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\wedgeb.

a\wedgeb\wedgec.

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

References

External links

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 Cℓn: 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

    v1\wedge\wedgevk

    and perform elementary column operations on these (factoring the pivots out) until the top block are elementary basis vectors of

    Fk

    . 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.