Blade (geometry)

Short description: Exterior product of vectors

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

In detail:[1]

* 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 a and b:

a∧b.
* A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors a, b, and c:

a∧b∧c.
* In a vector space of dimension n, a blade of grade n − 1 is called a pseudovector[2] or an antivector.[3]
* The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade.[4]
* In a vector space of dimension n, there are k(n − k) + 1 dimensions of freedom in choosing a k-blade for 0 ≤ k ≤ n, of which one dimension is an overall scaling multiplier.[5]

A vector subspace of finite dimension k may be represented by the k-blade formed as a wedge product of all the elements of a basis for that subspace.[6] Indeed, a k-blade is naturally equivalent to a k-subspace endowed with a volume form (an alternating k-multilinear scalar-valued function) normalized to take unit value on the k-blade.

## Contents

* 1 Examples
* 2 See also
* 3 Notes
* 4 References
* 5 External links

## 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

1. ↑ Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 981-02-4278-6. https://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3.
2. ↑ William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓn: Duals". Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0-8176-3257-3. https://books.google.com/books?id=oaoLbMS3ErwC&dq="pseudovectors+(grade+n+-+1+elements)"&pg=PA100.
3. ↑ Lengyel, Eric (2016). Foundations of Game Engine Development, Volume 1: Mathematics. Terathon Software LLC. ISBN 978-0-9858117-4-7.
4. ↑ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 978-1-84628-996-5. https://books.google.com/books?id=3VxZqfm3I_MC&dq=pseudoscalar+"highest+grade"&pg=PA85.
5. ↑ For Grassmannians (including the result about dimension) a good book is: Griffiths, Phillip; Harris, Joseph (1994), Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons, ISBN 978-0-471-05059-9 . The proof of the dimensionality is actually straightforward. Take the exterior product of k vectors v1∧⋯∧vk and perform elementary column operations on these (factoring the pivots out) until the top k × k block are elementary basis vectors of 𝔽k. The wedge product is then parametrized by the product of the pivots and the lower k × (n − k) block. Compare also with the dimension of a Grassmannian, k(n − k), in which the scalar multiplier is eliminated.
6. ↑ David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1. https://books.google.com/books?id=AlvTCEzSI5wC&pg=PA54.

## References

* David Hestenes; Garret Sobczyk (1987). "Chapter 1: Geometric algebra". Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics. Springer. p. 1 ff. ISBN 90-277-2561-6. https://books.google.com/books?id=yyjLeKEdt20C.
* Chris Doran; Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. ISBN 0-521-48022-1. https://books.google.com/books?id=nZ6MsVIdt88C&q="geometric+algebra+for+physicists".
* 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.
* R Wareham; J Cameron; J Lasenby (2005). "Applications of conformal geometric algebra to computer vision and graphics". Computer algebra and geometric algebra with applications. Springer. p. 329 ff. ISBN 3-540-26296-2. https://books.google.com/books?id=uxofVAQE3LoC&pg=PA330.

## External links

* A Geometric Algebra Primer, especially for computer scientists.

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