyle \pm 1}$⁠) formed from the basis ⁠${\displaystyle \{e_{1},\ldots ,e_{n}\}}$⁠. The dual basis vectors may be constructed as

${\displaystyle e^{i}=(-1)^{i-1}(e_{1}\wedge \cdots \wedge {\check {e}}_{i}\wedge \cdots \wedge e_{n})I^{-1},}$

where the ${\displaystyle {\check {e}}_{i}}$ denotes that the ⁠${\displaystyle i}$⁠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 ⁠${\displaystyle a}$⁠, scalar components ${\displaystyle a^{i}}$ can be defined as

${\displaystyle a^{i}=a\cdot e^{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components as

${\displaystyle a=\sum _{i}a^{i}e_{i}\ .}$

We can also define scalar components ${\displaystyle a_{i}}$ as

${\displaystyle a_{i}=a\cdot e_{i}\ ,}$

in terms of which ${\displaystyle a}$ can be separated into vector components in terms of the dual basis as

${\displaystyle a=\sum _{i}a_{i}e^{i}\ .}$

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

${\displaystyle J=(j_{1},\dots ,j_{n})\ ,}$

where ⁠${\displaystyle j_{1}<j_{2}<\dots <j_{n}}$⁠, we can write a basis blade as

${\displaystyle e_{J}=e_{j_{1}}\wedge e_{j_{2}}\wedge \cdots \wedge e_{j_{n}}\ .}$

The corresponding reciprocal blade has the indices in opposite order:

${\displaystyle e^{J}=e^{j_{n}}\wedge \cdots \wedge e^{j_{2}}\wedge e^{j_{1}}\ .}$

Similar to the case above with vectors, it can be shown that

${\displaystyle e^{J}*e_{K}=\delta _{K}^{J}\ ,}$

where ${\displaystyle *}$ is the scalar product.

With ${\displaystyle A}$ a multivector, we can define scalar components as^[30]

${\displaystyle A^{ij\cdots k}=(e^{k}\wedge \cdots \wedge e^{j}\wedge e^{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A^{ij\cdots k}e_{i}\wedge e_{j}\wedge \cdots \wedge e_{k}\ .}$

We can alternatively define scalar components

${\displaystyle A_{ij\cdots k}=(e_{k}\wedge \cdots \wedge e_{j}\wedge e_{i})*A\ ,}$

in terms of which ${\displaystyle A}$ can be separated into component blades as

${\displaystyle A=\sum _{i<j<\cdots <k}A_{ij\cdots k}e^{i}\wedge e^{j}\wedge \cdots \wedge e^{k}\ .}$

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 functions on multivectors, which can still be used when necessary. The geometric algebra of an ⁠${\displaystyle n}$⁠-dimensional vector space is spanned by a basis of ${\displaystyle 2^{n}}$ elements. If a multivector is represented by a ${\displaystyle 2^{n}\times 1}$ 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 by a ${\displaystyle 2^{n}\times 2^{n}}$ 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^[k] extension of the versor. If ${\displaystyle f}$ is a linear function that maps vectors to vectors, then its outermorphism is the function that obeys the rule

${\displaystyle {\underline {\mathsf {f}}}(a_{1}\wedge a_{2}\wedge \cdots \wedge a_{r})=f(a_{1})\wedge f(a_{2})\wedge \cdots \wedge f(a_{r})}$

for a blade, extended to the whole algebra through linearity.

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]

The even subalgebra of ${\displaystyle {\mathcal {G}}(2,0)}$ is isomorphic to the complex numbers, as may be seen by writing a vector ${\displaystyle P}$ in terms of its components in an orthonormal basis and left multiplying by the basis vector ⁠${\displaystyle e_{1}}$⁠, yielding

${\displaystyle Z=e_{1}P=e_{1}(xe_{1}+ye_{2})=x(1)+y(e_{1}e_{2}),}$

where we identify ${\displaystyle i\mapsto e_{1}e_{2}}$ since

${\displaystyle (e_{1}e_{2})^{2}=e_{1}e_{2}e_{1}e_{2}=-e_{1}e_{1}e_{2}e_{2}=-1.}$

Similarly, the even subalgebra of ${\displaystyle {\mathcal {G}}(3,0)}$ with basis ${\displaystyle \{1,e_{2}e_{3},e_{3}e_{1},e_{1}e_{2}\}}$ is isomorphic to the quaternions as may be seen by identifying ⁠${\displaystyle i\mapsto -e_{2}e_{3}}$⁠, ${\displaystyle j\mapsto -e_{3}e_{1}}$ and ⁠${\displaystyle k\mapsto -e_{1}e_{2}}$⁠.

