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Download as PDF; Printable version; In other projects ... In mathematics, a geometric algebra ... the main applications are the geometric algebra of Minkowski 3+1 ...
In the algebra of this space, based on the geometric product of vectors, such transformations correspond to the algebra's characteristic sandwich operations, similar to the use of quaternions for spatial rotation in 3D, which combine very efficiently. A consequence of rotors representing transformations is that the representations of spheres ...
In this article, certain applications of the dual quaternion algebra to 2D geometry are discussed. At this present time, the article is focused on a 4-dimensional subalgebra of the dual quaternions which will later be called the planar quaternions. The planar quaternions make up a four-dimensional algebra over the real numbers.
Plane-based geometric algebra is an application of Clifford algebra to modelling planes, lines, points, and rigid transformations. Generally this is with the goal of solving applied problems involving these elements and their intersections , projections , and their angle from one another in 3D space. [ 1 ]
Leo Dorst, Chris J. L. Doran, Joan Lasenby: Applications of geometric algebra in computer science and engineering, Birkhäuser, 2002, ISBN 0-8176-4267-6 Chris J. L. Doran: Geometric Algebra and its Application to Mathematical Physics , Sidney Sussex College, Dissertation submitted for the degree of Doctor of Philosophy in the University of ...
The universal geometric algebra (n, n) of order 2 2n is defined as the Clifford algebra of 2n-dimensional pseudo-Euclidean space R n, n. [1] This algebra is also called the "mother algebra". It has a nondegenerate signature. The vectors in this space generate the algebra through the geometric product.
In geometric algebra, the outermorphism of a linear function between vector spaces is a natural extension of the map to arbitrary multivectors. [1] It is the unique unital algebra homomorphism of exterior algebras whose restriction to the vector spaces is the original function.
The bivectors of the three-dimensional Euclidean space form the Lie algebra, which is isomorphic to the () Lie algebra. This accidental isomorphism allows to picture a geometric interpretation of the states of the two dimensional Hilbert space by using the Bloch sphere. One of those systems is the spin 1/2 particle.