Blade (geometry)
Template:Short description Script error: No such module "about". In the study of geometric algebras, a kScript error: No such module "Check for unknown parameters".-blade or a simple kScript error: No such module "Check for unknown parameters".-vector is a generalization of the concept of scalars and vectors to include simple bivectors, trivectors, etc. Specifically, a kScript error: No such module "Check for unknown parameters".-blade is a kScript error: No such module "Check for unknown parameters".-vector that can be expressed as the exterior product (informally wedge product) of 1-vectors, and is of grade kScript error: No such module "Check for unknown parameters"..
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 aScript error: No such module "Check for unknown parameters". and bScript error: No such module "Check for unknown parameters".:
- A 3-blade is a simple trivector, that is, it may be expressed as the wedge product of three vectors aScript error: No such module "Check for unknown parameters"., bScript error: No such module "Check for unknown parameters"., and cScript error: No such module "Check for unknown parameters".:
- In a vector space of dimension nScript error: No such module "Check for unknown parameters"., a blade of grade n − 1Script error: No such module "Check for unknown parameters". 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 nScript error: No such module "Check for unknown parameters". is an nScript error: No such module "Check for unknown parameters".-blade.[4]
- In a vector space of dimension nScript error: No such module "Check for unknown parameters"., there are k(n − k) + 1Script error: No such module "Check for unknown parameters". dimensions of freedom in choosing a kScript error: No such module "Check for unknown parameters".-blade for 0 ≤ k ≤ nScript error: No such module "Check for unknown parameters"., of which one dimension is an overall scaling multiplier.[5]
A vector subspace of finite dimension kScript error: No such module "Check for unknown parameters". may be represented by the kScript error: No such module "Check for unknown parameters".-blade formed as a wedge product of all the elements of a basis for that subspace.[6] Indeed, a kScript error: No such module "Check for unknown parameters".-blade is naturally equivalent to a kScript error: No such module "Check for unknown parameters".-subspace, up to a scalar factor. When the space is endowed with a volume form (an alternating kScript error: No such module "Check for unknown parameters".-multilinear scalar-valued function), such a kScript error: No such module "Check for unknown parameters".-blade may be normalized to take unit value, making the correspondence unique up to a sign.
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 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.
See also
Notes
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- ↑ Script error: No such module "citation/CS1".
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- ↑ Script error: No such module "citation/CS1".
- ↑ For Grassmannians (including the result about dimension) a good book is: Script error: No such module "citation/CS1".. The proof of the dimensionality is actually straightforward. Take the exterior product of kScript error: No such module "Check for unknown parameters". vectors and perform elementary column operations on these (factoring the pivots out) until the top k × kScript error: No such module "Check for unknown parameters". block are elementary basis vectors of . The wedge product is then parametrized by the product of the pivots and the lower k × (n − k)Script error: No such module "Check for unknown parameters". block. Compare also with the dimension of a Grassmannian, k(n − k)Script error: No such module "Check for unknown parameters"., in which the scalar multiplier is eliminated.
- ↑ Script error: No such module "citation/CS1".
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References
- Script error: No such module "citation/CS1".
- Script error: No such module "citation/CS1".
- 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.
- Script error: No such module "citation/CS1".
External links
- A Geometric Algebra Primer, especially for computer scientists.