The above definition of the geometric algebra is abstract, so we summarize the properties of the geometric product by the following set of axioms. An important property of the geometric product is the existence of elements having a multiplicative inverse. A nonzero element of the algebra does not necessarily have a multiplicative inverse. In this article, this identification is assumed. Thus we can define the inner product [b] of vectors as.
The antisymmetric part is the exterior product of the two vectors, the product of the contained exterior algebra :. The inner and exterior products are associated with familiar concepts from standard vector algebra. 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. 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. 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. With these, we can define a real symmetric matrix in the same way as a Gramian matrix. 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. The total number of basis vectors that square to zero is also invariant, and may be nonzero if the degenerate case is allowed. Every multivector of the geometric algebra can be expressed as a linear combination of the canonical basis elements. Using an orthogonal basis, a graded vector space structure can be established.
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. As a result:. Since the geometric product of two even multivectors is an even multivector, they define an even subalgebra.
Unit pseudoscalars are blades that play important roles in GA. It is sometimes possible to identify the presence of an imaginary unit in a physical equation. The dual basis vectors may be constructed as. It is common practice to extend the exterior product on vectors to the entire algebra.
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This may be done through the use of the grade projection operator:. This generalization is consistent with the above definition involving antisymmetrization. Another generalization related to the exterior product is 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 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 Dorst 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. Hestenes and Perwass. No consistent notation has emerged. Dorst 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,. Although versors are easier to work with because they can be directly represented in the algebra as multivectors, they are a subgroup of linear functions on multivectors and these can still be used when necessary.
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 its unique extension.
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. From Vectors to Geometric Algebra covers basic analytic geometry and gives an introduction to stereographic projection.
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Dotting the " Pauli vector " a dyad :. However, a useful inner product cannot be defined in the space and so there is no geometric product either leaving only outer product and non-metric uses of duality such as meet and join.
Nevertheless, there has been investigation of 4-dimensional alternatives to the full 5-dimensional CGA for limited geometries such as rigid body movements. Other useful references are Li and Bayro-Corrochano This allows all conformal transformations to be done as rotations and reflections and is covariant , extending incidence relations of projective geometry to circles and spheres. A fast changing and fluid area of GA, CGA is also being investigated for applications to relativistic physics.
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. 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 rotoreflections and rotations. The result of the reflection will be. A general reflection may be expressed as the composite of any odd number of single-axis reflections. We can also show that.
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The descriptions for rotations and reflections, including their outermorphisms, are examples of such sandwiching. The outermorphisms have a particularly simple algebraic form. Specifically, a mapping of vectors of the form. 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.
Clifford group, although Lundholm deprecates this usage.
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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. It could be any shape, although the volume equals that of the parallelotope. The mathematical description of rotational forces such as torque and angular momentum often makes use of the cross product of vector calculus in three dimensions with a convention of orientation handedness.
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