Blade (geometry)

In the study of geometric algebras, 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:

  ${\displaystyle a\wedge b.}$

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

  ${\displaystyle a\wedge b\wedge c.}$

• In a 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 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.^[5]

For an n-dimensional space, there are blades of all grades from 0 to n inclusive. 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]

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 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, and 2-blades are oriented area elements. 3-blades 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. 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 1-84628-996-3.
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, MR 1288523. The proof of the dimensionality is actually straightforward. Take k 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 k × k block are elementary basis vectors of ${\mathbb {F}}^{k}$. The wedge product is then parametrized by the product of the pivots and the lower k × (n − k) block.
6. 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 & 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.