Blade (geometry)
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In geometric algebra, a blade is a generalization of the notion of vectors and scalars to include simple bivectors, trivectors, etc. In detail:[1]
- A scalar or 0-blade of grade 0 is the inner product or dot product of two vectors a and b denoted as:
- A vector is a 1-blade of grade 1.
- A 2-blade of grade 2 is a simple bivector – (sums of 2-blades also are bivectors, but may not be simple), given by the wedge product of two vectors a and b:
- A 3-blade of grade 3 is a trivector, that is, a wedge product of three vectors, a, b and c:
- A k-blade is a blade of grade k.
- The highest grade element in a space is called the pseudoscalar.[2]
- In a space of dimension n, the blade of grade n−1 is called a pseudovector.[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 zero through n. A vector space of finite dimension n is related to a pseudoscalar of grade n.[5]
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[edit] 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.
[edit] See also
[edit] Notes
- ^ Marcos A. Rodrigues (2000). "§1.2 Geometric algebra: an outline". Invariants for pattern recognition and classification. World Scientific. p. 3 ff. ISBN 9810242786. http://books.google.com/books?id=QbFSt0SlDjIC&pg=PA3.
- ^ John A. Vince (2008). Geometric algebra for computer graphics. Springer. p. 85. ISBN 1846289963. http://books.google.com/books?id=3VxZqfm3I_MC&pg=PA85&dq=pseudoscalar+%22highest+grade%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=1#v=onepage&q=pseudoscalar%20%22highest%20grade%22&f=false.
- ^ William E Baylis (2004). "§4.2.3 Higher-grade multivectors in Cℓn: Duals". Lectures on Clifford (geometric) algebras and applications. Birkhäuser. p. 100. ISBN 0817632573. http://books.google.com/books?id=oaoLbMS3ErwC&pg=PA100&dq=%22pseudovectors+%28grade+n+-+1+elements%29%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=1#v=onepage&q=%22pseudovectors%20%28grade%20n%20-%201%20elements%29%22&f=false.
- ^ 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, MR1288523. 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 
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. - ^ David Hestenes (1999). New foundations for classical mechanics: Fundamental Theories of Physics. Springer. p. 54. ISBN 0792353021. http://books.google.com/books?id=AlvTCEzSI5wC&pg=PA54.
and perform elementary column operations on these (factoring the pivots out) until the top 
block are elementary basis vectors of 
. The wedge product is then parametrized by the product of the pivots and the lower [edit] 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 9027725616. http://books.google.com/books?id=yyjLeKEdt20C&printsec=frontcover&source=gbs_navlinks_s#v=onepage&q=&f=false.
- Chris Doran and Anthony Lasenby (2003). Geometric algebra for physicists. Cambridge University Press. ISBN 9780521480221. http://books.google.com/books?id=nZ6MsVIdt88C&printsec=frontcover&dq=%22geometric+algebra+for+physicists%22&lr=&as_drrb_is=q&as_minm_is=0&as_miny_is=&as_maxm_is=0&as_maxy_is=&as_brr=0&cd=1#v=onepage&q=&f=false.
- 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 3540262962. http://books.google.com/books?id=uxofVAQE3LoC&pg=PA330#v=onepage&q=&f=false.
[edit] External links
- A Geometric Algebra Primer, especially for computer scientists.
