Blade (geometry)

In the study of geometric algebras, a -blade or a simple -vector is a generalization of the concept of scalars and vectors to include ''simple'' bivectors, trivectors, etc. Specifically, a -blade is a -vector that can be expressed as the exterior product (informally ''wedge product'') of 1-vectors, and is of ''grade'' .

In detail:

*A 0-blade is a scalar.
*A 1-blade is a vector. Every vector is simple.
*A 2-blade is a ''simple'' bivector. Sums of 2-blades are also bivectors, but not always simple. A 2-blade may be expressed as the wedge product of two vectors and :
*:$a \wedge b .$
*A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors , , and :
*:$a \wedge b \wedge c.$
*In a vector space of dimension , a blade of grade is called a ''pseudovector'' or an ''antivector''.
*The highest grade element in a space is called a '' pseudoscalar'', and in a space of dimension is an -blade.
*In a vector space of dimension , there are dimensions of freedom in choosing a -blade, of which one dimension is an overall scaling multiplier.For Grassmannians (including the result about dimension) a good book is: . The proof of the dimensionality is actually straightforward. Take vectors and wedge them together $v_1\wedge\cdots\wedge v_k$ and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of $\mathbb^k$. The wedge product is then parametrized by the product of the pivots and the lower block. Compare also with the dimension of a Grassmannian, , in which the scalar multiplier is eliminated.

A vector subspace of finite dimension may be represented by the -blade formed as a wedge product of all the elements of a basis for that subspace. Indeed, a -blade is naturally equivalent to a -subspace endowed with a volume form (an alternating -multilinear scalar-valued function) normalized to take unit value on the -blade.

Examples

In two-dimensional space, scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades in this context known as pseudoscalars, in that they are elements of a one-dimensional space distinct from regular scalars.

In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, while 2-blades are oriented area elements. In this case 3-blades are called pseudoscalars and represent three-dimensional volume elements, which form a one-dimensional vector space similar to scalars. Unlike scalars, 3-blades transform according to the Jacobian determinant of a change-of-coordinate function.

See also

* Grassmannian
* Multivector
* Exterior algebra
* Differential form
* Geometric algebra
*Clifford algebra

Notes

References

*
*
A Lasenby, J Lasenby & R Wareham
(2004) ''A covariant approach to geometry using geometric algebra'' Technical Report. University of Cambridge Department of Engineering, Cambridge, UK.
*{{cite book , title=Computer algebra and geometric algebra with applications , year=2005 , page=329 ''ff'' , author=R Wareham , author2=J Cameron , author3=J Lasenby , name-list-style=amp , chapter=Applications of conformal geometric algebra to computer vision and graphics , chapter-url=https://books.google.com/books?id=uxofVAQE3LoC&pg=PA330 , editor1= Hongbo Li, editor2=Peter J Olver, editor2-link=Peter J. Olver, editor3=Gerald Sommer , isbn=3-540-26296-2 , publisher=Springer

External links

A Geometric Algebra Primer
especially for computer scientists.

Geometric algebra
Vector calculus