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 $b$ :
$a\wedge b.$
A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors $a$ , $b$ , and $c$ :
$a\wedge b\wedge c.$
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 $0 \leq k \leq n$ , 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[edit]

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.

Notes[edit]

^ 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.
^ 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.
^ Lengyel, Eric (2016). Foundations of Game Engine Development, Volume 1: Mathematics. Terathon Software LLC. ISBN 978-0-9858117-4-7.
^ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 978-1-84628-996-5.
^ 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 the exterior product of $k$ vectors $v_{1}\wedge \cdots \wedge v_{k}$ and perform elementary column operations on these (factoring the pivots out) until the top $k \times k$ block are elementary basis vectors of $\mathbb {F} ^{k}$ . The wedge product is then parametrized by the product of the pivots and the lower $k \times (n - k)$ block. Compare also with the dimension of a Grassmannian, $k (n - k)$ , in which the scalar multiplier is eliminated.
^ David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1.

References[edit]

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 (eds.). Computer algebra and geometric algebra with applications. Springer. p. 329 ff. ISBN 3-540-26296-2.

External links[edit]

A Geometric Algebra Primer, especially for computer scientists.

[Rodrigues-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.

[Baylis01-2] 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.

[Vince-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 the exterior product of $k$ vectors $v_{1}\wedge \cdots \wedge v_{k}$ and perform elementary column operations on these (factoring the pivots out) until the top $k \times k$ block are elementary basis vectors of $\mathbb {F} ^{k}$ . The wedge product is then parametrized by the product of the pivots and the lower $k \times (n - k)$ block. Compare also with the dimension of a Grassmannian, $k (n - k)$ , in which the scalar multiplier is eliminated.

[Hestenes-6] David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0-7923-5302-1.

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