Blade (geometry)
In the study of
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
- ISBN 981-02-4278-6.
- ISBN 0-8176-3257-3.
- ISBN 978-0-9858117-4-7.
- ISBN 978-1-84628-996-5.
- MR 1288523. The proof of the dimensionality is actually straightforward. Take the exterior product of k vectors 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.
- ISBN 0-7923-5302-1.
References
- 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; ISBN 3-540-26296-2.
External links
- A Geometric Algebra Primer, especially for computer scientists.