Blade (geometry)

In geometric algebra, a blade is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is any object that can be expressed as the exterior product (informally wedge product) of k 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. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades. A 2-blade may be expressed as the wedge product of two vectors a and b:

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

• In a space of dimension n, a blade of grade n − 1 is called a pseudovector.^[2]
• The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade.^[3]
• In a space of dimension n, there are k(n − k) + 1 dimensions of freedom in choosing a k-blade, of which one dimension is an overall scaling multiplier.^[4]

In a n-dimensional spaces, there are blades of grade 0 through n. 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.^[5]

Examples

For example, in 2-dimensional space scalars are described as 0-blades, vectors are 1-blades, and area elements are 2-blades known as pseudoscalars, in that they are one-dimensional objects distinct from regular scalars.

In three-dimensional space, 0-blades are again scalars and 1-blades are three-dimensional vectors, but in three-dimensions, areas have an orientation, so while 2-blades are area elements, they are oriented. 3-blades (trivectors) represent volume elements and in three-dimensional space, these are scalar-like—i.e., 3-blades in three-dimensions form a one-dimensional vector space.

See also

• Multivector
• Exterior algebra
• 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. 
2. ↑ William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓ_n: Duals". Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0-8176-3257-3. 
3. ↑ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 1-84628-996-3. 
4. ↑ 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, MR 1288523 . The proof of the dimensionality is actually straightforward. Take k vectors and wedge them together and perform elementary column operations on these (factoring the pivots out) until the top k × k block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower k × (n − k) block.
5. ↑ David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1. 

General 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. 
• Chris Doran and Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. ISBN 0-521-48022-1. 
• 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". In Hongbo Li, Peter J. Olver, Gerald Sommer. Computer algebra and geometric algebra with applications. Springer. p. 329 ff. ISBN 3-540-26296-2. 

External links

• A Geometric Algebra Primer, especially for computer scientists.