Every associative algebra has a matrix representation; replacing the three Cartesian basis vectors by the Pauli matrices gives a representation of ⁠${\displaystyle {\mathcal {G}}(3,0)}$⁠:

${\displaystyle {\begin{aligned}e_{1}=\sigma _{1}=\sigma _{x}&={\begin{pmatrix}0&1\\1&0\end{pmatrix}}\\e_{2}=\sigma _{2}=\sigma _{y}&={\begin{pmatrix}0&-i\\i&0\end{pmatrix}}\\e_{3}=\sigma _{3}=\sigma _{z}&={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}\,.\end{aligned}}}$

Dotting the "Pauli vector" (a dyad):

${\displaystyle \sigma =\sigma _{1}e_{1}+\sigma _{2}e_{2}+\sigma _{3}e_{3}}$ with arbitrary vectors ${\displaystyle a}$ and ${\displaystyle b}$ and multiplying through gives:

${\displaystyle (\sigma \cdot a)(\sigma \cdot b)=a\cdot b+a\wedge b}$ (Equivalently, by inspection, ⁠${\displaystyle a\cdot b+i\sigma \cdot (a\times b)}$⁠)

In physics, the main applications are the geometric algebra of Minkowski 3+1 spacetime, ⁠${\displaystyle {\mathcal {G}}(1,3)}$⁠, called spacetime algebra (STA),^[3] or less commonly, ⁠${\displaystyle {\mathcal {G}}(3,0)}$⁠, interpreted the algebra of physical space (APS).

While in STA, points of spacetime are represented simply by vectors, in APS, points of ⁠${\displaystyle (3+1)}$⁠-dimensional spacetime are instead represented by paravectors, a three-dimensional vector (space) plus a one-dimensional scalar (time).

In spacetime algebra the electromagnetic field tensor has a bivector representation ⁠${\displaystyle {F}=({E}+ic{B})\gamma _{0}}$⁠.^[32] Here, the ${\displaystyle i=\gamma _{0}\gamma _{1}\gamma _{2}\gamma _{3}}$ is the unit pseudoscalar (or four-dimensional volume element), ${\displaystyle \gamma _{0}}$ is the unit vector in time direction, and ${\displaystyle E}$ and ${\displaystyle B}$ are the classic electric and magnetic field vectors (with a zero time component). Using the four-current ⁠${\displaystyle {J}}$⁠, Maxwell's equations then become

Formulation	Homogeneous equations	Non-homogeneous equations
Fields	${\displaystyle DF=\mu _{0}J}$
	${\displaystyle D\wedge F=0}$	${\displaystyle D~\rfloor ~F=\mu _{0}J}$
Potentials (any gauge)	${\displaystyle F=D\wedge A}$	${\displaystyle D~\rfloor ~(D\wedge A)=\mu _{0}J}$
Potentials (Lorenz gauge)	${\displaystyle F=DA}$ ${\displaystyle D~\rfloor ~A=0}$	${\displaystyle D^{2}A=\mu _{0}J}$

In geometric calculus, juxtaposition of vectors such as in ${\displaystyle DF}$ indicate the geometric product and can be decomposed into parts as ⁠${\displaystyle DF=D~\rfloor ~F+D\wedge F}$⁠. Here ${\displaystyle D}$ is the covector derivative in any spacetime and reduces to ${\displaystyle \nabla }$ in flat spacetime. Where ${\displaystyle \bigtriangledown }$ plays a role in Minkowski ⁠${\displaystyle 4}$⁠-spacetime which is synonymous to the role of ${\displaystyle \nabla }$ in Euclidean ⁠${\displaystyle 3}$⁠-space and is related to the d'Alembertian by ⁠${\displaystyle \Box =\bigtriangledown ^{2}}$⁠. Indeed, given an observer represented by a future pointing timelike vector ${\displaystyle \gamma _{0}}$ we have

