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:

  ${\displaystyle a\wedge b.}$

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

  ${\displaystyle a\wedge b\wedge c.}$

• 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 ≤ 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, up to a scalar factor. When the space is endowed with a volume form (an alternating k-multilinear scalar-valued function), such a k-blade may be normalized to take unit value, making the correspondence unique up to a sign.

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.

• Grassmannian
• Multivector
• Exterior algebra
• Differential form
• Geometric algebra
• Clifford algebra

• 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.

• A Geometric Algebra Primer, especially for computer scientists.