Blade (geometry)

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

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

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 Cℓn: 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 978-1-84628-996-5.
  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 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. 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.

References
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