${\displaystyle \gamma _{0}\cdot \bigtriangledown ={\frac {1}{c}}{\frac {\partial }{\partial t}}}$

${\displaystyle \gamma _{0}\wedge \bigtriangledown =\nabla }$

Boosts in this Lorentzian metric space have the same expression ${\displaystyle e^{\beta }}$ as rotation in Euclidean space, where ${\displaystyle {\beta }}$ 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 are a representation of ⁠${\displaystyle {\mathcal {G}}(1,3)}$⁠, showing the equivalence with matrix representations used by physicists.

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 of ${\displaystyle n}$ dimensions, the rotors represent a set of transformations with ${\displaystyle n(n-1)/2}$ degrees of freedom, corresponding to rotations – for example, ${\displaystyle 3}$ when ${\displaystyle n=3}$ and ${\displaystyle 6}$ when ⁠${\displaystyle n=4}$⁠. Geometric algebra is often used to model a projective space, i.e. as a homogeneous 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 dimension ${\displaystyle n+1}$ have $n(n-1)/2+n$ degrees of freedom, the same as the number of degrees of freedom in the rotations and translations combined for an ⁠${\displaystyle n}$⁠-dimensional space.

This is the case in Projective Geometric Algebra (PGA), which is used^[33][34][35] to represent Euclidean isometries in 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 ⁠${\displaystyle {\mathcal {G}}(3,0,1)}$⁠. With a suitable identification of subspaces to represent points, lines and planes, the versors of this algebra represent all proper Euclidean isometries, which are always screw motions in 3-dimensional space, along with all improper Euclidean isometries, which includes reflections, rotoreflections, transflections, and point reflections.

PGA combines ${\displaystyle {\mathcal {G}}(3,0,1)}$ with a complement operator 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 line ${\displaystyle {\boldsymbol {e}}_{12}}$ is the line ⁠${\displaystyle {\boldsymbol {e}}_{03}}$⁠, because ${\displaystyle {\boldsymbol {e}}_{0}}$ and ${\displaystyle {\boldsymbol {e}}_{3}}$ are basis elements that are not contained in ${\displaystyle {\boldsymbol {e}}_{12}}$ but are contained in ⁠${\displaystyle {\boldsymbol {e}}_{03}}$⁠. In the PGA of 2-dimensional space, the complement of ${\displaystyle {\boldsymbol {e}}_{12}}$ is ⁠${\displaystyle {\boldsymbol {e}}_{0}}$⁠, since there is no ${\displaystyle {\boldsymbol {e}}_{3}}$ element.

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 see Plane-based geometric algebra for discussion of homogeneous models of elliptic and hyperbolic geometry compared with the Euclidean geometry derived from PGA.

Working within GA, Euclidean space ${\displaystyle \mathbb {E} ^{3}}$ (along with a conformal point at infinity) is embedded projectively in the CGA ${\displaystyle {\mathcal {G}}(4,1)}$ 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 rounds objects such as circles and spheres.

Specifically, we add orthogonal basis vectors ${\displaystyle e_{+}}$ and ${\displaystyle e_{-}}$ such that ${\displaystyle e_{+}^{2}=+1}$ and ${\displaystyle e_{-}^{2}=-1}$ to the basis of the vector space that generates ${\displaystyle {\mathcal {G}}(3,0)}$ and identify null vectors

${\displaystyle n_{\text{o}}={\tfrac {1}{2}}(e_{-}-e_{+})}$ as the point at the origin and

${\displaystyle n_{\infty }=e_{-}+e_{+}}$ as a conformal point at infinity (see Compactification), giving

${\displaystyle n_{\infty }\cdot n_{\text{o}}=-1.}$