Blade (geometry) explained

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:^[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 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^[2] or an antivector.^[3]
• The highest grade element in a space is called a pseudoscalar, and in a space of dimension is an -blade.^[4]
• In a vector space of dimension, there are dimensions of freedom in choosing a -blade for, of which one dimension is an overall scaling multiplier.^[5]

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.^[6] Indeed, a -blade is naturally equivalent to a -subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating -multilinear scalar-valued function), such a -blade may be normalized to take unit value, making the correspondence unique up to a sign.

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 that is 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

References

• Book: Clifford Algebra to Geometric Calculus: A Unified Language for Mathematics and Physics . David Hestenes . David Hestenes . Garret Sobczyk . Chapter 1: Geometric algebra . 1 ff . 90-277-2561-6 . 1987 . Springer .
• Book: Geometric algebra for physicists . Chris Doran . Anthony Lasenby . amp . 0-521-48022-1 . 2003 . Cambridge University Press .
• 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.
• Book: Computer algebra and geometric algebra with applications . 2005 . 329 ff . R Wareham . J Cameron . J Lasenby . amp . Applications of conformal geometric algebra to computer vision and graphics . https://books.google.com/books?id=uxofVAQE3LoC&pg=PA330 . Hongbo Li. Peter J Olver. Peter J. Olver. Gerald Sommer . 3-540-26296-2 . Springer.

External links

• A Geometric Algebra Primer, especially for computer scientists.

Notes and References

1. Book: Invariants for pattern recognition and classification . Marcos A. Rodrigues . §1.2 Geometric algebra: an outline . https://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3 . 3 ff . 981-02-4278-6 . 2000 . World Scientific.
2. Book: https://books.google.com/books?id=oaoLbMS3ErwC&dq=%22pseudovectors+%28grade+n+-+1+elements%29%22&pg=PA100 . 100 . William E Baylis . Lectures on Clifford (geometric) algebras and applications . 0-8176-3257-3 . 2004 . §4.2.3 Higher-grade multivectors in Cℓ_n: Duals . Birkhäuser.
3. Book: Foundations of Game Engine Development, Volume 1: Mathematics . Lengyel . Eric . Terathon Software LLC. 2016 . 978-0-9858117-4-7.
4. Book: Geometric algebra for computer graphics . John A. Vince . 85 . 978-1-84628-996-5 . 2008 . Springer.
5. For Grassmannians (including the result about dimension) a good book is: . The proof of the dimensionality is actually straightforward. Take the exterior product of vectors v _{1\wedge … \wedge} v _{k} and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of F ^{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.
6. Book: New foundations for classical mechanics: Fundamental Theories of Physics . David Hestenes . 54 . 0-7923-5302-1 . 1999 . Springer.