Blade (geometry)
In the study of geometric algebras, a blade is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a k-blade is any object that can be expressed as the exterior product (informally wedge product) of k 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. Linear combinations of 2-blades also are bivectors, but need not be simple, and are hence not necessarily 2-blades. 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 space of dimension n, a blade of grade n − 1 is called a pseudovector.[2]
- The highest grade element in a space is called a pseudoscalar, and in a space of dimension n is an n-blade.[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]
For an n-dimensional space, there are blades of all grades from 0 to n inclusive. 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.[5]
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 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, and 2-blades are oriented area elements. 3-blades 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.
See also
Notes
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ Lua error in package.lua at line 80: module 'strict' not found.
- ↑ For Grassmannians (including the result about dimension) a good book is: Lua error in package.lua at line 80: module 'strict' not found.. 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 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. - ↑ Lua error in package.lua at line 80: module 'strict' not found.
General references
- Lua error in package.lua at line 80: module 'strict' not found.
- Lua error in package.lua at line 80: module 'strict' not found.
- 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.
- Lua error in package.lua at line 80: module 'strict' not found.
External links
- A Geometric Algebra Primer, especially for computer scientists.
